International Journal of Fatigue 37 (2012) 112–122

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Experimental and numerical lifetime assessment of Al 2024 sheet

S. Khan, O. Kintzel, J. Mosler ⇑Helmholtz-Zentrum Geesthacht, Institute of Materials Research, Materials Mechanics, Max-Planck-Str. 1, 21502 Geesthacht, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 June 2011Received in revised form 22 September 2011Accepted 25 September 2011Available online 29 October 2011

Keywords:Low cycle fatigueContinuum damage mechanics (CDM)Aluminum alloysDuctile damage

0142-1123/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2011.09.010

⇑ Corresponding author.E-mail address: joern.mosler@hzg.de (J. Mosler).

In the present paper, a thorough analysis of the low-cycle fatigue behavior of flat sheets of aluminum Al2024-T351 is given. For that purpose, material characterization is combined with material modeling. Theexperimental analyses include monotonic and cyclic loading tests at high stress levels. For the assess-ment of microstructural characteristics, advanced imaging technology is used to reveal, e.g. crack initia-tion loci and particle sizing. The numerical simulation is done using a novel ductile–brittle damagemodel. Thereby, the model parameters are optimized by means of an inverse parameter identificationstrategy which, overall, leads to a very good agreement between experimentally observed and computa-tionally predicted data. For demonstrating the prediction capability of the novel coupled model also forcomplex engineering problems, a certain stringer assembly, as used in fuselage parts of airplanes, isanalyzed.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

High-performance metallic sheets show a broad range of indus-trial applications. For example, (i) stainless steel plates are used inship manufacturing and (ii) high-strength aluminum alloys areemployed in the aviation and automotive industry. More specifi-cally, high-strength aluminum alloys, such as Al 2024-T351, arewidely used for lightweight structural applications, especially intransportation vehicles. In the aerospace industry, they are utilizedin the form of sheets and plates in the fuselage or the lower wing.

Because of the growing demands on the critical performance ofmachines and constructions, and due to an emphasis on the econ-omy of production, it is generally accepted that many structuresneed not to be, or even cannot be, designed for infinite life. Thedesign must, however, ensure that the expected lifetime of thestructure is reached and the desired safety is achieved. In any case,knowledge of the prospective lifetime of a component is ofenormous practical interest. For the prediction of the lifetime,the consideration of low-cycle fatigue is of special significance,since repeated loading under high stress amplitudes limits the ser-vice lifetime of many highly-stressed components like those usedin the fuselage or gas turbine engines in aircrafts severely.

At the microstructural level, low-cycle fatigue is caused byaccumulation of micro-damage. In ductile metallic materials likesteels and aluminum, microvoid nucleation and growth have beenrecognized as key mechanisms of damage. Analyzing the elonga-tion of spherical or elliptical voids in a porous medium,

ll rights reserved.

Mc Clintock [1] and Rice and Tracey [2] initiated the research onvoid growth laws. Afterwards, many new developments in thisfield contributed to a number of ductile damage models or failurecriteria. Today, the approaches are numerous and can be sortedbroadly into three main classes: (i) abrupt failure criteria, (ii)porous metal plasticity, and (iii) continuum damage mechanics(CDM) approaches. For the first mentioned modeling approach,ductile failure is assumed when a micromechanical variable, forinstance the cavity growth, cf. Rice et al. [3], reaches a criticalcharacteristic material value. Regarding the second approach,damage effects are accounted for by a softening of the yield limitin dependence on the amount of porosity, cf. [4–6].

Finally, in the CDM approach, originated in the early works ofKachanov [7], softening effects associated with material damageare considered by means of thermodynamic internal state vari-ables. In this framework, in contrast to porous plasticity theory,damage effects are accounted for by a degradation of materialmoduli like the elastic stiffness. Mathematically speaking, thismethod is based on a smeared representation of the initiation,growth and coalescence of micro-defects and/or microcracks. Inte-grated within a thermodynamics framework, CDM models use asmany internal state variables as there are mechanisms of deforma-tion and material degradation to be accounted for, cf. [8,9].

Recently, a novel model suitable for the analysis of low cycle fa-tigue in high-strength aluminum alloys was proposed in [10]. Thismodel is based on careful experiments and material characteriza-tion of a thick plate of Al 2024-T351, see [11,12]. Within the resultsreported in [11,12], a broad variety of different damage mecha-nisms ranging from ductile to quasi-brittle could be observed forloading in S-direction (thickness direction). This complex mechan-

Table 1Composition (wt.%) of Al 2024-T351 by chemical analysis.

Cu Mg Mn Si Fe Cr Al

4.11 1.12 0.46 0.048 0.05 0.003 Rest

r 2356 10

160

30

10

S=4mm

Fig. 1. Specimen geometry used in the tests.

S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122 113

ical response is captured by the model discussed in [10] bycombining a ductile damage model originally advocated in [13]with a novel approach for quasi-brittle material degradation.

Although the material characterization of thick plates in S-direction is indeed important for understanding the complexmechanical response of Al 2024-T351, most practically relevantapplications are based on thin sheets. For this reason, such sheetsare analyzed in the present paper. More precisely, the present con-tribution addresses a microstructural analysis of flat specimens ofAl 2024-T351 under cyclic loading conditions, the identification offailure and damage mechanisms and their prediction by numericalmethods. The corresponding microstructural characterizations arecarried out with the aid of quantitative image analysis, yielding themorphology of grains and particles within the material. With frac-tography, the basic mechanisms of failure are identified. For theaccompanying numerical simulations, the model proposed in [10]is adopted. The respective material parameters are optimized bymeans of an inverse parameter identification strategy. For that pur-pose, standard flat tensile specimens are considered. Finally, thecalibrated material model is used for predicting crack initiationin a complex engineering structure. As a representative example,a stringer-skin connection of a fuselage is chosen.

2. Material characterization and experimental setup

The high-strength aluminum alloy considered in the presentstudy belongs to the Al–Cu alloy family. Such alloys are specificallydesigned for their superior mechanical properties for use in criticalstructural parts of aircrafts. They contain copper (Cu) as majoralloying element and hardening is achieved by aging. The materialused in this work is a hot-rolled sheet of Al 2024, as received in aT351 temper (solution heat treated, air-quenched, stress-relievedby cold stretching). The chemical composition is given in Table 1.

In order to characterize the alloy’s mechanical behavior, testshave been done on two different sources, a plate of 100 mm anda sheet of 4 mm thickness. Extensive work has already been doneon the plate, see [10–12], where in particular the short-transversal(thickness) direction was the main focus of interest. While thematerial response along the plate dimensions of the thick plate israther uniform and can be characterized as ductile, a quasi-brittlematerial behavior could be identified in thickness direction, see[10]. In the present paper, the damage mechanisms in flat sheetsof Al 2024-T351 are analyzed. For these analyses, specimens weremachined from a sheet of 4.0 mm thickness in two directions withrespect to the rolling direction, namely.

� L: Longitudinal.� T: Transversal, i.e. perpendicular to the rolling direction.

Load Cell

Upper Grips

Load Cell

Lower Grips

Nuts with controlled torque

Antibuckling guidStainless steel

4mm thick

Area for Δ lmeasuremen

Fig. 2. Anti-buckling support for LCF experiments, the extensometer

Within the cyclic loading experiments, symmetric strain ampli-tudes (R = �1) have been used. Consequently, large compressivestresses occur. For avoiding buckling under a negative strain ratio,the shape of the flat specimens has been carefully chosen, cf. [14].The shape and the dimensions of the specimen are depicted inFig. 1. After fabrication, the edges of the specimen were reworkedby hand using a cylindrical abrasive pencil and polished lengthwiseespecially around the notches using a rubbing compound toprevent initial edge cracks.

The monotonic tensile tests were performed under displace-ment control using a mechanical Zwick Roell 1484 testing machine(maximum force 200 kN) with a constant cross-head speed of0.5 mm/min, while the cyclic tests were done on a Schenck160 kN servo-hydraulic testing machine. Grips were carefullyaligned with the aid of an alignment fixture. An extensometer withgauge length of 8 mm, laterally attached to the specimen, was usedto measure length changes, since the measurement of cross-headtravel was not appropriate. Another reason for attaching a clip-gauge across the specimen is the possibility of obtaining high res-olution displacement data close to the region of interest. It shouldbe mentioned that the damage localization loci were always insidethe respective gauge length. Load–displacement responses wererecorded using DASYLabTM software. The stress was determinedas ratio between the clamp force to the (undeformed) cross-sec-tional area at the center of the specimen, i.e., engineering stresseswere used.

The cyclic tests were conducted at room temperature with tri-angular loading increments at a standard frequency 0.01 Hz withfully reverse control of the strain amplitude (R = �1). This very

e

Teflon sheets0.5mm thick

Flat Specimen 4mm thick

Bolts

t

(gauge length 8 mm) is attached to the edges of the specimen.

Fig. 3. Independence of the experiments of the anti-buckling guide: macroscopicbehavior during the first half-cycle of a cyclic test using the anti-buckling guidecompared to that of the monotonic tensile test without the anti-buckling guide.

Fig. 4. (a) Location and orientation of the specimens; (b) microstructure of Al 2024sheet material (magnified).

114 S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122

low frequency was chosen in order to obtain a better resolution ofthe load–displacement curve. During the experiments, the instantof mesocrack initiation was assumed at the moment of a suddendrop of the maximum cyclic peak load, cf. [12].

The determination of fatigue properties in thin sheets is extre-mely problematic due to a high risk of buckling of the sheet duringcompression, cf. [15,16]. Although the specimen’s geometry hasbeen chosen carefully, an anti-buckling guide was additionallyused to prevent buckling, allowing the specimen to ’’lean’’ againstthe guide without buckling. A floating guide technique was usedfor that purpose, see Fig. 2. A Teflon (PTFE) lubrication film of0.5 mm thickness was inserted between the buckling guides atboth sides of the specimen. By these means it could be ensured,that the axial freedom of movement of the specimen was con-strained as little as technically possible while at the same timebuckling could be prevented. For confirming this hypothesis, anexperimental test series was done where only the first half-cycleat the highest amplitude (both in T- and L-direction) was com-pared to a monotonic tensile test without the anti-buckling guide.As evident from Fig. 3, the macroscopic behavior is indeed not dis-turbed profoundly by the use of the anti-buckling guide.

Table 2Mechanical properties of Al 2024 T351 (monotonic loading, 3 samples considered inaveraging).

Tensiledirection

Yield stress(MPa)

Tensile strength(MPa)

Fracture strain(–)

L 365 473 0.31T 321 468 0.30

2.1. Microstructure

The mechanical anisotropy of metals is strongly related to thespatial distribution of grain orientations, grain shapes and sizes.While orientations of grains can in principle be determined by tex-ture measurements, sizes and shapes of the grains become visibleby polishing and etching plane cross-sections of the material. Pho-tographs have been taken of all three different surfaces usingpolarized light, having its normal vectors in L-, T- and S-direction,respectively. Fig. 4a shows the locations and orientations of thetest specimens taken and Fig. 4b displays the grain structure,showing the material anisotropy of the sheet with elongated grainsalong the L–T-planes. While the average grain sizes in L- and T-directions are 20 lm and 10 lm, respectively, the grain dimen-sions across the thickness of the sheet are reduced due to the roll-ing process and measure on average only 5 lm. Clearly, since theloading is applied across the plane of the sheet, the mechanicalproperties are primarily influenced by the orientation of the grainsalong this plane, having roughly an aspect-ratio of 2 with respectto the T- and L-directions.

Typically, the presence of intermetallic particles, which act asvoid nucleation sites, determines the damage tolerance of a mate-rial. Material inhomogeneities are common fatigue crack nucle-

ation sites in aluminum alloys, cf. [17,18]. Inhomogeneitiesinclude constituent particles and micropores as intrinsic propertiesas well as surface features like scratches. In thin sheet aluminum2024-T351, the rolling of the material has eliminated any micropo-rosity and in the absence of surface scratches, fatigue cracks initi-ate at constituent particles. Constituent particles generally range insize from 1 to 40 lm, see [12], and contribute to the strength of thealloy. During high strain processing of thick plates into sheet mate-rial, many of these particles break and the larger ones are brokeninto clusters of smaller particles. If the material is subjected tofatigue loading, cracks emanating from the constituent particlesgrow into the surrounding aluminum matrix and continue topropagate. Here, these heterogeneous particles in the range of10–20 lm are distributed spatially in clusters through the lengthand thickness of the material. In contrast to thick plates, theseparticles do not vary much in their dimensions, cf. [10–12].

2.2. Results – experiments

Table 2 summarizes the characteristic quantities obtained fromthe uni-axial tensile tests of the material in the following loadingdirections of the sheet: L (longitudinal) and T (transversal). Accord-ingly, the L-specimens show a higher value of yield stress, ultimatestress and total elongation. In all tests, the ductility is fairly similar.The strain at fracture lies between 0.31 and 0.30, where the highervalue has been measured in L-direction. For the related stressquantities, engineering stresses have been calculated, taking theminimum area at the center of the specimen.

Additionally, cyclic deformation experiments at constant strainrange D� were also performed. The complete characterization ofthe material response requires the continuous monitoring of thestress amplitude. During the initial phase of cyclic deformation,(rapid) cyclic hardening occurs, giving rise to an increase in peakstresses rp, followed by an extended regime of cyclic saturationduring which stresses and strains attain steady-state values D�pl,s

and Drs (Fig. 5b). Finally, fatigue damage softening sets in. In thisconnection, two essential stages can usually be distinguished, mes-ocrack initiation and crack propagation. The total fatigue life NR is

10080

6040

200

-400

-200

0

200

400

-0.02-0.01

0.000.01

0.02

Stre

ss [M

Pa]

Strain ΔεCycles

-0.01 0.00 0.01Δ

-600

600

σs

Δε

Stress, [MPa]

Strain, [-]

Δεpl,s

Fig. 5. (a) Cyclic tension–compression response for a specimen subjected to loading with a strain range of 0.035, lasting for 94 cycles. (For the sake of clarity, the cyclesbetween 10–30 and 50–75 have been omitted.) (b) Stabilized hysteresis for LCF experiments (schematically).

Fig. 6. (a) Stress–strain response for uni-axial monotonic tensile tests; (b) peak stresses within the LCF experiments.

40 65 90 115 140

0,005

0,01

0,015

0,02

0,025

Stra

in, [

-]

Number of Cycles

ND

vs Δεpl/2

NR

vs Δε/2

15 40 65 90 115 140

0,005

0,01

0,015

0,02

0,025

Stra

in, [

-]

Number of Cycles

ND

vs Δεpl/2

NR

vs Δε/2

Fig. 7. Coffin-Manson diagrams for the specimens according to Fig. 1 tested in: (a) L-direction; (b) T-direction (model parameters according to Table 3).

Table 3Coffin-Manson parameters for Al 2024-T351, see Eq. (2).

Orientation ��f (–) c (–) Strain range (%) Cycles (NR)

L 0.09838 �0.55 2.5–3.75 48–197T 0.06421 �0.34127 2.375–4.5 21–145

S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122 115

the sum of the number of cycles necessary to initiate damage ND

and the number of cycles necessary to propagate the crack until fi-nal failure NE (see Fig. 6), i.e.

NR ¼ ND þ NE: ð1Þ

Various semi-empirical models have been proposed for defininga relationship between the lifetime and certain other variables forlow cycle fatigue, cf. [19–21]. The Coffin-Manson plot, see Fig. 7, isthe most commonly used approach, which relates the total strainamplitude, D�/2, to the fatigue life NR and reads

D�=2 ¼ ��f ðNRÞc; ð2Þ

where ��f and c are the fatigue coefficient and exponent, respec-tively. The Coffin-Manson parameters, as could be identified as

mean values from the respective experiments, are given inTable 3 for the L- and the T-direction. Table 3 shows the interrela-tionships between the applied strain ranges and the correspondingfatigue lifes. Analogously to the standard Coffin-Manson plot, themoment of damage initiation, measured by the number of cyclesND, can be plotted in a similar manner. However, as will be ex-

Fig. 8. A 60� axial view of the fracture surface; failure after 133 cycles in L-direction, D� = 0.0275. The lower small circle indicates the crack initiation site.

Fig. 9. 3D rendered micro-structure of the fracture surface; the left lower circleindicates the crack initiation site; failure after 133 cycles in L-direction,D� = 0.0275.

116 S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122

plained later, damage initiation is assumed to be governed by theaccumulated plastic strain. Hence, ND is plotted as a function ofD�pl, see Fig. 7a and b.

An important observation is that this alloy/structure exhibits asudden loss of strength. In the monotonic tests upon reaching theultimate stress (Fig. 6a), minute softening is observed before finalrupture. Specimens under cyclic loading fail in a similar abruptmanner as well (Fig. 6b). Since in ductile materials, the softeningphase is longer, the aforementioned observation is an indicatorfor the brittleness of the material suggesting a rather quasi-brittlematerial response at the micro-level, see [10].

3. Fractography

After LCF failure of a specimen, the fracture surface morphologywas examined with a scanning electron microscope, JEOL JSM-6460LV, operated at an accelerating voltage of 20 kV. The visualiza-tions were made at different magnifications to identify the uniquefractographic features, showing the immediate vicinity of the crackinitiation site and discriminating the regions of stable and unstablecrack extension.

3.1. Uni-axial tension tests

Cracks initiate and propagate in regions where deformation anddamage are localized leading eventually either to a flat-faced (frac-ture surface perpendicular to loading) or slant (shear) fracture,depending on constraints and material. For the considered flatspecimens of Al 2024 under monotonic loading, shear rupture oc-curs. A dominant macrocrack propagates with relatively low neck-ing through the specimen’s thickness. The profile of the fracturesurface is of so-called slant-shear fracture-type (across the thick-ness). This kind of failure is typical for thin sheets and small-diam-eter rods, cf. [22]. The term slant-shear fracture is somewhatmisleading, because the angle between the principle axis and thefracture surface was measured on average to be 38�.

3.2. Cyclic tests

Typically, fatigue fracture surfaces of aluminum alloy 2024 havea chaotic wavy appearance, cf. [11]. Furthermore, the fracture sur-face is composed of relatively shiny smooth areas with periodicmarkings called tire tracks, because they often resemble the tracksleft by a tire. For the sake of clarity, the fracture surface is dividedinto two separate regions: that produced by stable growth and thatcorresponding to unstable crack growth (Fig. 8). The elliptical frontseparating such regions grows from the surface across the speci-

Fig. 10. (a) Transition path dividing stable (below) and unstable (above) crack growth rcycles in L-direction, D � = 0.0275.

men’s thickness. The damage initiation point (discussed in detailin a later section) is close to the center of the ellipse.

A 3D rendering of the images obtained from the SEM shows theinitiation point (Fig. 9). The stable crack growth region has adimple structure with smaller cavities, characteristic for ductiledamage in aluminum alloys. The stable crack grows nearly perpen-dicular to the loading direction and produces a flat profile. Theunstable region has a dimple structure with bigger cavities(Fig. 10a) which are formed during unloading/compression.Fatigue lines can be observed only in the stable crack growth re-

egions, shown in white; (b) crack initiation site with fatigue lines; failure after 133

S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122 117

gion (Fig. 10b) with small flat patches, also produced in the com-pression phase during unloading. The related unstable crackgrowth region is slanted like that one observed in monotonic load-ing for the considered flat specimens. The sizes of the unstable andstable crack growth regions vary with loading conditions. For avery high amplitude, restricting the lifetime to a few cycles, thestable crack growth region is smaller as compared to a loweramplitude with a larger number of cycles. There is no observedconnection between stable/unstable crack growth regions andtrans/intergranular crack propagation. Failure occurs always in amixed mode without any generalized trend.

3.2.1. Damage initiationThe nucleation of fatigue cracks represents an important stage

in the damage evolution process in cyclically loaded materials. Inhomogeneous materials without appreciable macroscopic defects,the surface of the material plays a prominent role in fatigue cracknucleation. As a matter of fact, the majority of fatigue cracks initi-ate at the surface, cf. [23]. If macroscopic and microscopic defects(e.g. inclusions, holes and non-coherent precipitates) are present,the interface between the defects and the matrix is a potential siteof crack nucleation.

Fig. 11 shows the profile analysis of a fracture surface. Largecavities were found at the surface of the broken specimen and acavity produced by a particle-up-rooting can be seen close to the

0 20 40 60 80 100 120 140 160 180 -30

-20

-10

0

10

Dep

th[

m]

Path length [ m]μ

μ

Fig. 11. (above) SEM fractograph of the failure surface after 133 cycles in L-direction, D� = 0.0275. The line defines the analyzed path. (below) Profile analysisassociated with the above path showing two big cavities at the surface.

initiation site of the fatigue crack. Area analysis (Fig. 11) revealedbig particles (see SubSection 2.1) near the surface. As expected,the lifetime is roughly inversely proportional to particle sizing. Thisimplies, the bigger the particle, the sooner damage initiates and theless resistant the material.

4. A ductile–brittle damage model

The experimentally observed quasi-brittle material response offlat sheets of Al 2024 can be naturally captured by a coupledductile–brittle damage model. Such a material model was recentlyproposed in [10,24]. For getting further insight into the mechanicalresponse of Al 2024-T351 accompanying finite element simula-tions based on this constitutive model are performed. Asintroduction, the thermodynamically consistent framework pro-posed in [10,24] will be briefly described in what follows. Through-out this section, a geometrically linear setting is used where theglobal strains � are decomposed according to � = �e + �p with �e

and �p being its elastic and plastic parts. As far as the notation isconcerned, the superscript ‘‘p’’ signals variables associated withplasticity, while the superscript ‘‘b’’ represents variables corre-sponding to quasi-brittle damage accumulation.

4.1. Stored energy

The model advocated in [10,24] is based on a combination ofthe by now classical approach advocated by Lemaitre and Desm-orat [13] and a novel material law for quasi-brittle damage accu-mulation. Assuming an additive decomposition of the Helmholtzenergy W into the different damage modes, W reads

W ¼ ð1� DÞ Y þ HCa2

C

2ð3Þ

with

Y :¼ �e : C : �e

2þ Hk

kakk2

2þ Hi

a2i

2

!: ð4Þ

Here, D 2 [0;Dcrit] is a scalar-valued damage parameter, Y is theHelmholtz energy of the un-damaged solid, C is the fourth-orderelasticity tensor, Hk is the kinematic hardening modulus, Hi is theisotropic hardening modulus, ak is the strain-like internal variableassociated with kinematic hardening, its counterpart for isotropichardening is denoted as ai, HC is the hardening modulus corre-sponding to quasi-brittle damage accumulation and aC is therespective internal variable. Assuming an additive decompositionof the driving force governing damage accumulation is equivalentto the split

D ¼ cp Dp þ cb Db; with cb :¼ 1� cp; ð5Þ

where cp and cb are the corresponding composition factors. Withassumption (5), the driving force energetically conjugate to D is gi-ven by Y = �@DW = Yp + Yb with

Yp ¼ �@DpW ¼ cp Y and Yb ¼ �@DbW ¼ cb Y: ð6Þ

4.2. Space of admissible stress states

In line with classical plasticity theory, the space of admissiblestates is defined by a yield function.

4.2.1. Plastic deformationConcerning plastic deformation, the von Mises-type yield

function

118 S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122

/p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

devð~r� eQ kÞ : devð~r� eQ kÞr

� ðeQ i þ Q eq0 Þ 6 0 ð7Þ

is adopted where Qeq0 is the initial yield limit. This yield function is

formulated in terms of effective stress-like variables indicated bythe tilde sign. Introducing the stresses r and the stress-like internalvariables conjugate to �p, ak, ai in standard manner as

r ¼ @�W ¼ �@�p w; Q k ¼ �@akW; Q i ¼ �@ai

W; ð8Þ

their respective effective counterparts are given by

~r ¼ rð1� DÞ ;

eQ k ¼Q k

ð1� DÞ ;eQ i ¼

Q i

ð1� DÞ : ð9Þ

It bears emphasis that within the original model by Lemaitre andDesmorat [13] only the stresses are used as effective quantities.

According to the material characterization given in the previ-ous sections, Al2024 shows an anisotropic mechanical responsein general. However, in the present paper focus is on the duc-tile-to-brittle transition of material failure for uniaxial loadingstates. In this case, the isotropic von Mises yield function (7)represents an admissible choice. Certainly, different sets of mate-rial parameters have to be used for the different loadingdirections.

4.2.2. Quasi-brittle damage accumulationAnalogously to the plastic deformation, quasi-brittle damage

accumulation is also governed by a yield-type function. In line withEq. (7), this function is assumed as

/b ¼ jðYbÞN � Cj

S2� Q b0 6 0: ð10Þ

According to Eq. (10) this function is formulated in terms of energy-like variables such as the energy release rate Yb. C is a so-called shifttensor, which defines an alternate threshold, while Qb0 represents aconstant threshold, both defining damage activation. N and S2 arematerial parameters.

4.3. Evolution equations

In case of fully elastic states, the stresses can be directly com-puted by using Eq. (8)1. Otherwise, the updated plastic deforma-tion and the updated damage accumulation have to bedetermined first. They are based on the evolution equations dis-cussed next.

4.3.1. Plastic deformationIf plastic loading is signaled by /p = 0 and _/p ¼ 0, where the

superposed dot denotes the material time derivative, the evolutionequations of the internal variables associated with ductile defor-mation are derived from the convex plastic potential

�/p ¼ /p þ Bk

Hk

eQ k : eQ k

2þ Bi

Hi

eQ 2i

2þ ðYpÞM

MS1ð1� DÞ : ð11Þ

The additional quadratic terms, depending on the stress-likeinternal variables eQ k and eQ i, are associated with non-linear kine-matic and isotropic hardening of Armstrong-Frederick-type, whilethe last term corresponds to the evolution law of the damage-re-lated variable Dp. In Eq. (11), Bk, Bi, M and S1 are material param-eters. With Eq. (11) and following the framework of generalizedstandard materials (see [25]), the evolution equations are postu-lated to be

_�p ¼ kp @r�/p; _ak ¼ kp @Q k

�/p; _ai ¼ kp @Qi�/p ð12Þ

and

_Dp ¼ kp @Yp �/p: ð13Þ

Here, kp P 0 is the plastic multiplier. As shown in [10,24], convexityof the plastic potential �/p guarantees a positive dissipation in caseof plastic loading. Thus, the second law of thermodynamics is auto-matically fulfilled. The crack closure effect is accounted for in caseof compression by reducing the damage evolution (13) by a factorof 0.2, cf. [25].

4.3.2. Quasi-brittle damage accumulationConsidering again the framework of generalized standard mate-

rials, cf. [25], the evolution equations are derived from the convexdamage potential

�/b :¼ /b þ BC

HC

C2

2ð14Þ

where BC is an additional material parameter. Using Eq. (14), theevolution of the quasi-brittle damage related internal variablesare assumed to be of the type

_Db ¼ kb @�/b

@Yb¼ kb signððYbÞN � CÞ

S2N ðYbÞN�1

; _aC ¼ kb @�/b

@Cð15Þ

if loading is signaled by /b = 0 and _/b ¼ 0. In Eq. (15), kb is the dam-age multiplier. For guaranteeing a positive damage evolution, dam-age growth is considered for a half-cycle only, i.e., in case of apositive sign of (Yb)N � C.

4.4. Damage initiation and the formation of a mesocrack

While plastic deformation occur, whenever the loading condi-tions /p = 0 and _/p ¼ 0 are fulfilled, damage initiation is governedby an additional criterion. Following [13], damage is assumed toinitiate when the energy associated with cold plastic work ws

reaches the threshold wD, cf. [13,26]. This assumption complieswell with the empirical Coffin-Manson rule in which the fatiguelifetime depends on the amplitude of the (plastic) strains. Themodified energy related to cold plastic work ws proposed in[13,26] reads

ws ¼Z tnþ1

0R1ð1� e�brÞ A

mr

1�mm _r

� �dt þ Hk

2ðak : akÞ ð16Þ

where A and m are material parameters, r is the accumulated plasticstrain and R1 and b are related to isotropic hardening, cf. [10].Based on the driving force ws (strictly speaking, a driving energy),damage initiates, if

ws > wD: ð17Þ

The threshold wD is a material parameter.Once criterion (17) is met, damage evolution is governed by the

yield functions (7) and (10) as well as by the evolution Eqs. (13)and (15)1. Physically speaking, damage initiation corresponds tothe formation of microcracks. These microcracks propagate andcoalesce leading eventually to a mesocrack. Analogously to [13],it is assumed that such a mesocrack forms, if the damage variableD reaches the threshold Dcrit. Within the numerical simulations,Dcrit has been set to Dcrit = 0.23.

4.5. A variationally consistent reformulation

Since the evolution equations of the constitutive model de-scribed in the present section fall into the framework of general-ized standard materials (see [25]), the resulting overall model isthermodynamically consistent and fulfills the second law of ther-modynamics. Although the underlying evolution equations arenon-associative ð�/p –/p; �/b –/bÞ it was recently shown in [24]that the model possesses even a variational structure. Withoutgoing too much into detail, this structure allows to compute all

S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122 119

state variables jointly and conveniently from minimizing an incre-mentally defined energy E which turns out to be the stress power,i.e.

_X ¼ arg inf_XE; with X ¼ f�p;ak;ai;aC;Dg ð18Þ

and

r ¼ @ _� inf_XE: ð19Þ

Based on this variational structure, efficient finite element imple-mentations can be derived, cf. [24]. Such an implementation hasalso been employed for the numerical analyses presented in the fol-lowing sections. In particular, the tangent stiffness matrix resultingfrom the variational constitutive update is always symmetric. Thatis in sharp contrast to conventional implementations such as thoserelying on the return-mapping scheme.

5. Material parameter identification strategy

The identification of the model parameters is accomplished infive tasks, following a step-by-step procedure. This method is usedfor the T- as well as for the L-direction. In the following, the stepsdefining the identification strategy are briefly described.

1. The elastic material parameters and the yield limit (E,m,ry) areinferred from the monotonic tensile curves. Concerning theevolution of quasi-brittle damage, the admissible assumptionsN = 1.25 and S2 = N are made, cf. [10]. Further following [10],Qb0 can be computed from Qb0 = [cb(rf)2/2E]N/N. Taking an aver-age value of rf = 62.5 MPa, Qb0 is computed as 0.0009122838(MJ/m3)N (cb = 1,E = 70,000 MPa as average value was takenfor both T- and L-directions, N = 1.25). Further details are omit-ted here. They can be found in [10].

2. The isotropic hardening (Hi,Bi) and non-linear kinematic hard-ening parameters (Hk,Bk) are calibrated considering the experi-mental data for a strain amplitude of D� = 0.0375 (beforedamage initiation). For that purpose, the respective leastsquares problem is solved.

3. For determining the set of parameters A, m and wD defining theplastic stored energy (16) driving damage initiation, the objec-tive function

Table 4Calibrated material parameters for Al 2024-T351 sheets, 4 mm thickness.

Orient. E (MPa) m (–)

ElasticityL 71,000 0.3T 70,000 0.3

Orient. Qeq0 (MPa) Hi (MPa) Bi (–) Hk [MPa] Bk (–)

PlasticityL 345 501.252 5.01252 3333.33 90.0T 280 1050.766 11.0606 16666.66 265.0

Orient. m (–) A (–) wD (MJ/m3)

Stored plastic energyL 4.964 9.7e�03 0.7247T 2.479 10.6e�03 1.41

M (–) S1 (–) HC (–) BC (MPa) cb (–) Dcrit (–)

Damage evolution1.0511 2.12 0.0515 200.5 0.45 0.230.9475 2.07 0.115 43.39 0.45 0.23

A;m;wDf g ¼ arg minA;m;wDf g

Xn

i¼1

absðlog NexpD;i � log Nsim

D;i Þ !

ð20Þ

is minimized, where NexpD;i are the experimentally observed cycles

and NsimD;i are the numerically simulated counterparts at the onset

of damage initiation, respectively. n is the number of experi-ments with different strain amplitudes.

4. The material parameters governing ductile damage (M and S1)are found by comparing the numerically predicted lifetimes totheir experimentally observed counterparts. Within the respec-tive optimization problem

M; S1f g ¼ arg minfM;S1g

Xn

i¼1

absðlog NsimR;i � log N exp

R;i Þ !

; ð21Þ

quasi-brittle material degradation is ignored (cb = 0). As a conse-quence, step 1 up to step 4 can also be used for calibrating the bynow classical model by Lemaitre and Desmorat [13].

5. Finally, the composition factor cb as well as the parameters BC

and HC defining the quasi-brittle damage model are computedfrom the optimization problem

BC;HC; cb� �

¼ arg minBC ;HC ;cbf g

Xn

i¼1

absðlog N simR;i � log Nexp

R;i Þ !

:

ð22Þ

The material parameters obtained from the described materialparameter identification strategy are summarized in Table 4.

6. Numerical analysis of LCF in Al 2024-T351

Based on the calibration of the material parameters, the predic-tive capabilities of the constitutive model suitable for the analysisof sheets of high-strength aluminum alloys are demonstrated inthe present section. While SubSection 6.1 is associated with the flatspecimens experimentally analyzed in Section 2, a more complexengineering structure is considered in SubSection 6.2. As a proto-type of such a structure, a stringer-skin connection of a fuselageis chosen.

6.1. Flat specimens

The lifetimes of the flat specimens experimentally analyzed inSection 2 are computed here by using the ductile–brittle damagemodel discussed in the previous section. For the respective finiteelement simulations, a three-dimensional model representing aquarter of the specimen’s gauge section is considered imposing atwofold symmetry condition (X1 and X2). Despite the fact thatthe specimen’s thickness is comparably small, the simulationsare done using a three-dimensional discretization. A coarse meshis used with mesh refinement at the expected regions of stress con-centration which allows a good simulation of necking and con-straints caused by damage localization. Tri-linear finite elements(3D bricks with 8 nodes) are used in the discretization. The trian-gulation consists of eight elements across the thickness. A typicaldimension of a finite element is 0.2 mm in longitudinal and trans-versal and 0.5 mm in thickness-direction. The specimens areloaded under displacement control with a prescribed linear ampli-tude function for cyclic loading (Fig. 12).

The 3D specimens are meshed with a geometrical imperfection.A slight hourglass shape is used for localizing damage at the mid-dle-plane. Clearly, mesh size and design are critical for damagecalculations with softening, cf. [6,27,28]. As mesh-defining quan-tity, the total fracture energy is related to the damage process inthe respective zone, cf. [27,29]. Hence, the mesh size can be inter-

u

X3

X2

X1

l

uΔ

Δ Δ

10 mm

Time

8m

m

4 mm

Fig. 12. Sketch of the specimen and the loading conditions used in the finiteelement simulations.

Table 5The lifetime prediction (CDM) compared to experiments in the LCF regime (for both L-and T-directions). Nexp

E is the number of loading cycles during the softening regime.

Orientation De (%) Number of cycles

Experiment Simulation – NsimE

NexpE

Ductile Coupled

L 3.75 4 12.0 10.03.5 8 11.26 9.623.0 8 13.59 10.52.75 10 17.755 11.0

T 3.25 5 7.0 6.03.0 6 17.22 16.02.5 7 23.405 20.452.375 9 25.82 21.4875

120 S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122

preted as some micro-structural characteristic length scale (e.g.inter-particle spacing or grain size).

Within the computations, the calibrated material parameterssummarized in Table 4 are used. The numerically predicted life-times are given in Table 5. For the sake of comparison, a purelyductile damage model in the spirit of [13] is also considered. Sinceboth models are equivalent before damage accumulation, only thenumbers of cycles within the softening regime ðN sim

E Þ are given.According to Table 5, a purely ductile damage model leads to a sig-nificant overestimation of the lifetime. By way of contrast, the fullycoupled ductile–brittle approach captures the underlying physical

0,026 0,028 0,030 0,032 0,034 0,036 0,03840

60

80

100

120

140

160

Num

ber

of c

ycle

s

Δε

ExperimentsCoupled ductile-brittle damageDuctile damage

Fig. 13. Lifetime prediction for the flat specimen (L-direction), see Fig. 12.

processes in a more realistic manner and thus, leads to betteragreements with the experiments. A graphical illustration of thenumerical results for the L-direction, together with the experimen-tally observed counterparts, is shown in Fig. 13. As evident, thecoupled model yields more realistic lifetimes.

6.2. Stringer-skin connection of a fuselage

The whole visible part of an airplane is constituted of stiffeningpanels, i.e. an outer thin sheet (skin) with generally orthogonallyarranged integral or fastened stiffeners. The stringers and framesin a fuselage shell are sketched in Fig. 14. The fastened stiffenerscan be riveted, adhesive bonded or welded to the skin [31]. Clearly,the design of such structural components requires the estimationof their lifetimes. In the past, different, mostly ad hoc, conceptshave been considered for that purpose. Since it was experimentallyobserved that the stringer failure is driven by a low-cycle fatiguemechanism [32], the lifetime can be naturally estimated by usingthe constitutive model described in the previous sections.

Although the stringer is bonded to the skin, the whole assemblyis considered as homogeneous and no special contact conditionshave been defined within the respective finite element simula-tions. Despite the fact that a bonded stringer-skin assembly per-forms better with regard to damage tolerance, the purpose ofthis simulation was not to evaluate/judge the bonding strength.In industrial practice and at the laboratory testing stage onlythrough-the-thickness cracks are used, i.e., the crack front is al-ways considered straight. To model such a situation, an ellipticalthrough-thickness notch is taken here as pre-crack ahead of thestringer, see Fig. 15. The assumption of such a blunt notch reducingstress singularities is very practical from a modeling point of view.Otherwise damage would already evolve from the beginning.Furthermore, it is important to note that the numerical analyseswere not intended to obtain the crack propagation profile. Moreexplicitly, focus was on damage initiation (propagation of micro-cracks).

The structure shown in Fig. 15 has been cyclically loaded dis-placement controlled under symmetric load condition (R = �1).A very fine mesh especially around the elliptical notch has beengenerated. With one exception, the material parameters given in

Fig. 14. Fuselage stiffened panel with a longitudinal skin crack over a broken frame[30].

u = 0.5 mm

ws0.00+8. -15e 01+1.63e+00

Δ

Δ

u = 0.5 mm

Fig. 15. Distribution of the stored plastic energy ws driving damage initiation.Lower left hand side: crack front at the elliptical notch (top view); lower right handside: profile of plastic stored energy (side view). The relative position of theanalyzed stringer-skin connection within the fuselage section is marked by thecircle in Fig. 14.

S. Khan et al. / International Journal of Fatigue 37 (2012) 112–122 121

Table 4 have been used for the numerical analysis. This exceptionis the plastic energy threshold (wD) which has been altered to avalue of 1.62. This value has been taken from the uni-axial simula-tions (L-direction).

The results of the finite element simulation are shown in Fig. 15.The coupled damage model predicts microcrack initiation in theassembly after 60 load cycles. Subsequently, damage evolves fastto the critical value.

Without doubt, crack propagation is also important for estimat-ing the lifetime of structural components. For that purpose, how-ever, the constitutive model has to be modified. Such anextension is beyond the scope of the present paper.

7. Conclusions

In the present paper, the low-cycle fatigue behavior of flatsheets of aluminum Al 2024-T351 was carefully analyzed. For thatpurpose, material characterization was combined with materialmodeling. For allowing symmetric strain amplitudes (R = �1) with-in the cyclic loading experiments, a floating anti-buckling guidehas been proposed which successfully prevents buckling at highcompressive loading without affecting adversely the mechanicalbehavior of the specimen. Based on the subsequent fractographyit was shown that Al 2024 as a sheet exhibits a mixed-mode frac-ture behavior ranging between brittle and ductile. Furthermore,damage initiates at bigger particles/inclusions at the surface ofthe specimen and propagates with a spherical crack front. A tran-sition zone between the fracture mechanism was also observed(stable/unstable crack growth regions). Additionally, experimentalmacroscopic results confirmed a rapid damage evolution up to fail-ure. All these observations suggested that a ductile damage modelalone is not sufficient for predicting the lifetime of this alloy accu-rately enough.

For capturing the complex ductile–brittle damage accumulationof aluminum Al 2024-T351 sheets, a recently published coupled

approach was considered. Within this approach, a modified modeloriginally advocated by Lemaitre [13] suitable for the analysis ofductile damage was combined with a model for quasi-brittle dam-age evolution. The parameters of that model were optimized bymeans of a staggered identification strategy. Comparisons betweenthe predictions computed by the final model and experiments forflat specimens showed a very good agreement. In particular, theestimated lifetimes are more realistic than those based on a purelyductile damage model. That confirmed the quasi-brittle nature ofdamage accumulation in flat sheets of Al 2024-T351.

As an outlook and for demonstrating the robustness and effi-ciency of the finite element model, a complex engineering problemwas also analyzed. More precisely, damage initiation in a stringer-skin connection of a fuselage was considered. While in the past,different, mostly ad hoc, concepts have been considered for design-ing such structures to reach a desired lifetime, the combination ofcareful material characterization and physically sound modeling asdiscussed in the present paper allows to put this design process onmore solid grounds.

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