Journal of Composite Materials On the onset of the asymptotic stable fracture region in the mode II fatigue delamination growth behaviour of composites

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DOI: 10.1177/0021998314525482 published online 5 March 2014Journal of Composite Materials

Calvin David Rans, Jeremy Atkinson and Chun Libehaviour of composites

On the onset of the asymptotic stable fracture region in the mode II fatigue delamination growth

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JOURNAL OFC O M P O S I T EM AT E R I A L SArticle

On the onset of the asymptotic stablefracture region in the mode II fatiguedelamination growth behaviourof composites

Calvin David Rans1,2, Jeremy Atkinson1 and Chun Li3

Abstract

An experimental investigation aimed at identifying the presence and onset of an asymptotic stable fracture region in the

mode II fatigue delamination growth behaviour of composites is presented. The study is motivated by the possibility that

experimental data sets, particularly those obtained at high R-ratios, may unknowingly contain data points within the

asymptotic stable fracture region that influence its perceived log-linear behaviour and the fit of various log-linear

delamination growth models. Results from the experimental investigation indicate that the asymptotic stable fracture

region can extend to GIImax values as low as 0.7GIIc. The implications of this result on characterizing and fitting of various

delamination growth models (including models assuming log-linear behaviour) to delamination growth behaviour

are discussed.

Keywords

Mode II, fatigue, delamination growth, asymptotic behaviour

Introduction

The damage tolerance certication requirement for civilaircraft necessitates the ability to detect and repairstructural damages before they become critical to theintegrity of the aircraft. Two approaches to thisrequirement are typically employed. In the slow-growth approach, a slow and predictable fatiguegrowth life for possible damage scenarios must bedemonstrated. Inspection intervals are set to ensure suf-cient opportunity to detect and repair potential dam-ages over their detectable growth life. In the no-growthapproach, potential incidental damages (i.e. delamin-ations due to impact) must have a demonstrable lackof growth under service fatigue loads. Inspection inter-vals are based on the probabilities of specic damagecausing incidents occurring over the lifetime of thestructure. Regardless of the approach used, sucientknowledge of the damage propagation behaviourunder static and fatigue loading is required.

Delaminations are a particular damage type of con-cern for composite structures. Manufacturing defectsand in service impact events have the potential tocause delaminations that signicantly reduce the

strength and stiness of composite structure.Understanding the static and fatigue propagationbehaviour of delaminations is thus critical for the appli-cation of composite structures in aircraft. A recent crit-ical review of the literature presented by Pascoe et al.1

demonstrates the wide variety of modelling approachesthat have been employed to help further that under-standing on fatigue propagation behaviour.

It is generally accepted that fatigue delaminationgrowth, analogous to metal fatigue crack growth, exhi-bits a sigmoidal behaviour when plotted against strainenergy release rate on a loglog scale as illustrated in

1Department of Mechanical & Aerospace Engineering, Carleton

University, Ottawa, Canada2Faculty of Aerospace Engineering, Delft University of Technology, Delft,

The Netherlands3Institute for Aerospace Research, National Research Council of Canada,

Ottawa, Canada

Corresponding author:

Calvin David Rans, Delft University of Technology, Kluyverweg 1, Delft,

2629 HS, the Netherlands.

Email: [email protected]

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Figure 1. Numerous models for predicting growth focuson the log-linear region of the curve. This region iscommonly referred to as the Paris region based onthe early work of Paris and Erdogan2 and Paris et al.3

These models take a general form of the power law

da

dN C f G n 1

where da/dN is the delamination growth rate, C and nare empirical constants, assumed to be material con-stants, and f G is a formulation of the strain energyrelease rate. Common formulations include Gmax andG as discussed in Pascoe et al.1

The asymptotes for the threshold and stable fractureregions of Figure 1 are generally accepted to be denedby a threshold strain energy release rate or range (Gth orGth) and the critical strain energy release rate forstatic failure (Gc), respectively. Several modicationsto the Paris relation in equation (1) have been proposedin the literature to capture the asymptotic behaviour inthese regions. Martin and Murri4 proposed multiplyingthe Paris relation by a sigmoidal factor dependant onGth and Gc. A similar approach was also adopted byKinloch and Osiyemi5 and Abdel Wahab et al.6

Alternatively, modifying f G in the Paris relation tointroduce the sigmoidal behaviour has been adoptedin models proposed by Jones et al.,7 Andersons et al.8

(for threshold behaviour only), and in metal crackgrowth models proposed by Forman et al.9 Bothapproaches rely on assuming a sigmoidal behaviour apriori and including factors to enforce this behaviourthat are often viewed as parameters that are used toensure that the entire range of data ts the equations7

rather than physics-based quantities. Furthermore,these factors, dependent on the implementation, canchange the underlying meaning of the material depend-ent constants C and n.

Identication of the onset of each asymptotic regionis important for the characterization and understandingof fatigue delamination behaviour. A relevant casestudy to illustrate this point is the study of R-ratioeects on fatigue delamination growth behaviour. Inorder to quantify the inuence of R-ratio, fatiguedelamination growth curves similar to that illustratedin Figure 1 are experimentally determined for constantR-ratios. Several curves are generated for dierent R-ratios and compared. When testing higher R-ratios,higher maximum fatigue loads are required to achievecomparable fatigue amplitudes as tested for lower R-ratios. In other words, Gmax is necessarily increased toachieve a comparable DG. Thus, the threshold asymp-tote dependent on DG and the static failure asymptotedependent on Gmax will move closer to each other, nar-rowing the log-linear region of Figure 1, as the R-ratio

is increased. As this narrowing eect increases, there isa risk that data points obtained from experiments thatappear to maintain the log-linear trend when plottedactually lie within the asymptotic regions of the delam-ination growth curve.

The possibility of the above case occurring has beenidentied by the authors in three data sets from theliterature. A recent paper by Rans et al.10 discussesthe formulation of the cyclic strain energy release raterange for delamination growth characterization andadvocates the use of (G)2 as strain energy releaserate range consistent with similitude of the cyclic cracktip stress range. Several case studies presented by Ranset al.10 demonstrated that use of this denition elimi-nated the apparent R-ratio eects observed when usingG or Gmax as the independent variable for character-izing mode II fatigue delamination growth behaviour.The present authors have applied this denition of thestrain energy release rate range to analysing three newdata sets from the literature.1113 The results of theseanalyses are presented in Figures 2 through 4, withsubgures (a) depicting the data as a function ofGIImax/GIIc, and subgures (b) depicting the data as afunction of (GII).2 Results in subgures (a) illus-trate the higher magnitudes of GIImax tested at higherR-ratios and how the results from each R-ratio appearto be log-linear. Plots of the data reanalysed using(GII)2 in subgures (b) indicate another possibility.The data sets correlate well with each other using thisparameter except for a few data points indicated withlabel A. These data points all correspond to tests run athigher R-ratios and at the highest GIImax values exam-ined in their respective studies, and for all the studies

IThreshold

IILog-Linear

IIIStable Fracture

Critical G(G

c)

Thre

shol

d G

Rang

e (

Gth

)

log(da

/dN)

log(G)

( )( )nda C f GdN =

Figure 1. Sigmoidal behaviour of fatigue delamination growth

versus strain energy release rate.

2 Journal of Composite Materials 0(0)



fall within the range GIImax >0.7GIIc. It is possible thatthese data points lie within the asymptotic stable frac-ture region, but that this is masked by a combination ofa lack of additional data points in this region, scatter inthe data, and the ability for log-log scales to masksubtle deviations in trends.

The results in Figures 2 through 4 motivate the needto study the onset of the stable fracture region for fati-gue delamination growth. This paper presents anexperimental study on mode II delamination growthin a unidirectional carbon/epoxy material in responseto this need.

Test program

Based upon the above evidence from the literature, atest program to investigate the onset of the asymptotic

stable fracture region in the mode II fatigue delamin-ation growth behaviour for composite materials wascarried out. This test program consisted of mode IIfatigue delamination growth tests performed on a uni-directional carbon/epoxy material under constant amp-litude loading at various R-ratios and applied loads.This section summarizes the test program, includingspecimen conguration, measurement methods, anderror analysis.

The central cut-ply specimen

Fatigue delamination growth tests were carried outwith the so-called central cut-ply specimen used in pre-vious studies of mode II fatigue delamination growth incomposite materials.11,1315 A similar specimen cong-uration, with a discontinuous metal layer, has also been

GIImax/GIIc

da/d

N[m

m/c

ycle

]

0.1 0.3 0.5 0.7 0.910-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2 R = 0.2R = 0.5R = 0.6

T800H/#3631GIIc = 0.820 kJ/m

2

(GII)2 [kJ/m2]

da/d

N[m

m/c

ycle

]

0.05 0.1 0.15 0.2 0.25 0.310-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

R = 0.2R = 0.5R = 0.6

T800H/#3631 A

(a) (b)

Figure 3. Experimental fatigue delamination growth data for T800H/#363112; label A indicates data points where GIImax >0.7GIIc.

GIImax/GIIc

da/d

N[m

m/c

ycle

]

0.1 0.3 0.5 0.7 0.910-8

10-7

10-6

10-5

10-4

10-3

10-2 R = 0.1R = 0.3R = 0.5

IM7/8552GIIc = 1.18 kJ/m

2

(GII)2 [kJ/m2]

da/d

N[m

m/c

ycle

]

0.1 0.2 0.3 0.4 0.5 0.610-8

10-7

10-6

10-5

10-4

10-3

10-2

R = 0.1R = 0.3R = 0.5

IM7/8552

A

(a) (b)

Figure 2. Experimental fatigue delamination growth data for IM7/855211; label A indicates data points where GIImax >0.7GIIc.

Rans et al. 3



used extensively to study mode II fatigue delaminationgrowth in hybrid bre metal laminate materials.1622

The test specimen consists of a series of discontinuous(or cut) plies laminated between continuous plies in asymmetric layup.

Four delamination fronts initiate from the discon-tinuity in the laminate producing planar delaminationsalong the two interfaces between the continuous anddiscontinuous plies (Figure 5). A pure mode II sheareld is applied to the delamination fronts through a fareld tensile force, P, applied to the ends of the speci-men. An expression for the strain energy release rateassociated with each of the four delamination frontswas determined by Allegri et al.,11 following theapproach outlined by Williams,23 and is summarizedas follows

GII P2

4B2Et

1

2

where denotes the ratio of the number of cut pliesto the total number of plies in the specimen, B and t arethe specimen width and thickness respectively, and E isthe Youngs modulus of the material.

Equation (2) holds only if the delamination state issymmetric across the specimen mid-plane, and if all

four delaminations grow at the same rate. The delam-ination state does not need to be symmetric with respectto the cut-ply location, as illustrated in Figure 5,because the strain energy release rate is independentof delamination length.

The independence of equation (2) on delaminationlength is a desirable property for fatigue delaminationgrowth characterization. It permits delaminationgrowth measurements at a constant strain energyrelease rate through simple constant amplitude fatiguetests. This strain energy release rate solution, however,is based upon simple beam theory and the assumptionthat the delaminated plies are fully unloaded.11 Theseassumptions break down as the delamination lengthapproaches zero, resulting in a region where equation(2) is not valid. Finite element (FE) simulations per-formed by Kawashita et al.15 have shown that thisregion encompasses delamination lengths up to alength of 510 times the thickness of the cut plies. Alltests carried out in the present test program do notconsider delamination growth within this region.

Delamination growth measurement approach

Monitoring of delamination length during the fatiguetest can be achieved by monitoring changes in specimencompliance using an extensometer. As the delamin-ations in the specimen grow, the overall specimen sti-ness reduces. This stiness reduction can be modelledby dividing the specimen into a series of beams. Pristineregions of the specimen will have a stiness equal to thenominal specimen stiness, while delaminated regionswill have their stiness reduced by a factor of (1 ).The total deformation of the beam model is thus afunction of delamination length, applied load, and thenominal specimen stiness. Equating the deformation

(GII)2 [kJ/m2]

da/d

N[m

m/c

ycle

]

0.1 0.2 0.3 0.4 0.5 0.610-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

R = 0.1R = 0.5

E-glass/913A

0.05GIImax/GIIc

da/d

N[m

m/c

ycle

]

0.1 0.3 0.5 0.7 0.910-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

R = 0.1R = 0.5

E-glass/913GIIc = 1.08 kJ/m

2

(a) (b)

Figure 4. Experimental fatigue delamination growth data for E-glass/91313; label A indicates data points where GIImax >0.7GIIc.

2a

1

2

3

4PP

cut plies

Figure 5. Illustration of four delamination fronts in a side view

of central cut-ply specimen.




from the model to the deformation measured by theextensometer, the size of the delamination present inthe specimen can be estimated. For brevity, the nalexpression for delamination length is as follows

2a Lgauge 1

EBt

P" 1

3

where Lgauge and "* are the extensometer gauge lengthand strain measurement, respectively. A detailedaccount of the derivation was presented by Allegriet al.11 This equation is valid for the scenario whereboth planar delaminations of length 2a are symmetricacross the laminate mid-plane.

Direct application of equation (3) to determinedelamination length is highly sensitive to the relativezeroing of the extensometer and load cell and doesnot account for the compliance of the resin-rich areaat the cut plies. As a result, the absolute delaminationlength calculated by this equation can be inaccurateand sometimes nonsensical (i.e. negative values).

These sources of error can be removed by consider-ing the change in delamination length rather than abso-lute length. By xing the load, P, at which theextensometer strain reading is surveyed, it becomes pos-sible to relate the delamination growth rate to the slopeof the extensometer strain plotted versus the number offatigue cycles, N, by taking the derivative of equation(3) with respect to N

da

dN 1

EBtLgauge2P

d"

dN4

Due to the strain energy release rate being independ-ent of delamination length for the central cut-ply spe-cimen, a constant delamination growth rate is expectedfor constant amplitude fatigue loading. Similarly,according to equation (4), a linear "* versus N behav-iour is also expected. This behaviour enables a simplemeans of calculating delamination growth rates for a

single strain energy release rate by applying a linear tto the extensometer readings of a constant amplitudefatigue test.

Fatigue tests

Constant amplitude fatigue tests were carried out oncentral cut-ply specimens fabricated from unidirec-tional M30SC/DT120 carbon/epoxy prepreg material.The nominal specimen dimensions are shown inFigure 6. Properties of the prepreg material are pre-sented in Table 1.The overall laminate consisted of 10plies of unidirectional material with the two centralplies being cut along the entire specimen width.Articial delaminations were not added during manu-facturing. Aluminium tabs were bonded to the ends ofthe specimens to facilitate gripping in the test frame.

Fabrication of the specimens began with the produc-tion of four 160mm 300mm rectangular carbon/epoxy laminates containing the central cut plies.These panels were cured at a temperature of 120Cand a pressure of 6 bar. Aluminium tabs were second-ary bonded to the rectangular laminates after curing.Specimens with a nominal width of 15mm were cutfrom the rectangular panels using a lubricated diamondblade saw. A shorter specimen containing no cut plies

SEE DETAIL A

DETAIL A

15.0

100 0404

2 CUT PLIES

1.56

10TO

TAL

PLIE

S

Figure 6. Cut-ply specimen details.

Table 1. Mechanical properties for unidirectional M30SC/

DT120 prepreg.

Property

Manufacturers

data Measured values

E11 (GPa) 155 145 2.5E22 (GPa) 7.8

G12 (GPa) 5.5

n12 0.27

GIIc (kJ/m2) 1.50 0.05a

aCalculated using the measured Youngs modulus.

Rans et al. 5



was also cut from each panel and used to measure theYoungs modulus of the material using a clip extens-ometer, which was found to be slightly below the manu-facturers quoted value (see Table 1).

Static tests were performed in order to determine thecritical mode II strain energy release rate, GIIc, for thematerial. Two central cut-ply specimens from each ofthe four manufactured panels were statically loadeduntil failure. Prior to static loading, each specimenwas pre-fatigued (R 0.1, GIImax&GIIc) to initiate nat-ural delamination fronts at the four delamination initi-ation sights. Since the exact length of the initialdelaminations was not needed to calculate GIIc, theywere not measured. However, the delaminations werevisually observable in all cases, and thus deemed to beof sucient length to be free of any boundary conditioneects of the cut-ply region. Equation (2) was used toestimate GIIc by substituting P with the failure load,dened as the maximum static load observed in thetest prior to a signicant drop in load. An averagevalue of 1.50 0.05 kJ/m2 was found for GIIc usingthis approach and the measured value of Youngsmodulus.

Table 2 denes the test space and number of speci-mens for the fatigue test program. The test spacefocused on the GIImax/GIIc range of 0.70.9 in orderto identify the presence of an asymptotic stable fractureregion in the delamination growth behaviour. A singlespecimen was used for each instance in the test matrixto avoid potential inuences of variable amplitudeloading on a specimen and to maximize the amountof delamination growth present for each calculationof growth rate. Delamination growth was calculatedusing equation (4) and the peak strain readings froman MTS 634.12E-24 clip extensometer recorded overthe duration of each test. The gauge length of theextensometer was increased to 4

00(101.6mm) using

an MTS 634.15B-31 gauge length extender.Fatigue tests were carried out on an MTS 810 servo-

hydraulic test frame containing hydraulic wedge gripsand a 100 kN load cell (model MTS 661.20E-03). Themaximum and minimum forces for each test were cal-culated using equation (2), the measured specimengeometry, and the measured GIIc value from Table 1.

Tests were performed at frequencies varying between 1and 5Hz, where lower frequencies were primarily usedfor tests atGIImax/Gc 0.850.9 to increase the durationof the test, enabling visual inspections. Previous studieshave shown that mode II delamination growth behav-iour for bre reinforced epoxy materials is not signi-cantly inuenced by variations in frequencies in thisrange.13

Verification of delamination growth measurements

For a select number of tests, optical measurements ofdelamination lengths were made to verify the assump-tions behind equation (4) and validate its accuracy.This equation assumes that delamination state is sym-metric across the specimen mid-plane and all fourdelaminations grow at the same rate.

Regarding these assumptions, Pascoe et al.24

observed that non-symmetrical disbond growth couldoccur in thick bonded metal specimens of a similar con-guration to the cut-ply specimen. This observed non-symmetry was found to aect the local strain energyrelease rate, due to secondary bending, at individualdisbond fronts in such a way as to promote furthernon-symmetry. Furthermore, not accounting for thelocal variation in strain energy release rate resulted insignicant scatter in the delamination growth behav-iour that masked the true behaviour of the interface.

Despite these observations by Pascoe, the samebehaviour was not observed in the present test programusing thin specimens. Once initiated, all delaminationsgrew together and at the same overall rate. Initiationbehaviour, however, was not equal for all of the delam-inations. It was observed that one pair of delaminationstended to initiate rst, resulting in the resin-rich pocketat the cut-ply interface to act as an adhesive llet, delay-ing the initiation of the second pair of delaminations.This process is illustrated in Figure 7. Delaminationswere observed to always initiate in pairs, in adjacentdelamination interfaces, growing in the same directionand symmetry in the delamination state across thelaminate mid-plane was maintained. As a result, theapplication of equations (3) and (4) remains valid forthis test program.

These observations are clearly illustrated in a repre-sentative extensometer data set shown in Figure 8.Region I of this curve corresponds to the initiationphase for the four delaminations. For this particularinstance, the strain begins to linearly increase early inthe life, indicating that the rst delamination pairinitiated very early in the test. The slope of this linearregion is half that of region II, consistent for the behav-iour of one delamination pair growing. Within regionII, all four delaminations are growing. The slope of thestrain versus fatigue cycles in this region is used to

Table 2. Fatigue test matrix.

R

GIImax/GIIc

0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.85 0.9

0.1 3 2 3 3 3 3 3 3 2

0.3 3 2 3 3 3 3 3 3 2

0.5 3 3 5 2 3 3 3




calculate the delamination growth rate as per equation(4). Finally, in region III, the delaminations begin togrow beyond the measurement region of the extensom-eter, resulting in a levelling o of the extensometerreading.

A second axis has been added to Figure 8 in order toillustrate the above described issues related to

calculating crack length directly from extensometerdata using equation (3). Using this approach, a negativeinitial delamination length is calculated. Furthermore,calculations of delamination length are signicantlylower compared to optical measurements made forthis particular test. However, despite the error in abso-lute value of the delamination length, using the slope ofthe extensometer data combined with equation (4) topredict delamination growth rate produces good agree-ment with the optical measurements.

Based on these results, all remaining delaminationgrowth rates in this paper were calculated using equa-tion (4) using the slope from the linear region II of theextensometer readings from the test. All tests were runsuciently long such that either the region III behav-iour could be identied in the extensometer data oruntil visual conrmation of the growth of all fourdelamination could be made. The latter condition wasonly used in tests where delamination growth was suf-ciently slow that permitting growth of the delamin-ations beyond the extensometer gauge length wouldbe prohibitive in testing time (i.e. >3 million cycles).

Measurement error analysis

An often under-discussed topic with respect to the pres-entation of fatigue delamination growth data is theinuence of measurement error on the results. The par-ameters of interest, namely delamination growth rateand strain energy release rate, are seldom measured dir-ectly; they are calculated parameters dependent on sev-eral measured quantities. Thus, although individualmeasurement errors may be small, their cumulativeeects must be considered.

In the present study, the parameters of interest arethe strain energy release rate, G, and delaminationgrowth rate, da/dN, which are calculated using equa-tions (2) and (4), respectively. Consider rst the calcu-lation of strain energy release rate. This parameter isdependent on measurements for specimen width andthickness, the Youngs modulus of the material, andapplied load. The quantity is exact and does nothave any measurement error associated with it. Thus,in terms of measured quantities

GII / P2

B2Et5

The relative error, , for GII can thus be expressed assum of the relative errors of the measured quantities as

GII 2P 2B E t 6

where the subscript refers to the quantity the relativeerror applies to. The relative error is dened by the

nominal location of cut in plies

1

2

resin-rich pocket

1

2

resin-rich pocket

3

4

2a

(a)

(b)

(c)

Figure 7. Delayed initiation of second delamination pair due to

the formation of a resin-rich adhesive fillet.

N [cycles]

stra

in

a[m

m]

0 10000 20000 300006000

6500

7000

7500

8000

-10

0

10

20

30

40extensometer readingoptical measurement

I II III

slope = 0.0847 strain/cycle(2.63x10-3 mm/cycle)

slope = 2.810x10-3 mm/cycle

Figure 8. Comparison of extensometer-based and optical-

based measurements of crack length and growth for a specimen

tested at R 0.1 and GIImax/GIIc 0.4.

Rans et al. 7



absolute error divided by the measured quantity. Forinstance, the Youngs modulus of the material wasmeasured as 145 2.5GPa. The relative error for thisquantity is thus 2.5/145 or 1.7%.

The relative errors for B, t, and P will not be xedvalues as the measured quantities vary from test to test.All length values were measured using a set of calliperswith an accuracy of 0.05mm, and the load cell usedduring testing had a repeatability of 0.03% of the fullrange of the load cell or 30N. Using these quantities,equation (6) can be expressed as

GII 230N

P

2 0:05mm

B

1:7% 0:05mm

t

7

For illustrative purposes, using the nominal speci-men dimensions from Figure 6 and the load range of1733 kN, the relative error for GII is 5.85.9%. Thisload range corresponds to the range of maximum loadsapplied in the test program. It is clear from this exam-ple, although the individual measurement errors aresmall, their cumulative eect on the calculated param-eter, GII, can be signicant.

A similar analysis can be done for the crack growthrate. Following the same approach, the relative errorcan be dened as

da=dN E B t Lgauge 2P d"=dN 8

Dening a relative error for the quantity d"

dN is notstraight-forward as this quantity represents a trend inmeasured quantities. However, the linearity of thecurve in Figure 8 (and other similar measurements) sug-gests that this error is small. Neglecting this term, therelative error for the delamination growth rate is in thesame order of magnitude as for GII. The magnitude ofthis error is much smaller than the scatter in delamin-ation growth rates observed during the test.

Based on the results of this analysis, it was decidedthat the relative error in GII, as expressed by equation(7), should be indicated on all results using error bars.Conversely, for delamination growth rates, error barsare not included. The scatter observed in repeated testswas found to be more signicant than the relative cal-culation error. One must keep in mind the nature of aloglog scale when visually assessing scatter in the pre-sented plots. A constant measurement accuracy fordelamination length would imply a constant scatterband for delamination growth rate (if all scatter canbe attributed to this xed accuracy). However, whenplotted on a log scale, this constant scatter bandwould visually appear larger at the low end of thelog scale.

Results

Onset of the asymptotic stable fracture region

Delamination growth results from the test program aresummarized in Figure 9. Delamination growth ratesand strain energy release rates were calculated usingequations (4) and (2), respectively. As discussed in theprevious section, uncertainty due to measurement erroris indicated by horizontal error bars for the strainenergy release rate and by scatter in the various datapoints for delamination growth rate. Each data pointrepresents the results from one test specimen accordingto the test matrix in Table 2.

The delamination growth data in Figure 9 exhibits anon-linear behaviour on a loglog scale. To improvevisualization of this, the data set for R 0.1 has beenrescaled and plotted in Figure 10. From this gure, the

GIImax [kJ/m2]

da/d

N[m

m/c

ycle

]

0.3 0.5 0.7 0.9 1.1 1.3 1.510-4

10-3

10-2

10-1

100

R = 0.1

M30SC/DT120

> 0.7 GIIc

Figure 10. Experimental fatigue delamination growth results

for R 0.1 rescaled to visualize non-linear behaviour on a loglogscale.

GIImax [kJ/m2]

da/d

N[m

m/c

ycle

]

0.3 0.5 0.7 0.9 1.1 1.3 1.510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

R = 0.1R = 0.3R = 0.5

M30SC/DT120> 0.7 GIIc

Figure 9. Experimental fatigue delamination growth results for

MS30SC/DT120.




delamination growth data follow the typical linear loglog scale trend for GIImax below approximately 0.7GIIc.Above this limit, the observed delamination growthrates rapidly increase, deviating from the previouslinear trend, consistent with expectations for an asymp-totic behaviour. This trend is clearly observed for theR 0.1 and 0.3 data sets; however, it is dicult toascertain for R 0.5 due to the limited number ofdata points below the 0.7GIIc limit and the large gradi-ent in delamination growth behaviour below this limit.

It should be noted that there is no theoretical basisfor the 0.7GIIc limit discussed in this study. It is anobserved limit consistent with the data from thisstudy and the data from the literature given inFigure 2 through 4. Its value, however, is plausible inrelation to the onset of a stable fracture mechanism. Asdiscussed in Rans et al.,10 the strain energy release rate,G, is proportional to the square of the stress intensityfactor, K (or square of the applied stress). This is alsoevident from the proportionality between GII and P

2 inequation (2). Thus, the limit 0.7GIIc is equivalent to0.84KIIc or 0.84Pc, where 0.840.7. Given thelocal discontinuities in a composite laminate, it is plaus-ible that localized unstable static fracture could occurthis close to the fracture toughness of the material,resulting in accelerated delamination growth and theonset of the asymptotic stable fracture region.

This proportionality between G and K2 also hasimplications on the denition of a cyclical strainenergy release rate range. The cyclical stress intensityfactor range, K, typically used to characterize fatiguecrack growth in metals, is a measure of the cyclic amp-litude of the stress state ahead of a crack tip. It isdependent on the amplitude of this cyclic stress butindependent of the mean stress, or in other words,dependent on the cyclic nature of the stress cycle butindependent of its monotonic nature. The cyclic strainenergy release rate, G, often used as an analogy forK in delamination growth, does not have this samebehaviour due to the above proportionality. From thisproportionality, it follows that

G / K2max K2min 2 K Kmean 9

Thus, DG is dependent on the cyclic nature andmean (or monotonic) nature of the fatigue stressstate. A more accurate analogy with K would be todene the cyclical strain energy release rate range as(G)2, as is discussed in Rans et al.10 This discussionis not meant to discredit the potential use of G as asimilitude parameter for characterizing delaminationgrowth, but draw attention to its meaning, and theimplication of its use. The authors, however, promotethe use of the parameters (G)2 and Gmax as simili-tude parameters to characterize, respectively, the cyclic

and monotonic nature of delamination growthindependently.

Figure 11 shows the delamination growth data fromthis study plotted as a function of (GII)2, with alldata points above the 0.7GIIc limit highlighted in white.Here, the deviation of the delamination growth from alinear trend on a loglog scale becomes more apparent.Furthermore, the underlying trend for all R-ratiosappears to be the same when plotted against(GII)2, indicating an absence of mean stress, ormonotonic, eects on the delamination growth behav-iour. This observation is consistent with previous obser-vations discussed at the beginning of this paper and inRans et al.10

Above the 0.7GIIc limit, the delamination growthbehaviour does not converge to a single trend whenplotted against (GII)2 (Figure 11) or GIImax(Figure 9). This suggests that both cyclic and mono-tonic fracture mechanisms are present in this region.Indeed, it is logical that the introduction of localstatic fracture into the fracture process as GIIc isapproached would not completely remove the cyclicfracture mechanism. Curiously, if the delaminationgrowth data from this study are plotted against G,as shown in Figure 12, the data in this region appearto collapse to a single trend. This suggests that thebehaviour in this region is related to the specic com-bination of monotonic and cyclic eects described byequation (9). Further study of this observation is neces-sary before conclusions about its general validity can bemade.

Correlation with existing models

In the previous section, the observed presence of astable fracture region in the mode II delaminationgrowth curve for the tested material system was

(GII)2 [kJ/m2]

da/d

N[m

m/c

ycle

]

0.1 0.3 0.5 0.7 0.9 1.1 1.310-7

10-6

10-5

10-4

10-3

10-2

10-1

100

R = 0.1R = 0.3R = 0.5Gmax 0.7 GcGmax > 0.7 Gc

MS30SC/DT120

Figure 11. Fatigue delamination growth data plotted as a

function of (GII)2.

Rans et al. 9



discussed. In this section, the implications of this regionon the t of common delamination growth models fromthe literature will be examined.

The rst model examined was proposed by Allegriet al.11 as a simplied two-parameter semi-empiricalmodel capable of predicting the inuence of R-ratioon mode II delamination growth. This model takesthe basic form of a power relation, analogous to thewell-known Paris Law, with the normalized maximumstrain energy release rate, GIImax/GIIc, as the drivingforce for delamination growth. R-ratio eects areaccounted for in the exponent of the power relation,as follows

da

dN C GIImax

GIIc

b1R 2 10

Comparison to this model was chosen for illustrativepurposes only. The authors acknowledge that Allegriet al. explicitly state that their model extrapolates theso-called Paris Region of the delamination growthcurve to the point of static failure, thus neglecting thepresence of an asymptotic stable fracture region.11 As aresult, comparison of the model with the present datacan provide an unfair impression of its capabilities.However, the prevalence of fatigue delaminationgrowth models based on extrapolating the ParisRegion of the delamination growth curve, particularlywithout explicit limits on the range of validity of thisextrapolation, necessitates discussion with such a com-parison. This is meant to draw awareness to implica-tions of having some data within the static asymptoticregion on such models, not to criticize such models.

As a regression method for equation (10), Allegriadvocates a linear regression of the data plotted on aloglog scale, a log-linear regression. This simplied

regression approach is commonly adopted in ttingParis-type relations for damage growth. Althoughsimple in execution, this regression approach minimizesthe squares of the residuals between the logarithms ofpredicted and observed delamination growth ratesrather than minimizing the overall residuals.Practically, this lowers the weighting factors of resi-duals at large delamination growth rates, relative tothose at smaller delamination growth rates, due to theproperties of a logarithmic function. As a result, a log-linear regression of equation (10) will tend to generatelarger errors at high delamination growth rates com-pared to a non-linear regression technique.

This potential pitfall is clearly illustrated in theregression of the data set from this study using equation(10) presented in Figure 13. The log-linear regressionwas performed as outlined in Allegri et al.,11 while thenon-linear regression was carried out using the non-linear bisquare tting routine available in theMATLAB tool SFTOOL. Details of the tting par-ameters are given in Table 3. The log-linear regressionappears to give a reasonable t to the data as plotted ona loglog scale; however, the large residuals at thehigher delamination growth rates are masked by theloglog scale. The non-linear regression, on the otherhand, is heavily inuenced by the data points in theregion of GIImax >0.7GIIc where the onset of asymptoticbehaviour was observed.

Which regression technique is more appropriate inthis instance is debatable. Indeed, Allegris model is notintended to capture the asymptotic behaviour of thedelamination growth curve. Thus, it could be arguedthat the weighting eect of a log-linear regression isfavourable in this instance. The presence of a fewdata points within the asymptotic region of the delam-ination growth curve would not adversely aect the

GII [kJ/m2]

da/d

N[m

m/c

ycle

]

0.3 0.5 0.7 0.9 1.1 1.3 1.510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

R = 0.1R = 0.3R = 0.5Gmax 0.7 GcGmax > 0.7 Gc

MS30SC/DT120

Figure 12. Fatigue delamination growth data plotted as a

function of GII.

GIImax [kJ/m2]

da/d

N[m

m/c

ycle

]

0.3 0.5 0.7 0.9 1.1 1.3 1.510-7

10-6

10-5

10-4

10-3

10-2

10-1

100 R = 0.1R = 0.3R = 0.5nonlinear regressionlog-linear regression

M30SC/DT120

> 0.7GIIc

Figure 13. Results of equation (10) fitted to the experimental

data set using log-linear and non-linear regression.




apparent t (i.e. the data sets in Figures 2 through 4).The preferable approach is to identify the limit thatdenes the onset of the asymptotic region and constrainthe application of the model beyond this limit. Basedon the results from the present study, it could be arguedthat the literature data presented in Figure 2 throughFigure 4 contain data points already within this asymp-totic region that could aect the t of non-asymptoticmodels.

The second model examined is a variant of themodel proposed by Forman et al.9 In order to accountfor the asymptotic stable fracture region and R-ratioeects observed in metal fatigue crack growth,Forman extended the classical Paris Law as follows

da

dN C K

b

1 R Kc Kmax 11

where the factor of (1R) in the denominatoraccounted for R-ratio eects and the factor(KcKmax) introduced the asymptotic behaviour inthe stable fracture region. C and b are material depend-ent parameters.

In order to apply this equation to mode II delamin-ation growth data in this study, the stress intensityfactor values (KcKmax) and K are substituted with(GIIcGIImax)2 and (GII)2, respectively.Additionally, the term (1R) is dropped due to theabsence of R-ratio eects for mode II delamination asdiscussed in Rans et al.10 and as demonstrated for thisdata set in Figure 11. The resulting modied Formanrelation is thus given by

da

dN C

GII

p 2bGIIc

p GIImaxp 2 12

where the material dependent parameter C and b aredierent than those in equation (11).

Results from the regression of the data from thisstudy using equation (12) are plotted in Figure 14. Aswith the previous model, log-linear and non-linearregression analyses were performed. Details of the

tting parameters from these analyses are given inTable 3. The closer agreement between the two regres-sion analyses for this model highlights its ability to cap-ture the observed behaviour at higher delaminationgrowth rates; the eect of reducing the weighting ofresiduals for the higher delamination growth rates inthe log-linear regression is minimal. Furthermore,both regression analyses produce a good t to theexperimental data.

The authors contend that the form of this model ismore appropriate for characterizing mode II delamin-ation growth than other models based on maximumstrain energy release rate. Use of (GII)2 as a simili-tude parameter for characterizing fatigue delaminationgrowth correlates well with experimental observationsand links the damage process to the amplitude of thefatigue stress state at the damage, a cyclic loading pro-cess. Deviations from this behaviour can be explainedby the onset of a stable fracture mode, resulting in anasymptote as the fracture toughness of the material isapproached a monotonic loading process. Althoughequation (12) captures both of these processes, it stillrelies on material dependent parameters lacking anyknown physical meaning. Furthermore, there is nophysical basis for the intensity of the asymptote gradi-ent imposed by the 1/(GIIc GIImax)2 factor or itsimplied onset. These shortcomings may be acceptablefor the purposes of an engineering model, but solutionsto them should be sought and their limitationsunderstood.

Conclusions

A series of fatigue tests have been performed to inves-tigate the onset of the asymptotic stable fracture regionin the mode II fatigue delamination growth behaviour

(GII)2 [kJ/m2]

da/d

N[m

m/c

ycle

]

0.1 0.3 0.5 0.7 0.9 1.1 1.310-7

10-6

10-5

10-4

10-3

10-2

10-1

100

R = 0.1R = 0.3R = 0.5nonlinear regressionlog-linear regression

M30SC/DT120

Figure 14. Results of equation (12) fitted to the experimental

data set using log-linear and non-linear regression.

Table 3. Summary of fitting parameters.

Equation

Log-linear regression Non-linear regression

C (mm/cycle) b C (mm/cycle) b

(10) 0.1895 4.208 0.9834 8.847

(12) 0.002940 4.121 0.001625 3.122

Rans et al. 11



of a unidirectional carbon/epoxy material. Based uponthe results of these tests, the following conclusions canbe made:

. An asymptote in the delamination growth rateversus maximum strain energy release rate behaviourof the material system under investigation wasobserved. This asymptote was most evident in thedata sets obtained at fatigue stress ratios of 0.1and 0.3.

. The onset of the asymptotic region appears to occurat approximately 0.7GIIc. Based on the strain energyrelease rate being proportional to the square of thestress intensity factor, this correlates with 0.84KIIc.This represents a plausible value for the onset oflocalized stable static delamination growth underfatigue loading given the local discontinuitieswithin a composite laminate.

. Mode II fatigue delamination growth behaviourbelow the observed asymptote shows a good correl-ation with (G)2, consistent with previous nd-ings of the authors10 and consistent with otherstudies12,25,26 utilizing an analogous stress intensityfactor description, K.

Based upon these results, existing mode II fatiguedelamination growth models from the literature weretted to the data obtained in this study to investigatethe impact and potential dangers of ignoring thisasymptotic behaviour on delamination growth predic-tions. The data from the study were found to correlatewell with a modied Forman equation based on theclassical Paris Law with (G)2 as the similitude par-ameter for damage growth and a correction factor of 1/(GIIc GIImax)2 to dene the onset of the asymp-totic stable fracture region. Although good correlationwas observed, it is noted that there is no physical basisfor 1/(GIIc GIImax)2 dening the onset of thisregion, thus its applicability for other material systemsneeds to be investigated further.

Finally, this study limited itself to positive R-ratios.For negative R-ratios, it is possible for the criticalenergy release rate to be approached by both the max-imum and minimum portions of the fatigue load cycle.This could result in a variation in the apparent onsetand/or strength of the asymptote, however, an in-depthinvestigation into this is required. Usage of the cyclicstrain energy release rate range (G)2 for negativeR-ratios is further discussed in Rans et al.10

Funding

This work was supported by the Netherlands Organizationfor Scientic Research Technical Science Foundation [Venigrant 10674].

Conflict of interest

None declared.

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Appendix 1

Notation

a crack or delamination length (mm)B specimen width (mm)E Youngs modulus (GPa)G strain energy release rate (kJ/m2)G strain energy release rate range (kJ/m2)

(G)2 alternative definition of strain energyrelease rate range (kJ/m2)

K stress intensity factor (MPamm)K stress intensity factor range

(MPamm)Lgauge extensometer gauge length (mm)

N number of fatigue cyclesP applied load (N)R fatigue stress ratiot specimen thickness (mm)

"* extensometer strain (mm/mm)Z relative error (%) ratio of cut plies to total plies

Subscripts

1 fibre direction2 transverse fibre directionII mode IIc critical (in relation to static failure)

max maximum (in relation to fatigue cycle)min minimum (in relation to fatigue cycle)

Rans et al. 13


Documents

Journal of Composite Materials On the onset of the asymptotic stable fracture region in the mode II fatigue delamination growth behaviour of composites