International Journal of Mechanical Sciences 52 (2010) 1125–1135

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

An equivalent ellipse method to analyse the fatigue behaviour following‘multi-surface initiations’

Ali Mkaddem �, Mohamed El Mansori

Arts et Metiers Paris Tech, LMPF, Rue Saint Dominique, B.P. 508, 51006 Chalons-en-Champagne, France

a r t i c l e i n f o

Article history:

Received 30 March 2009

Received in revised form

15 September 2009

Accepted 17 September 2009Available online 24 September 2009

Keywords:

Fatigue

Crack growth

Multi-surface initiations

Metal–matrix composites

03/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ijmecsci.2009.09.023

esponding author. Tel.: +33 3 26699135; fax:

ail address: ali.mkaddem@ensam.fr (A. Mkadd

a b s t r a c t

This paper discusses the mechanisms of fatigue when cracks synergetically initiate in multiple sites at

the surfaces of specimen. Metal–matrix composites such as aluminium matrix composites reinforced by

silicon carbide particles are good candidates to accelerate fatigue failures following multi-surface

initiations (MSIs). Closure effects of MSIs on the variation of fatigue behaviour were explored for various

applied stress in two states of composite used: pre-treated state and non-pre-treated state. Using an

equivalent ellipse method (EEM), it is found that the quality of surface finish of the specimen is of great

role in crack initiation and growth. It is revealed that total lifetime of specimen is sensitive to heat

treatment. Considering no transition from small to long crack, predictive formula used leads to

identification of the Paris law exponent. It is found that this exponent is sensitive to both the applied

load and the value of crack growth rate below threshold.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, the importance of metal–matrix composite (MMC) inthe development of engineering components in automotive andaerospace sectors has been pointed out by many researchesthroughout the world. In early publication, Lim and Dunne [1]studied effects of reinforcement volume fraction on mechanicalbehaviour of aluminium–silicon carbide particle composite. Inthis context, Dermakar [2] found that the reinforcement volumefraction is among the principal parameters that govern thelifetime of MMC. The addition of silicon carbide (SiC) reinforce-ments can increase fracture toughness and thermal shockresistance of those materials [3–6]. Aluminium-matrix-basedcomposites have significant potential for structural applicationsdue to their outstanding combination of large specific strength,stiffness and density. They have the advantage of being processedby conventional means such as forging, rolling, extrusion andparticularly machining [7–10]. In such a material, it was foundthat fatigue resistance is directly influenced by two mainparameters; (i) increase of volume fraction and (ii) decrease ofthe size of particles. In order to improve the performance and useof such innovative materials, an analysis of fatigue behaviourbecomes necessary. Obviously, fatigue life of materials is a goodindicator of the working life of engineering components. Mechan-isms of fatigue failure must be investigated in detail to produce

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+33 3 26699176.

em).

large-quality parts. In spite of the interest of researchers to large-cycle regime, many details on this topic are remaining to beexplored [11,12]. This is mainly due to the large time for tests. Thecosts to develop sophisticated testing devices remain too high. Toanalyse fatigue mechanisms, the major investigations referred totension–compression tests using metallic specimens, i.e. steel,cast iron and aluminium alloys [13–18]. These metals have theadvantage of being homogeneous compared with other materialsas metal–matrix composites, where reinforcements, i.e. particlesor fibres, are mainly considered as the origin of stress concentra-tion within the matrix. Thus, the fatigue life as a random variablewill be inevitably influenced by microstructure heterogeneity[19,20]. Literature has given many explanations about failuremechanisms in very-large-cycle fatigue, for which the distributionof fatigue life depends on different mechanisms, i.e. initiation,crack growth and propagation generated in the material. Theircomprehension is extremely important for correlating failure withthe associated fatigue life regime. The most common ideaof observations is that two initiation mechanisms are acting:(i) surface crack initiation that is found up to 106 cycles and(ii) probable appearance of internal initiation sites beyond thisvalue. Series of recent researches have been conducted by Marineset al. [21–24] to design sophisticated experiments in order toexamine material behaviour in very-large-cycle fatigue regime.The authors have pointed out the decrease of material strength upto a large number of cycles. A remarkable event between 106 and108 cycles has been noted. In this range of cycle number, crackinitiation may switch location from the specimen surface to aninterior initiation termed as fish eye. Particularly for steel,

SiCp

Fig. 1. Microstructure of Al/SiCp composite: as-received state.

Fig. 2. Geometry and dimensions of the specimen used for fatigue test.

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–11351126

experimental findings showed that fatigue fracture can occur inthis range as a consequence of initiation at a certain distance fromthe surface. Bayraktar et al. [25] detailed failure mechanisms ofdifferent metallic alloys. They focused on effects of internaldefects such as inclusions, pores and nature of microstructure ormetallurgical impurity on crack initiation in very-large-cyclefatigue range. They concluded that the initiation site, whichdepends on a defect, is not always located in the depth ofspecimens at very-large level of cycles, since the probability offinding a defect at the surface depends on the number anddistribution of defects in a given volume of the specimen.Observing internal initiation, Paris et al. [26–28] questionedthemselves about the significance of crack growth portion on lifeof a fish eye. This was answered by developing an estimatingformula using the Paris–Hertzberg–McClintock crack growth ratelaw:

da

dN¼ bg

DKeff

Effiffiffiffiffibg

p !3

ð1Þ

where a is the crack length, N the cycle number, bg the magnitudeof the Burger’s vector, DKeff the effective stress intensity factorand E the Young’s modulus. The authors showed the effectivenessof the above relationship as a predictor of threshold corner. Onone hand, they demonstrated that crack growth life of fish-eyefailures is not a significant portion compared with the total fatiguelife for several steels. On the other hand, they found that the crackgrowth portion of semi-circular surface flaws is small if theexperimental lifetime exceeds 107 cycles. Here, it is veryinteresting to mention that major estimations were computedfor steels and aluminium alloys, where the mechanisms of failureare associated to a ‘single initiation’. Thus, one can wonder howthe power law of Paris et al. [27] is capable of presenting thelifetime of material when the fatigue behaviour is governed bymechanisms of ‘multi-surface initiations’. The objective of thiswork is to investigate the effects of simultaneous initiations,randomly appearing at the surface in a metal–matrix compositereinforced with silicon carbide particles, on the estimation offatigue lifetime. The Paris power law and some considerationsmade by Marines et al. [23] will be retained for this study.Experimental findings will prove that the use of an equivalentellipse method (EEM) for modelling the contribution of eachinitiation on failure is very satisfactory.

2. Experimental procedure

The material studied is an extruded composite. It is a 2009aluminium matrix reinforced with discontinuous silicon carbideparticles (Al/SiCp-MMC). The average dimension of the SiCp

particles is about 5–8mm (typical microstructure is shown inFig. 1). The reinforcement is arbitrarily distributed into thelongitudinal cross-section of the material. Contrary to long- fibrereinforced materials, where orientation is easily detected, theconsidered reinforcement does not exhibit any preferredorientation.

The parameters used are cutting speeds of 90 m min�1 andfeed rates of 0.15 mm rev�1 and 0.3 mm rev�1. Pre-treated speci-mens were manufactured after heat treatment; the surface ofspecimens was not polished. The depth of cut is kept constant at1.25 mm. For ensuring good comparisons between the results offatigue, the finishing operation uses a new tool for each specimen.As recommended in literature [29,30], an uncoated tungstencarbide tool (WC) with clearance angle of 51, cutting edge angle of351, length edge of 9.525 mm and nose radius of 0.4 mm was used.

Lubrication was kept active during machining. The specimen usedfor fatigue test has the standard geometry of Fig. 2.

Fatigue tests were performed using a SIMPLEX bendingmachine equipped with four-point fixtures devices. The rotationspeed of the machine was fixed at 3500 rev min�1. Allexperimental tests were carried out at the same load ratio,R=�1, and at room temperature. Two states of materialwere used: non-pre-treated state and pre-treated state. Thematerial was heated up to 498 1C in a furnace and held at thattemperature for 240 min before quenching in water to roomtemperature. This heat treatment has the advantage of enhanc-ing mechanical homogeneity within the matrix material byrelaxing residual stresses introduced at the particle–matrixinterface from prior elaboration process of the composites.Thermal properties of ceramics are known to be significantlydifferent from there of metal phases of composite matrices, whichmight result in the appearance of residual stresses and disloca-tions within composites on cooling from the fabrication tempera-ture. Table 1 summarises the mechanical properties, chemicalcomposition of the material and specifications of heat treatmentused.

3. EEM-based approach

Paris et al. [27] modelled the fatigue crack growth as a semi-elliptical surface geometry with a transversal direction 2b andradial dimension a. This model was developed for a single crackinitiation such as the case in steels and aluminium alloys. Here,the aim is to evaluate the portion of life of each site when fatiguecracks initiate in more than one regions at the surface of loaded

Table 1Chemical composition, mechanical properties and heat treatment conditions used.

Composition (wt%) Mechanical properties (MPa) Heat treatment

Composites (Al/SiCp-MMC) Young’s modulus (E) 80�103 T (1C): 498

Yield stress (sy) 300 Holding time: 4 h

Ultimate tensile strength (UTS) 500 Cooling: water

Matrix (Al-2009) Silicon: 0.25max Young’s modulus (E) 70�103

Iron: 0.2max Yield stress (sy) 260–300

Copper: 3.2–4.4 Ultimate tensile strength (UTS) 390–440

Chromium: 1.0–1.6

Oxygen: 0.1max

Titanium: 0.6max

Reinforcement SiC (Particles): 15 Young’s modulus (E) 425�103

Yield stress (sy) 1172

Ultimate tensile strength (UTS) 3900

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–1135 1127

specimen. For a system composed of n initiation sites with n

associated propagations, the growth of the crack number i, can beassimilated to an ellipse with minor axis 2a0i and major axis 2b0i.Similarly, the propagation associated with single initiation will beassumed to have similar shape with axis 2ai and 2bi, respectively.Furthermore, three interesting assumptions were formulated andmaintained for the proposed EEM:

�

the total surface of n initiation sites is approximated by anellipse with perfect shape, in the manner:

p Sn

i ¼ 1a0ib0i ¼ pa0eqb0eq ¼ S0eq ð2Þ

a0eq and b0eq are the half-minor and half-major axes of theequivalent ellipse, respectively;

� the equivalent ellipse has to be included in the cross-section of

the specimen: the major axis of the ellipse cannot be over thespecimen radius r.The second condition verifies the transformation of multi-initiations to unique initiation due to physical growth of cracksthat cannot propagate out of the cross-section of the specimen.This results in the following equation, which uses the relationcos 2g+sin 2g=1 (Fig. 3):

a20eq � 2ra0eqþb2

0eq ¼ 0 ð3Þ

Combining Eqs. (2) and (3) leads to

a40eq � 2ra3

0eqþS0eq

p

� �2

¼ 0 ð4Þ

In the same way, the equation of propagation contribution canbe established for the equivalent ellipse:

a4eq � 2ra3

eqþSeq

p

� �2

¼ 0 ð5Þ

Eqs. (4) and (5) determine the dimensions of the equivalentellipse for multiple initiation sites and their associatedpropagations. From experimental observations, it is proventhat (a0i/ro0.5) whereas (ai/ro0.5) is not always verified,especially for a large number of cycles. Thus, the followingrelations should be used for computing the equivalent surfaceof the propagation portion:

Si ¼ paibi forai=rr0:5andgr p2

ð6Þ

Si ¼ pr2 � pðr � aiÞbi forai=r40:5andg4 p2

ð7Þ

The crack growth from a0 to a is considered depending on the

� material state and surface finish state of each specimen. It was

assumed that cracks grow according to a constant propagationrate. This was formulated as follows:

a0eq

aeq¼ ra ¼

a01

a1¼ � � � ¼

a0i

ai¼ � � � ¼

a0n

an

b0eq

beq¼ rb ¼

b01

b1¼ � � � ¼

b0i

bi¼ � � � ¼

b0n

bn

8>>><>>>:

ð8Þ

ra and rb are constant ratios that physically represent thelateral and radial rates of fatigue crack growth, respectively.They are material- and loading-conditions-dependent con-stants. The number of cycles ðNi

SI�totalÞ associated with crackgrowth of a site i is the portion of the total experimentallifetime ðNeq

SI�totalÞ. Referring to the surface ratio, the crackgrowth lifetime of the site i is estimated using the followingformula:

NiSI�total ¼

Si

pPn

i ¼ 1

aibi

NeqSI�total ¼

a0ib0i

rarb

1

aeqbeqNeq

SI�total ð9Þ

Fig. 3 shows the principle of EEM for characterising the crackgrowth when cracks occur at single site or multiple initiationsites. The subscript ‘SI’ denotes surface initiation.

In the diagram of Fig. 3, aeq (Ea for single-initiation case ofmetals) denotes the exponent of fatigue crack growth rate belowthe threshold corner [23,24,26,27] in which the multi-surfaceinitiations (MSIs) of cracks were reduced to an equivalent ellipse,verifying the above assumptions of the EEM.

4. Estimation of crack growth life from the (da/dN) curve

Marines et al. [21–24] presented a summary of a historicalreview on crack growth estimation referring to Paris law. Fromearly observations, Hertzberg [31,32] and Paris et al. [26] havenoted that the threshold corner is obtained at

da

dN¼ bg3DKeff ¼ E

ffiffiffiffiffibg

q¼DK0 ð10Þ

The reliability of this relationship for predicting the crack growthrate and the corner has been proved in literature. Therefore, thisobservation has been basically considered for developing pre-dictive formula available for estimating very-large-cycle fatiguefailures. Unfortunately, no study has been conducted to explainthe mechanisms of fatigue crack propagation following multi-surface initiations. This paper explores closure effects on thisissue. Integrations of fatigue crack growth rates will be separatedfor equivalent crack growth below threshold Neq

SI�int, small cracks

Real approach: observations on MSI of cycled Al/SiC pcomposites

Evaluation of the contribution of each initiation site on the total fatigue lifetime:iNSI−Total

Global approach for idealised model:Application of EEM

Evaluation of a0eq, b0eq, aeq, beq for equivalent ellipse

r

a0eq

beqb0eq

aeq

Local approach for idealised model:

Computation of ra and rb

Updating of aira

a0i= and birb

b0i=

Updating of Si associated at each site

Initiation sites

Propagation zone

Final fracture zone

�

Fig. 3. Principle of the local and global approaches of the EEM.

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–11351128

NeqSI�a0eq-aieq

and long cracks NeqSI�aieq-aeq

. The addition of the threecontributions gives the total estimated lifetime.

NeqSI�total ¼Neq

SI�intþNeqSI�a0eq-aieq

þNeqSI�aieq-aeq

ð11Þ

When there is no transition from small to long crack (aeqrai eq), Eq. (11) will be reduced to

NeqSI�total ¼Neq

SI�intþNeqSI�a0eq-aeq

ð12Þ

4.1. Estimation of crack growth life portion below threshold corner

As noted by Paris et al. [27], the growth curve shouldpass through the threshold corner but prior to the corner in sucha manner that the fatigue law below threshold is approxi-

mated by

da

dN¼ bg

DK0

Effiffiffiffiffibg

p !a

a

a0

� �a=2

¼ bga

a0

� �a=2

ð13Þ

where ab3, DK0 is the crack growth rate threshold verifyingDK0 ¼ E

ffiffiffiffiffibg

pat the corner, and a0 is the half-minor axis of the

initial flaw of the crack. Then, the number of cycles belowthreshold might be determined for better estimation of the crackgrowth cycles at this stage. Eq. (13) leads to

dN¼1

bg

a0

a

� �a=2

da ð14Þ

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–1135 1129

The integration of the above expression from an initialequivalent flaw aint eq to an equivalent semi-elliptical flaw ofmulti-initiation system with size a0eq leads to the followingrelation:

NeqSI�int ¼

ða0eqÞaeq=2

bg

Z a0eq

aint eq

1

aaeq=2da¼

a0eq

bg

1

ððaeq=2Þ � 1Þ

a0eq

aint eq

� �ðaeq=2Þ�1

� 1

" #

ð15Þ

where aeq is the exponent of fatigue behaviour of equivalent semi-elliptical flaw.

Furthermore, the initial crack flaw of a single initiation can becharacterized as given in [24] by the following expression:

a0 ¼bg

pE2

Y2ðDsÞ2ð16Þ

where Y is a factor depending on the ratio a/2r in the case ofsingle-initiation flaw and is written as follows:

Y ¼ð1:84=pÞ½tanðpa=4rÞ=ðpa=4rÞ�0:5

cosðpa=4rÞ0:752þ2:02

a

2r

� �þ0:37 1� sin

pa

4r

� �n o3� �

ð17Þ

For an equivalent ellipse of a multi-initiation system, a0

becomes a0eq. Thus, the factor Y and the flaw size a should bereplaced in Eq. (17) by Yeq and the equivalent flaw size aeq,respectively. Therefore, combining Eqs. (15) and (16) leads to

NeqSI�int ¼

1

pE2

Y2eqðDsÞ

2

1

ððaeq=2Þ � 1Þ

a0eq

aint eq

� �ðaeq=2Þ�1

� 1

" #ð18Þ

4.2. Estimation of crack growth life portion for small crack

To determine the crack growth portion beyond thresholdcorner, when no transition from small to long crack is considered,Paris law will be integrated from the equivalent initial crack flawa0eq until the semi-elliptical flaw aeq. Using a=3, the Paris lawgiven by Eq. (13) leads to

NeqSI�a0eq-aeq

¼ða0eqÞ

3=2

bg

Z aeq

a0eq

1

a3=2da¼ 2

a0eq

bg1�

a0eq

aeq

� �1=2" #

ð19Þ

Using Eq. (16), the above relation may be rewritten as follows:

NeqSI�a0eq-aeq

¼2

pE2

Y2eqðDsÞ

21�

a0eq

aeq

� �1=2" #

ð20Þ

4.3. Estimation of total crack growth lifetime

Combining Eq. (18) and (20) leads to an estimation of the totalcrack growth lifetime needed for a system of multi-surfaceinitiations with an equivalent semi-elliptical surface failure. Thus,it can be written as

NeqSI�total ¼

2

pE2

Y2eqðDsÞ

2

1

2ððaeq=2Þ � 1Þ

a0eq

aint eq

� �ðaeq=2Þ�1

þaeq � 3

aeq � 2�

a0eq

aeq

� �1=2" #

ð21Þ

For this, it was considered that aint eq=0.9a0eq [21–24] andNeq

SI�total ¼Nf�exp. Thus, the following equation that should benumerically resolved for the variable aeq, is obtained:

Nf�expp2

Y2eqðDsÞ

2

E2¼

1

2ððaeq=2Þ � 1Þ

1

0:9

� �ðaeq=2Þ�1

þaeq � 3

aeq � 2�

a0eq

aeq

� �1=2" #

ð22Þ

Nf�exp denotes the fatigue lifetime measured in experimentaltests. Analysis of Eq. (22) leads to the identification of the Parislaw exponent aeq required for estimating the crack growth lifeportion below threshold. The last expression is more general than

the one developed for single initiation in [21–24,27,32] but itsapplication for multi-initiations needs, at an early stage, theapplication of the EEM.

5. Results and discussion

5.1. Influence of surface finish on Nf: ‘S–N’ curves

The turning operations were carried out on a numericallycontrol turning machine. Lubrication is considered for all opera-tions in order to delay damaging effects of SiCp particles oncutting tools. The fatigue behaviour of metal–matrix compositesdepends on several factors, including particle type, size andvolume fraction, matrix microstructure, particles–matrix inter-faces characteristics and ultimately conditions of cutting, i.e. feedrate. As known in machining, the finished surface is better as feedrate decreases. Moreover, it is extremely probable that friction attool–work surface interface varies with the variation of the feedrate.

Locally, at the material–tool interfaces, the friction effects notonly depend on lubrication conditions but also on the cutting timeof the tool. It is obvious that thermal and mechanical properties attool–surface interface change with cutting parameters, especially,with feed rate. As a consequence, the layers, i.e. subsurface, closeto the turned surface will be inevitably affected. Thus, mechan-isms of crack initiations activated at finished surfaces underalternate mechanical loadings, i.e. fatigue load, evolve withmicrostructure and mechanical properties at the subsurface. Forrevealing the surface property effects on fatigue lifetime, twospecimens with different surface finish were considered forfatigue tests (Fig. 4). For the used Al/SiCp-MMC, where thematrix was made from an aluminium alloy, micrographicexaminations proved that the microstructure does not presentinterfacial defects between matrix and reinforcements.

Generally, processing conditions that are used for producingAl/SiCp-MMC enhance the bonding at particles–matrix interfaces.For these reasons, it has been assumed that failure under fatigueload essentially occurs as a consequence of global fatiguemechanisms of the composite.

From the micrographs of Fig. 4, it can be noted that failureoccurring under fatigue load is localised at the median zone of onestreak generated beforehand by the tool. Microscopic inspectionshave shown that failure was always caused at this zone and neverat the board of streaks. When the tool operates in the material,temperature increases and reaches its maximum value at the noseof the tool, where lubricant circulation is technically hard toensure, contrary to the lateral zones, where streak boards aregenerated. Thus, thermal effects, rapidly attenuated by the actionof lubricant behind the tool once the tool advances, result in akind of local treatment that strengthens microstructure at themedian zone of streak and causes its cracking earlier than thestreak boards.

Microscopic observations did not explicitly show effects ofsurface finish quality on fatigue behaviour. Therefore, the S–N

curves were plotted for 0.15 and 0.30 mm rev�1 feed rates. Figs. 5aand b clearly illustrate a drop induced in the material lifetimewhen the feed rate increases. This result has been confirmed forboth pre-treated state and non-pre-treated states of the material.A long cutting time should affect the surface more than arelatively short time as recorded for large feed rates.

The mechanical properties of the subsurface should beenhanced as a consequence of local thermal-induced effectscaused by the operating tool. As known, thermal effects, i.e. heattreatments, are highly time-dependent phenomena. Hence, use ofsmall feed rates (long cutting time) presents a good opportunity

Pre-treated, f=0.15 mm.rev-1

Pre-treated, f = 0.30 mm.rev-1

Non pre-treated, f = 0.15mm.rev-1

Non pre-treated, f = 0.30 mm.rev-1

Streak bounds Median zone

Fig. 4. Crack propagation paths in cycled Al/SiCp-MMC: (a, b) pre-treated states, and (c, d) non-pre-treated states. (Ds=280 MPa, R=�1).

160

180

200

220

240

260

280

300

320

App

lied

stre

ss σ

(MPa

)A

pplie

d st

ress

σ (M

Pa)

Pre-treated: 0.30mm/rev

Non pre-treated: 0.30mm/rev

160

180

200

220

240

260

280

300

320

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6

Pre-treated: 0.15mm/rev

Non pre-treated: 0.15mm/rev

Fatigue life Nf (x106 cycles)

Fatigue life Nf (x106 cycles)

1

0 0.2 0.4 0.6 0.8 1.2 1.4 1.61

Fig. 5. S–N curves obtained for test specimens with surface finish prepared using

two feed rates: (a) f=0.15 and (b) f=0.30 mm rev�1 (R=�1).

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–11351130

to ensure the effectiveness of these effects at least at subsurfacelayers. This phenomenon plays a large role in increasing thematerial resistance, recorded over remarkably large fatigue life-times.

5.2. Estimation of growth area and initiation area of cracks:

efficiency of EEM

The physical limits of initiation and propagation zones havebeen studied through microscopic inspections (Figs. 6a and b).Investigations of failure surfaces show the difference betweeninitiation site states and propagation site states of cracks.Initiation sites of cracks which are known as zones thatconsume the greatest contribution of cycle number, result insmooth failure aspect as seen in Fig. 6c. There zones occasionallyexhibit a preferred orientation whereby cracks grow as can beobserved in the micrographs of Fig. 6d.

The failure at the end of the lifetime generates a rapidseparation of materials with rough aspect as can be observedin the large delimited area of the micrographs of Figs. 6e and f.The final failure occurs as a consequence of the growth of micro-cracks initiating around particles. From the failure area of Fig. 6f,micro-cracks having a parallel direction seem to be weakerthan those observed in Fig. 6e. This is essentially due to heattreatment, which probably seeks to homogenise properties at theinterface between the matrix and particles in a manner so as toenhance the progress of separation mechanisms, and to ensuremore regular shape and scatter of micro-cracks. This explains thefact that only pre-treated material states lead to large-cyclefatigue.

The composite material discussed here is chosen in a mannerso as to favour multi-initiation mechanisms. From experimentalresults, it has been noticed that only pre-treated material statecan lead to large-cycle fatigue. Consequently, fatigue tests of onlypre-treated specimens will be explored in this section. Table 2summarises the measured dimensions of each crack initiation and

Crack orientationGrowth

Pre-treated

Non pre-treated

Fig. 6. Failure areas (a, b) with larger magnification showing initiation zones (c, d), and final separation zones (e, f). (Ds=300 MPa, R=�1, f=0.15 mm rev�1).

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–1135 1131

propagation area and the fatigue lifetime recorded for eachapplied load.

A numerical study of constitutive equations leads to theidentification of the fatigue law exponent aeq (Table 3). Smallerexponent values are obtained for lower number of cycles. Thisexponent is found to be sensitive to the testing load; the exponentseems to increase as fatigue lifetime increases and the ratio ra

increases. From measurements, the mechanisms of initiationseem arbitrary. For the same applied testing load, fatigue crackmay initiate from single surface site or multi-surface sites, whichproves that the mechanisms of initiation depend only on thematerial state and constituents of microstructure. It is of greatinterest to note that when cracks initiate at surface from morethan one sites, the number of cycles needed for initiation probablydecreases with respect to other fatigue conditions and appliedload. The fall noted in the number of cycles for initiationmechanisms is especially true for large fatigue loads, for whichNf�exp�104 cycles, i.e. for Ds=300 MPa. It was seen here in testsin the large-cycle regime, i.e. Nf�exp4105 cycles, that initiationsare the most time-consuming part of the total experimental life ascompared with crack growth (Table 3).

5.3. Estimation of the Paris law exponent: ‘a–S’ curves

For estimating the crack growth life portion below thresholdcorner, Marines et al. [22–24] have considered a constantexponent of Paris fatigue law. They always used three values ofa: 25, 100 and 200. The choice of these values has not beensufficiently discussed in open literature. The analysis here has theobjective to enlighten on the range of variation of the Paris lawexponent with used conditions.

The curves of Fig. 7 show the evolution of the estimatedexponent with testing load. A first observation proves thesensitivity of the exponent to fatigue load for both 0.15 and0.3 mm rev�1 feed rates. From Eq. (22), the increase of theexponent aeq with the number of cycles, which sensitively varieswith load, seems to be obvious. The highest specimen lifetime isrecorded for the lowest load, which mathematically requires thelargest values of the exponent as can be implicitly deduced fromthe predictive formula of the total crack growth lifetime. Thedifference between the numbers of cycles consumed at each feed

results in marked deviations in the values of aeq. Even if the slopesof the curves are close, the Paris exponents identified for0.15 mm rev�1 are about 37% higher than those found at0.30 mm rev�1. Thus, it is of great interest to note that theexponent of Paris law is not constant and not the only material-dependent parameter either.

The fatigue mechanisms did not evolve from the beginning ofinitiation to the end of initiation (beginning of propagation) inthe same way. Generally, crack initiation consumes the largestportion of total life. From the starting point to the thresholdpoint, the crack rate mechanisms should have enough timeto change. This can be implicitly observed through the variationof the exponent a. Any change in the mechanisms of fatigue,results in changes of crack life. So, from multi-scaling analysis,it can be assumed that the coefficient a, which mostly dependson material properties, implicitly varies with crack life, whichis related to the applied load. Scatter in the computed slope fora given condition (fixed load and material state) may be observedas a consequence of the deviation in number of cycles for thesame condition (e.g. for Ds=300 MPa and f=0.15 mm re-v�1)aeq=44 for Nf=90,200 cycles and, aeq=34 for Nf=61,900

cycles). The drastic change in the exponent may be explained bythe evolution of failure rate mechanisms with increasing cracklifetime.

5.4. Sensitivity of the Paris law exponent to the crack flaw:

‘a� aint=a0’ curves

The variation of exponent aeq with crack growth rate below thethreshold corner is given in Figs. 8a and b for different appliedloads.

The curves show the same trend and slightly increase withcrack growth rate. When the feed rate increases, the estimatedvalues decrease. When the crack growth rate below thresholdpasses from 0.9 to 0.96 for f=0.15 mm rev�1, the exponentincreases about 3 times for the lowest load, whereas it reaches4.2 times for the highest load. For f=0.30 mm rev�1, these ratiosare 3.1 and 5.2, respectively. This observation agrees with thechoice of Paris exponent value, which was taken as large as thecrack growth rate increases [23].

Table 2Fatigue lifetime and measured dimensions of crack initiations and propagations at each site of failure cross-section of pre-treated Al/SiCp-MMC material.

f

(mm rev�1)

Ds(MPa)

Test

number

Number of

initiation sites

a01

(mm)

b01

(mm)

a02

(mm)

b02

(mm)

a03

(mm)

b03

(mm)

a04

(mm)

b04

(mm)

S0eq (mm2) a1

(mm)

b1

(mm)

a2

(mm)

b2

(mm)

a3

(mm)

b3

(mm)

a4

(mm)

b4

(mm)

Seq

(mm2)

Nf�exp

(cycles)

0.15 300 1 2 0.511 2.695 0.418 1.859 6.771 1.534 4.461 21.493 90,200

300 2 1 0.883 3.811 10.570 1.580 4.415 21.913 61,900

280 3 1 1.208 3.996 15.170 1.905 4.740 28.372 101,600

280 4 1 0.976 3.346 10.258 1.766 4.647 25.780 98,200

260 5 1 0.929 3.904 11.398 2.138 4.647 31.208 174,650

260 6 2 0.651 3.346 0.186 0.651 7.219 1.394 4.182 0.256 1.255 19.325 234,250

260 7 1 0.836 3.346 8.792 1.673 4.182 21.981 223,200

240 8 2 0.744 2.138 0.976 3.253 14.966 1.952 4.647 28.494 190,500

240 9 1 1.255 3.904 15.387 2.091 4.461 29.308 156,800

240 10 2 0.797 3.721 0.487 2.747 13.526 1.993 4.607 0.576 2.835 33.982 328,600

210 11 1 1.506 3.455 16.350 1.949 4.607 28.211 837,200

210 12 1 1.152 4.164 15.067 2.082 4.430 28.976 1,237,800

190 13 1 1.506 3.987 18.865 2.436 4.696 36.602 1,352,400

0.30 300 14 2 0.976 3.811 0.279 2.324 13.718 1.766 4.554 0.674 3.346 32.348 48,700

300 15 4 0.354 3.012 0.487 2.569 0.532 1.772 0.221 1.063 10.986 1.905 4.607 0.576 1.772 0.443 1.063 0.310 1.107 33.335 57,700

300 16 2 0.465 2.695 0.302 2.231 6.052 1.859 4.647 0.744 2.509 32.999 73,900

280 17 1 1.208 4.182 15.875 1.812 4.554 25.930 58,450

280 18 2 0.697 3.067 0.511 2.788 11.194 1.348 4.740 20.068 66,100

280 19 2 1.115 3.904 0.139 1.394 14.288 2.138 4.740 0.325 2.509 34.396 80,300

260 20 1 1.019 3.810 12.195 2.126 4.607 30.776 105,150

260 21 2 0.558 3.067 0.790 2.556 11.717 1.139 3.253 1.022 2.928 21.038 121,000

240 22 1 1.063 3.455 11.541 2.259 4.607 32.700 171,100

240 23 1 0.886 2.304 6.412 2.348 4.786 35.305 174,100

220 24 1 1.162 3.160 11.533 2.184 4.601 31.567 317,700

220 25 1 1.240 3.721 14.500 2.502 4.696 37.561 298,100

200 26 1 1.882 3.811 22,531 2.742 4.461 42.578 473,100

A.

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tion

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5–

1135

113

2

Table 3Fatigue behaviour parameters computed using the EEM for multi-surface initiation sites of pre-treated Al/SiCp-MMC material ða0eq=aint eq ¼ 1=0:9Þ.

f

(mm rev�1)Ds

(MPa)

Test

number

Number of

initiation

sites

Eq. (4) Eq. (5) Eq. (8) Eqs. (6) and (7) Eq.

(17)

Eq.

(22)

Eq. (9) Eq. (18) Eq. (20)

a0eq

(mm)

aeq

(mm)

ra

(mm)

S1

(mm2)

S2

(mm2)

S3

(mm2)

S4

(mm2)

Yeq aeq N1SI�total

(cycles)

N2SI�total

(cycles)

N3SI�total

(cycles)

N4SI�total

(cycles)

NeqSI�int

(cycles)

NeqSI�a0eq-aeq

(cycles)

0.15 300 1 2 0.812 1.829 0.444 21.493 0.79 44.06 57,663 32,537 33,188 57,012

300 2 1 1.106 1.855 0.596 21.913 0.79 34.02 23,290 38,610

280 3 1 1.425 2.242 0.636 28.372 0.85 63.52 67,385 34,215

280 4 1 1.083 2.089 0.518 25.780 0.83 53.05 48,105 50,095

260 5 1 1.165 2.407 0.484 31.208 0.88 75.84 118,903 55,747

260 6 2 0.849 1.694 0.501 18.318 1.007 0.77 77.46 221,921 12,329 164,766 69,483

260 7 1 0.973 1.859 0.523 21.981 0.80 78.23 160,906 62,294

240 8 2 1.412 2.249 0.628 28.494 0.85 74.86 63,558 126,942 142,898 47,602

240 9 1 1.440 2.297 0.627 29.308 0.86 68.73 109,962 46,838

240 10 2 1.314 2.568 0.512 28.853 5.129 0.91 93.59 226,456 102,144 271,371 57,229

210 11 1 1.503 2.232 0.673 28.211 0.85 108.89 783,135 54,060

210 12 1 1.418 2.277 0.623 28.976 0.86 118.55 1,175,432 62,391

190 13 1 1.665 2.718 0.613 35.942 0.94 120.25 1,287,081 65,324

0.30 300 14 2 1.327 2.473 0.537 25.265 7.083 0.89 22.94 41,474 7226 12,881 35,819

300 15 4 1.136 2.530 0.449 27.570 3.206 1.480 1.079 0.90 27.55 17,614 20,658 15,542 3886 14,545 43,155

300 16 2 0.752 2.511 0.299 27.137 5.862 0.90 26.55 48,052 25,848 14,203 59,697

280 17 1 1.472 2.097 0.702 25.930 0.83 39.47 29,505 28,945

280 18 2 1.151 1.740 0.661 20.068 0.78 34.63 39,660 26,440 28,350 37,750

280 19 2 1.366 2.591 0.527 31.832 2.564 0.92 53.67 76,868 3432 40,231 40,069

260 20 1 1.222 2.382 0.513 30.776 0.88 54.55 52,607 52,543

260 21 2 1.188 1.801 0.660 11.635 9.403 0.79 59.25 55,490 65,510 77,881 43,119

240 22 1 1.176 2.494 0.471 32.700 0.90 69.86 106,073 65,027

240 23 1 0.782 2.644 0.296 35.305 0.93 65.95 85,350 88,750

220 24 1 1.175 2.428 0.484 31.567 0.89 85.13 240,471 77,229

220 25 1 1.380 2.773 0.498 36.908 0.95 88.02 233,699 64,401

200 26 1 1.893 3.058 0.619 38.426 1.02 100.74 423,679 49,425

A.

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2(2

01

0)

112

5–

1135

113

3

0

20

40

60

80

100

120

140

160

160

Pre-treated: f = 0.15mm/revPre-treated: f = 0.30mm/revLinear regression: f = 0.15mm/revLinear regression: f = 0.30mm/rev

0.74

0.74

�eq

Applied stress ΔσΔσ (MPa)180 200 220 240 260 280 300 320

Fig. 7. Variation of the fatigue law exponent aeq versus the applied stress Ds.

0

50

100

150

200

250

300

350

400300MPa280MPa260MPa240MPa220MPa200MPa

0

50

100

150

200

250

300

350

400

0.89aint-eq /a0eq

300MPa280MPa260MPa240MPa210MPa190MPa

0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97

0.89aint-eq /a0eq

0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97

� eq

� eq

Fig. 8. Sensitivity of the fatigue law exponent aeq to the ð1=ðða0eqÞ=ðaint eqÞÞÞ ratio:

(a) f=0.15 mm rev�1 (maximal standard deviation: 12.9) and (b) f=0.30 mm rev�1

(maximal standard deviation: 9.8).

A. Mkaddem, M. El Mansori / International Journal of Mechanical Sciences 52 (2010) 1125–11351134

6. Conclusions

The estimation of crack initiation and growth lives consumedby multi-surface initiation (MSI) mechanisms at large-cycleregime has been discussed herein for aluminium matrix compo-sites. This is why a new approach based on equivalent ellipse

method has been proposed for characterising initiation andgrowth area when cracks initiate at surface from more than onesites. The estimation of fatigue life portion below and beyondthreshold corner has been performed without considering thetransition from small to long cracks, using Paris law. The use ofEEM does not depend on the nature of the material and shows itseffectiveness especially for the composite considered wherecracks might initiate from several sites. Thus, the followingremarks can be drawn.

�

The quality of surface finish plays a great role in the fatigue lifeof specimen. This observation is true for pre-treated as well asfor non-pre-treated states of the considered material. Specifi-cally, good surface state implies higher fatigue lifetime of thespecimen. � Application of the EEM for the predictive formula makes

identification of the fatigue law for each portion of crack lifepossible. MSI mechanisms influence the initiation portion onlyat the largest testing load. These mechanisms act in a mannerso as to reduce the number of cycles consumed at initiationwhen Nf�exp�104 cycles. The initiation portion remains themost time-consuming contribution of the total experimentallife as compared with crack growth period if tests are in thelarge-cycle regime (Nf�exp4104 cycles).

� The Paris law exponent varies with experimental considera-

tions, which proves that it is not just a material-dependentparameter. The identification of aeq by applying Paris law fortotal fatigue life of the cracks proves that the exponent issensitive to the applied testing load and the crack growth ratebelow threshold corner. For surfaces finished at both 0.15 and0.30 mm rev�1, the exponent linearly decreases when theapplied load increases and when the crack growth rate belowthreshold decreases.

Acknowledgements

The authors gratefully acknowledge the technical supportof FORGES DE BOLOGNE group. Also, they gratefully acknowledgeB. Favre (engineer) for his valuable help during this work.

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