International Journal of Fatigue 36 (2012) 155–162

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International Journal of Fatigue

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

A cohesive model for fatigue failure in complex stress-states

Deepak Jha, Anuradha Banerjee ⇑Department of Applied Mechanics, Indian Institute of Technology, Madras Chennai-36, India

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

Article history:Received 15 February 2011Received in revised form 19 July 2011Accepted 28 July 2011Available online 4 August 2011

Keywords:CZMMulti-axial fatigueBiaxialityCohesive zoneStress state

0142-1123/$ - see front matter � 2011 Published bydoi:10.1016/j.ijfatigue.2011.07.015

⇑ Corresponding author.E-mail addresses: dkjha143@gmail.com (D.

(A. Banerjee).

A stress-state dependent traction–separation law is combined with an irreversible damage parameter toformulate a cohesive model relevant for life assessment under complex stress-states. Evolution of dam-age is based on continuum damage laws requiring two cohesive fatigue parameters. For two representa-tive metals, model is able to reproduce typical stress-life response, mean stress effects and sequencingeffects under variable amplitude loads. Control of cohesive fatigue parameters on characteristics of thestress-life curve is established, followed by predictions for proportional, cyclic stress-states. Withincreasing constraint, initiation and growth of damage is more rapid such that higher constraint resultsin lower life expectancy.

� 2011 Published by Elsevier Ltd.

1. Introduction

Micro-structural damage due to repetitive sub-critical loadsleading to fatigue failure is a key issue in design and assessmentof structural integrity of components such as rotating parts ofautomobiles, fuselage and other components of aircrafts, compres-sors, pumps, and turbines. Classical approaches to design againstfatigue failure of a specific material involve characterisation of to-tal fatigue life to failure in terms of cyclic stress range (S–N curve).For defect tolerant design of the same material, the region of stea-dy crack growth under cyclic load of an initially cracked specimenas shown in Fig. 1 is represented by the Paris law [1]. Both, stress-life approach and Paris law, are empirical in nature and thoughthey describe the fatigue behavior of the same material, they usean independent set of parameters. As a result, the prediction of fa-tigue life at different stress states requires determination of a largeset of parameters that can be used for prediction only for a limitedrange of conditions. The primary difference between the conditionsof failure in a material without any nominal defect and a materialwith a macroscopic defect is the comparatively high triaxiality ofthe state of stress ahead of the crack tip. Therefore, it is of signifi-cance to understand the similarities and differences between theprogressive degradation of the material in these different stressstates due to cyclic loads. In the proposed work, a model in theframework of cohesive zone concept that is able to incorporatethe dominant role of stress-states in prediction of failure due to fa-tigue damage is presented.

Elsevier Ltd.

Jha), anuban@iitm.ac.in

Initially, owing to the involved complexities and expenses incontrolled multi-axial fatigue experiments, it was usual to dependon uniaxial fatigue test for life predictions. Subsequently, fatiguefailure in complex stress states has been the focus of several inves-tigations and numerous models have been compared and reviewed[2–4]. To the best of authors’ knowledge none of these models areuniversally accepted and applicable for simulations of crack prop-agation. Of the different possible combinations of multi-axialstress-state generated by conditions of bending–torsion, torsion–tension, tension–compression, etc. the case of tension–tension isthe most critical one as it belongs to the first quadrant of the prin-cipal stress space. However, carrying out controlled experimentswith fixed biaxiality of the applied stress has been a challengeand very little data is available, specially for plane strain conditions[5,6].

In simulation of the non-linear damage processes prior to frac-ture, owing to its simplicity and few parameters, cohesive zonemodel finds wide acceptance in numerous applications as re-viewed in [7]. The model assumes that the damage mechanismsdue to excessive plastic deformation or micro-cracking that leadto failure are usually localized in a thin layer or process zone[8,9]. The model describes the constitutive behavior of this thinlayer as a traction–separation law (TSL) in which the cohesive trac-tion, Tn, is a defined function of the separation between the zoneboundaries, dn, as:

Tn ¼ rmaxf ðdnÞ; ð1Þ

where rmax is the peak value of traction. The function f(dn) definesthe shape of the traction–separation law and the area under thecohesive law curve is the work of separation or cohesive energy, C.

Fatigue test specimen

Plastic zone

Fatigue crack growth specimen

Fig. 1. Schematic diagram of the various stages in fatigue failure of a componentand the specimens used to characterize the failure processes [1].

156 D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162

In prediction of fatigue crack growth, as an alternative to usingParis law which has limited applicability as a predictive tool, irre-versible cohesive laws have been proposed and applied to crackgrowth in different materials including metals and polymers [10–13]. Using an exponential traction separation law for the mono-tonic behavior and by introducing an irreversible damage parame-ter that evolves and updates the traction–separation law, thecohesive model has been able to incorporate the progressive dam-age of an interface due to sub-critical repetitive fatigue loads[14,15]. By being able to capture most of the significant featuresof the fatigue failure data in uniaxial loading as well as the crackgrowth curves, the model establishes a connection between thefailure processes between significantly different stress states. Morerecent studies have focussed on incorporating creep-fatigue inter-action in a single crystal super alloy and retardation effects of over-loads on fatigue crack growth [16,17]. However, in these studiesthe cohesive parameters, cohesive strength and cohesive energy,have been assumed to be constant and therefore the effect ofstress-state on the cohesive law has not been accounted for.

In metals, the highly stressed region ahead of a crack tip hasdamage mechanisms that typically involve processes which nucle-ate micro-cracks or micro-voids, leading to failure by cleavage, in-ter-granular cracking or micro-void coalescence. The stress-stateplays a crucial role in the damage growth till failure. Correspondingmechanisms under cyclic loads involve the repetitive blunting andresharpening of crack tip involving localized plastic deformationand therefore must depend on the stress-state as well. For astress-state with significant hydrostatic component, it has beenestablished that the cohesive law is narrower and steeper. Thiscentral role of the stress-state on the failure process has been ac-counted for in the cohesive model, but only with relevance tomonotonic fracture [18,19].

Can the stress-state dependent model for monotonic fracture bedeveloped further to be applicable for fatigue failure investigationsas well? In the present work, a cohesive model for plane strain fa-tigue failure that is sensitive to the state of stress is presented inprediction of typical fatigue response of representative structuralmetals, ferritic steel and aluminum alloy. The overall mechanicalbehavior and the corresponding cohesive properties for monotonicstress-state dependence of the traction–separation law are takenfrom [19]. Two additional parameters are included in the growthof a defined irreversible cyclic damage parameter. The resultingevolution of the stress-state dependent cohesive law is able to ac-

count for irreversible damage due to sub-critical cyclic loads. In-sight into the basis of the model is presented followed by itspredictions of fatigue failure under basic uniaxial, fully reversed,cyclic load. Other typical features, such as mean stress effect andsequencing effects in variable amplitude loading. are also reported.A detailed discussion is developed on the role of constraint, as perthe model, on the fatigue failure curves and its effectiveness inreproducing the trends observed in some of the experimental liter-ature [6].

2. Stress-state dependent cohesive model for fatigue

Monotonic ductile fracture mechanism involves nucleation,growth and coalescence of micro-voids at the second phase parti-cles. The stress-state characterized by a triaxiality parameter, H,defined as the ratio between the hydrostatic stress and effectiveor mises stress, plays a key role in the fracture process. Corre-sponding mechanisms in the fatigue process are also driven bylocalized plastic deformation and there is experimental evidencethat the stress-state has a significant effect on fatigue as well [6].The model presented in this study is applicable for ductile metalsin which the fracture process is localized in a thin layer whichhas thickness of a void spacing, D. The parent material is taken tobe power-law strain hardening such that the uniaxial stress–straincurve is represented by:

� ¼ rE

r 6 ry; ð2Þ

¼ ry

Erry

� �1n

r > ry; ð3Þ

where E is the elastic modulus, ry is the initial yield stress and n isthe strain hardening exponent. In the cohesive zone framework, theeffect of stress state has been incorporated to be able to simulatefracture in a range of geometries which develop widely differentstress states [19]. For the present study the focus is retained onstress based applied load and thus only the traction–separationlaw up to the peak stress is used to represent the undamaged con-stitutive behavior of a process zone of a void spacing thickness, D:

Tn;o ¼ 1þffiffiffi3p

Heff

� �2E3

d̂n 0 < d̂n 6 d̂n1; ð4Þ

¼ 1þffiffiffi3p

Heff

� � ryffiffiffi3p 2Effiffiffi

3p

ry

d̂n

!n

d̂n1 6 d̂n 6 d̂n2: ð5Þ

where d̂n is the opening displacement normalized with D. The de-tails of the concept of an effective triaxiality parameter, Heff, andother model parameters, C and S, are presented in the Appendix.The linear behavior of the cohesive zone exists till the separationlimit d̂n1 ¼

ffiffiffi3p

ry=ð2EÞ is reached. Further separation results instrain-hardening up to d̂n2.

d̂n2 ¼ffiffiffi3p

2Ce�1:5Heff þ ry

E

� �ð6Þ

represents the triaxiality dependent shape parameter indicating on-set of rapid material deterioration leading to failure. The cohesivestrength of the undamaged process zone, rmax,o, is the traction cor-responding to d̂n2. As observed in [19] it strongly depends on thestress-state and therefore cannot be taken to be a material constant.For the remaining manuscript dn2 is denoted by do.

Two representative ductile metals used in the present study toillustrate the model behavior are ferritic steel and aluminum alloyas in [15]. The mechanical properties and the model parameterswhich are used to describe the stress-state dependent traction sep-aration law in monotonic conditions are tabulated in Table 1. Thecorresponding constitutive response of the cohesive layer without

Table 1Mechanical and cohesive failure properties of the representative metals.

Property Symbol Steel Aluminum

Elastic modulus E 210 GPa 70 GPaElastic limit ry 420 MPa 217 MPaPoisson’s ratio m 0.3 0.3Hardening exponent n 0.1 0.0565Saturation cohesive parameter S 20 5Failure locus parameter C 1 1

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

Tn

/ σy

δn / D

Steel, α = 0.2Steel, α = 0.8

Aluminum, α = 0.2Aluminum, α = 0.8

Fig. 2. Stress-state dependent cohesive law of steel and aluminum alloy.

-1

-0.5

0

0.5

1

1.5

0 0.01 0.02 0.03 0.04 0.05 0.06

Tn,o / σmaxo

Tn / σmaxo

σmax / σmaxo

δn / D

Fig. 3. Evolution of the traction–separation behavior with fatigue damage incor-porated in the cohesive law.

D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162 157

any fatigue damage is shown in Fig. 2 for the extreme cases of biax-iality ratio, a = 0.2 and 0.8. As pointed out in [15], for higher biax-iality ratio due to constrained plasticity, the traction–separationresponse has higher peak stress and lower work of separation forboth the materials.

To incorporate the damage due to fatigue, a cyclic damage var-iable is introduced that updates the constitutive behavior of theprocess zone such that the current traction is given by:

Tn ¼ Tn;oð1� DcÞ: ð7Þ

The evolution of damage is taken to obey the continuum dam-age evolution laws: (i) fatigue damage accumulation begins if anaccumulated deformation measure exceeds a critical threshold;(ii) increment of the damage is related to the increment of defor-mation weighted by the current load level; and (iii) there existsan endurance or fatigue limit which is a stress level below whichcycles can proceed indefinitely without failure. Here it is assumedthat macroscopically the separation of the material occurs in theplane of maximum principal stress, therefore the contribution ofshear traction on damage is ignored in the proposed model. Thecurrent state of damage is described by the evolution equation asproposed in [15]:

_Dc ¼j _dnjdR

Tn

rmax� rF

rmax;o

� �H

Dn

do� 1

� �and ð8Þ

_Dc P 0: ð9Þ

Here, two additional fatigue parameters are introduced, cohe-sive endurance limit, rF, and an accumulated cohesive length, dR.The accumulated opening displacement, Dn ¼

Rj _dnjdt, is the accu-

mulated deformation measure that is used to describe the onsetof fatigue damage accumulation through the Heaviside functionin the evolution equation. rmax = rmax,o(1 � Dc) is the current max-imum stress that can be supported by the process zone post onsetof damage accumulation. Effectively it represents the currentstrength of the cohesive layer which keeps decreasing withincreasing fatigue damage.

Since the applied load is stress based, failure is taken to occurwhen during a cycle traction exceeds the current peak stress. Be-sides the stress-state dependent traction–separation law, and theevolution equation for irreversible fatigue damage, a descriptionof the cohesive zone behavior under unloading and reloading isprovided by a linear function similar to the linear part of TSL, mod-ified to retain opening displacement continuity and use the currentelastic modulus as:

Tnðt þ DtÞ ¼ TnðtÞ þ 1þffiffiffi3p

Heff

� �2E3ð1� DcÞ d̂nðt þ DtÞ � d̂nðtÞ

� �:

ð10Þ

For a typical cyclic applied stress of constant amplitude, theevolution of the cohesive strength and the constitutive responseof the cohesive layer till final failure is shown in Fig. 3. The pro-posed model’s applicability is for the first quadrant of the principalstresses space, i.e., biaxial tension. Thus, the tangential stresses inthe finite element implementation are expected to be fairly lowand an elastic stiffness can be taken in the tangential direction asin [20] to avoid any numerical instabilities. Initially the cyclicstresses are lower than the initial cohesive strength. With accumu-lation of fatigue damage the cohesive strength progressively de-creases until the applied traction equals the cohesive strength. Asexpected the maximum increments of the separation take placenear the tensile peak of the load cycle and the extent of separationincreases rapidly near failure. It is to be noted that even thoughwith accumulation of damage the slope of the unloading–reloadingcurve decreases, in Fig. 3 the change is not obvious because thefailure occurs due to the applied stress exceeding cohesivestrength. This occurs at a reasonably low value of Dc = 0.2 in theparticular case considered for the figure.

3. Results and discussion

Initially the focus was on finding the stress-state dependent fa-tigue cohesive model’s capabilities in obtaining the characteristicfeatures of basic uniaxial fatigue response, that of an S–N curve.Thecomputational results presented in this section are for a materialpoint in the process zone and not of a structure. For a constant-amplitude, uniaxial fatigue load with the fatigue parameters atrF = 0.25 rmax,o and dR = 4 do, the life till failure is obtained for awide range of sinusoidal applied stress and shown in Fig. 4. Themodel is able to predict the typical features of fatigue failure data:decrease in life, Nf, for increasing stress amplitude, S, and a limitinglevel of stress below which the material is insensitive to the cyclicload. The slope of the linear part of the S–N curve being theBasquin’s exponent, is found to be �0.2 and �0.23 for steel and

1.6

1.8

2

2.2

2.4

2.6

2.8

3

2 2.5 3 3.5 4 4.5 5

S

SteelAluminum

Log10 (Nf)

Fig. 4. Stress-life curves for representative steel and aluminum alloy.

1.5

2

2.5

3

3.5

2 2.5 3 3.5 4 4.5

σmean/σmaxo = 0.0σmean/σmaxo = 0.2σmean/σmaxo = 0.4

Log

10 (S

)

Log10 (Nf)

Fig. 6. Effect of mean stress on stress-life (S–N) curve for aluminum.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

S / S

o

σmean / σmaxo

N = 10

N = 103

N = 105

Fig. 7. Effect of mean stress on the permissible stress amplitude for a constantfatigue life for steel.

158 D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162

aluminum respectively which is with in the acceptable range ob-served in available experimental literature for these metals.

It is known that the mean level of the imposed fatigue cycleplays a crucial role in the fatigue failure behavior of engineeringmaterials. Using the proposed model, the mean stress effect is ob-tained by comparing the stress-life curves of completely reversedcyclic load corresponding to no mean stress with that of for meanstress of 0.2 rmax,o and 0.4 rmax,o as shown in Figs. 5 and 6. As themean stress is increased, for the same applied amplitude the lifedecreases as well as the endurance limit is lowered. Higher meanstress implies not only the stresses are higher which increase thedamage contribution for every increment within a cycle but alsoa larger fraction of the cycle is higher than the threshold stressand therefore contributes to the damage compared to the com-pletely reversed cycle. The model, thus, is able to reproduce theexperimentally well established facts of reduction in life and low-ering of endurance limit for a positive value of mean stress for boththe representative materials.

A quantified summary of the mean stress effect can be repre-sented through a constant-life diagram, where different combina-tions of stress amplitude and mean stress resulting in a constantlife are plotted as shown in Fig. 7 for the steel. A range of cyclesto failure, N = 10, 103 and 105, are considered. It is observed thatwith the increase in applied mean stress the permissible stressamplitude, for a given life of N = 10, decreases linearly from thestress amplitude at fully reversed load So to zero at mean stressequivalent to the cohesive strength. Note, So is 811, 463 and217 MPa for the chosen cases of constant-life. This Goodman-dia-gram like curve for low cycle failure changes for higher cycle fail-ure, for instance, for the given life of N = 105 the decrease in stressamplitude is still linear but significantly more rapid and reaches

1.5

2

2.5

3

3.5

2 2.5 3 3.5 4 4.5

σmean/σmaxo = 0.0σmean/σmaxo = 0.2σmean/σmaxo = 0.4

Log10 (Nf)

Log

10 (S

)

Fig. 5. Effect of mean stress on stress-life (S–N) curve for steel.

near zero at approximately mean stress of 0.25 rmax,o. Implyingany higher mean stress cannot be applied with a non-zero stressamplitude without causing failure. The appearance of the curvefor higher cycle failure is more like Soderberg’s curve, as at nearzero amplitude it has the mean stress as the endurance limit whichfor rF in the range 0.45–0.55 would be very close to the yieldstrength of the steel considered for the present study.

In prediction of failure under variable amplitude loading, one ofthe major drawbacks of a simple linear damage accumulation ruleis that it does not account for the effect of the sequence in whichthe cyclic loads are applied. To test the fatigue cohesive model’sability in incorporating the sequencing effects, the model is appliedto make predictions for various combinations of two blocks of con-stant amplitude. The first load block is applied for N1 fully reversedcycles followed by a second block of a different but again a con-stant amplitude load till failure. The corresponding number of cy-cles, N2, is calculated by subtracting N1 from the total number ofcycles till failure. From the summary presented in Fig. 8 for thecase when the stress amplitudes for the two blocks are 0.5 rmax,o

and 0.3 rmax,o, there is a clear deviation observed from the lineardamage rule as previously observed in [15]. The effect of thehigh-low sequencing being that failure occurs for

PðNi=NfiÞ < 1.

This effect is most pronounced near N1/Nf1 = 0.5. The deviation iseven stronger when the lower limit of the imposed stress is re-duced to near endurance limit at 0.26 rmax,o. This deviation dueto the higher load being first in the sequence has been describedto be largely due to the contribution of the N1 cycles in initiatinga crack or significant damage process towards the initiation [1].For the reverse sequence in which the application of the loweramplitude load precedes, the failure occurs for

PðNi=NfiÞ > 1.

1.6

1.8

2

2.2

2.4

2 2.5 3 3.5 4 4.5 5

Log

10(S

)

σF /σmax,o = 0.0 σF /σmax,o = 0.1 σF /σmax,o = 0.25

Log10 (Nf)

Fig. 10. Effect of cohesive endurance limit on stress-life curve of Aluminum.

2

2.2

2.4

2.6

2.8

3

2 2.5 3 3.5 4 4.5 5

Log10 (Nf)

δΣ = 2δoδΣ = 4δoδΣ = 6δo

Log

10 (S

)

Fig. 11. Effect of cohesive length parameter on life for steel.

1.6

1.8

2

2.2

2.4

2 2.5 3 3.5 4 4.5 5

δΣ = 2δoδΣ = 4δoδΣ = 6δo

Log10 (Nf)

Log

10 (S

)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

N1 /

Nf1

N2 / Nf2

0.5-0.260.26-5

linear rule0.5-0.30.3-0.5

Fig. 8. Effect of sequencing in variable amplitude loading for steel.

D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162 159

The accumulation of micro-structural damage due to fatigue isincorporated in the model through two cohesive fatigue parame-ters, rF and dR. To understand their role in prediction of the S–Ncurve, first the stress-life curve for fully reversed uniaxial loadingis obtained for a range of rF keeping dR at 4do as shown in Figs. 9and 10. For rF = 0, there is no endurance limit observed for eithermaterial. In the model this is because any positive stress contrib-utes to fatigue damage, however, for non-zero rF, out of the posi-tive part of the cycle some portion is ignored in evolution offatigue damage. As rF is increased further the extent of ignoredportion of the positive cycle increases further thereby increasingthe life. Also, for higher cohesive endurance limit the S–N curvereaches the endurance limit in fewer cycles.

The stress-life predictions of the model for variations in thecohesive length parameter, dR, keeping rF constant at 0.25 rmax,o

are illustrated in Figs. 11 and 12. Unlike rF which is not only di-rectly related to the endurance limit and the number of cyclesfor which the limit is reached but also influences the slope of thestress-life curve, Figs. 11 and 12 show there is no effect of increas-ing dR observed on the slope of the stress-life curve.

Higher dR for the same increment in dn, results in lesser damageas it scales down the damage which manifests as a parallel shift inthe stress-life curve. The implication being that both the endurancelimit and the slope of an S–N curve are entirely characterized bythe stress parameter, and therefore, as per the model directlydependent on each other, while the intercept of the S–N curve islargely characterized by the length parameter.

The fatigue damage mechanisms are primarily driven by local-ized plasticity and therefore, dependence on the stress-state is nat-ural. For the present study, the parameter chosen to quantify thestress-state is the biaxiality ratio, a, which is directly related to

2

2.2

2.4

2.6

2.8

3

2 2.5 3 3.5 4 4.5 5 5.5

σF /σmax,o = 0.0 σF /σmax,o = 0.1

σF /σmax,o = 0.25

Log10 (Nf)

Log

10 (S

)

Fig. 9. Effect of cohesive endurance limit on stress-life curve of steel.

Fig. 12. Effect of cohesive length parameter on life of Aluminum.

the effective triaxiality parameter in [19]. Two extreme cases,low biaxiality ratio a = 0.2 and high biaxiality of a = 0.8, are exam-ined in detail. The traction separation law without any fatiguedamage, for the specific stress-states are shown in Fig. 2.

To observe the effect of biaxiality ratio on fatigue failure, thesimple case of rF = 0 is considered. For the representative metalsconsidered at an applied stress amplitude of S = 0.8 rmax,o the evo-lution of the damage parameter is shown in Figs. 13 and 14. As ex-pected until the normalized accumulated separation reaches unitythere is no damage accumulation. With further cycles, damagegrows such that average increase in damage per cycle is initiallylinear. As the magnitude of the damage parameter increases, it be-comes highly non-linear before final failure which occurs with inone cycle error of the theoretical value, Dc = 0.2. The figure also

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5

2

2.5

Δ n/ δ0

0

0.05

0.1

0.15

0.2

0.25

N

Biaxial (α =0.8)

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

0

0.05

0.1

0.15

0.2

0.25

N

Uniaxial (α=0)

Separation

Damage

Dc

Dc

Δ n/ δ0

Fig. 13. Evolution of accumulated separation and damage until failure in steel.

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

20

0.05

0.1

0.15

0.2

0.25

N

Biaxial (α =0.8)

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

0

0.05

0.1

0.15

0.2

0.25

Dc

N

Uniaxial (α=0)

Separation

Damage Dc

Δ n/ δ0

Δ n/ δ0

Fig. 14. Evolution of accumulated separation and damage until failure in aluminum.

160 D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162

shows that the average rate of growth of damage is higher for high-er biaxiality. The damage curve for every cycle has a rise corre-sponding to fraction of cycle with tensile stresses and a plateaureflecting the lack of damage accumulation during compression.The rate of rise is much more rapid in case of higher biaxiality ratioimplying fewer cycles to reach failure. The reason being that as perEq. (6), do decays exponentially with increasing constraint. Thus athigher a, do is small which implies lesser cycles to initiation of fa-tigue damage. Also since do is used for scaling the incremental sep-aration in calculation of incremental damage parameter it alsoimplies rapid increase in the damage parameter.

A clearer picture of the effect of constraint on fatigue failure lifeemerges in Figs. 15 and 16. Fully reversed imposed loads rangingbetween 60–100 MPa are considered for both the materials. With

a increasing from the uniaxial conditions up to 0.2, there is no sig-nificant effect on the life. Further increase of biaxiality ratio showsa significant drop in the number of cycles to fatigue failure till itsaturates as the critical value of a is reached. The effect of a indecreasing life is more pronounced for lower loads correspondingto fatigue at higher cycles.

A comprehensive comparison of the fatigue failure curves at dif-ferent stress-states for a = 0, 0.5 and 0.8 is shown in Figs. 17 and 18for steel and aluminum respectively. It is predicted that at higher afor a given life of very few cycles, material is able to withstandhigher amplitudes of imposed cyclic load which is consistent withthe TSLs of monotonic fracture. Also, the number of cycles to tran-sition decreases with increasing a. Post transition for a given life ofhigher cycles, the effect of higher constraint is to restrict the

0.0 E 0

5.0 E 3

1.0 E 4

1.5 E 4

2.0 E 4

2.5 E 4

3.0 E 4

3.5 E 4

0 0.2 0.4 0.6 0.8 1

Nf

α

S = 100 MPaS = 80 MPaS = 60 MPa

Fig. 15. Effect of biaxiality ratio on the overall life for ferritic steel.

0.0 E 0

1.0 E 3

2.0 E 3

3.0 E 3

4.0 E 3

5.0 E 3

6.0 E 3

0 0.2 0.4 0.6 0.8 1

Nf

α

S = 100 MPaS = 80 MPaS = 60 MPa

Fig. 16. Effect of biaxiality ratio on the overall life for aluminum alloy.

2

2.2

2.4

2.6

2.8

3

3.2

1 1.5 2 2.5 3 3.5 4 4.5 5

Log

10 (S

)

Log10 (Nf)

α = 0.0α = 0.5α = 0.8

Fig. 17. Effect of biaxiality ratio on stress-life curves of ferritic steel at differentstress states.

2

2.2

2.4

2.6

2.8

3

1 1.5 2 2.5 3 3.5 4 4.5 5

α = 0.0α = 0.5α = 0.8

Log10 (Nf)

Log

10 (S

)

Fig. 18. Effect of biaxiality ratio on stress-life curves of aluminum alloy at differentstress states.

D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162 161

permissible loads that can be applied. This effect has been attrib-uted to the rapid growth of damage at higher constraints as shownpreviously in Figs. 13 and 16, implying reduction in life for mostloads except when the applied loads are comparable to the cohe-sive peak strength as the failure is monotonic like.

Notably, while the previous studies were able to predict S–Ncurves and other characteristics of fatigue, they were insensitiveto the effect of stress-state on the traction–separation law in thecontext of fatigue failure predictions. Therefore, such models if ap-plied for cases of non-zero biaxiality would have predicted the

same S–N curve as in case of zero biaxiality. In contrast, the pro-posed model by incorporating effect of stress-state in the formula-tion is able to reproduce the experimentally observed reduction inlife due to biaxiality of the applied load as reported in literature [6].In the present study, the additional feature of effect on enduranceor fatigue limit at higher biaxiality has not been investigated as itrequires an understanding of the stress-state dependence of thecohesive endurance limit, rF, which is part of an ongoing study.Also, the computational results presented here are for stress con-trolled cyclic load. In the implementation of this model through auser defined displacement based element for a multi-elementgeometry would result in strain-controlled load for the element.This will require inclusion of the softening part of the stress-statedependent model of [19] and then traction exceeding the cohesivestrength would result in softening of the element leading to loadtransfer to neighboring elements. The implementation of the pro-posed model for multi-element geometries is in progress.

4. Concluding remarks

A cohesive model applicable for various stages of fatigue failureunder mode I plane strain loading is presented. The basis of themodel is introduction of an irreversible damage parameter to anexisting stress-state dependent cohesive law. Thus, the model issensitive to both cyclic damage as well as variations in thestress-state. Two representative metals with basic mechanicaland cohesive properties as described in [19] are chosen for thestudy. The model is able to predict the typical features of uniaxialfatigue failure data represented as stress-life curve: reduction inlife with increasing stress amplitude, an endurance limit belowwhich the material is insensitive to the cyclic load. For a given fa-tigue life, the S–N curve for steel has higher stress amplitude thanaluminum and the Basquin’s exponents obtained are in the typicalranges observed in literature. The model is also able to account forthe well established effect of the mean level of the imposed cyclicload reproducing constant life curves that fall with in Goodmanand Soderberg’s curves. For a variable amplitude applied load themodel is able to capture the effect of sequence. Higher loads pre-ceding lower loads are shown to cause failure at

PðNi=NfiÞ < 1

and vice versa.Based on continuum damage laws, the evolution of the irrevers-

ible damage parameter involves two cohesive fatigue parametersas in [15]. These parameters control the characteristics of thestress-life curve. The cohesive endurance limit parameter controlsthe endurance limit and also influences the Basquin’s exponent forthe stress-life curve. In contrast, the cohesive length parameter hasno effect on the slope of the stress-life curve, but increasing it re-sults in a parallel shift in the S–N curve such that for a given life

162 D. Jha, A. Banerjee / International Journal of Fatigue 36 (2012) 155–162

higher stress amplitudes can be applied. For cohesive limit in therange of 40–50% of the cohesive peak stress for steel and 20–30%for aluminum alloy gives Basquin’s exponent within the reasonablerange reported in literature. From the parametric study one canconclude that the cohesive limit parameter corresponds to theendurance limit of the material. For finding the cohesive lengthparameter, once the functional dependence of the intercept ofthe linear part of the logarithmic S–N curve is established, the cor-responding intercept of the experimental curve will indicate anappropriate choice of cohesive length parameter for the finite ele-ment simulations. However, these correlations need to be furtherexplored with detailed experimental investigations.

The model’s ability to incorporate the role of state of stress inthe damage accumulation is considered for a range of biaxiality ra-tios. At very low cycle failure, the conditions are close to mono-tonic. Thus, for a stress controlled loading, the higher theconstraint the higher the peak stress of the traction–separation,therefore, higher the stress amplitudes that can be applied. Other-wise, for most of the S–N curve, higher constraint is found to beconducive to faster initiation of damage as well as its rapid growthin both the materials such that for a given stress, life is shortenedat higher constraints as observed in available literature. While themodel captures the experimentally observed trends in both uniax-ial and biaxial fatigue response, the applicability of the modelneeds to be further verified by experimental validation for a mate-rial tested under fatigue load at different stress-states.

Acknowledgement

Author AB gratefully acknowledges funding support from Aero-nautics R&D Board.

Appendix A

It is well established that a single failure locus of the effectiveplastic strain at fracture initiation, ��pl, as a function of the triaxial-ity parameter, represents ductile fracture for different specimensof the same material. The triaxiality dependent failure locus being

��pl ¼ Ke�1:5H; ðA:1Þ

where K is a material dependent non-dimensional parameter. For aproportionate biaxial state of stress under conditions of planestrain, the triaxiality developed during elastic deformation can becalculated from elastic constitutive relations to be:

Hel ¼ð1þ mÞð1þ aÞ

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 þ 1Þðm2 � mþ 1Þ þ að2m2 � 2m� 1Þ

p ; ðA:2Þ

where m is the Poissons ratio. In the limit of incompressible defor-mation (m = 0.5), the triaxiality parameter reaches its extremevalue, Hpl. For any deformation beyond onset of yield the triaxialityin absence of any damage to the material would increase from theelastic value to its extreme value, Hpl. However, for large biaxialityratios (a > 0.5), after the onset of yield as the triaxiality, H, increasesfrom its elastic limit, the void nucleation and growth would corre-spondingly increase and that would imply that the extreme value oftriaxiality, Hpl, would never be realized as the material will fail at alower value of triaxiality. To incorporate this effect it was proposedin [19] that there is a saturation limit of the triaxiality parameter,Hsat, for which the equivalent plastic strain for failure due to voidfrom Eq. (A.1) is some multiple of the elastic strain at yield suchthat

Hsat ¼ �23

lnSry

CE

� �; ðA:3Þ

where S is a non-dimensional multiplicative factor and C is a non-dimensional material constant of the order of the material constant,K, in the failure locus of Eq. (A.1). Authors argued that the ductilefailure of material is not at the extreme value of triaxiality param-eter, Hpl, but at an effective triaxiality parameter, Heff. The effectivetriaxiality parameter was taken to be such that at low biaxiality itfollows Hpl while for higher biaxiality it saturates to the saturationlimit of Eq. (A.3), and is of the form:

Heff ¼12

HI

pl þ Hsat

� �� 1

2HI

pl � Hsat

� �tanh

a� ac

0:1

� �; ðA:4Þ

where HI

pl ¼ ð1þ 2aþ 2a2 þ 2a3Þ=ffiffiffi3p

, is the binomial expansion ofHpl(a) up to only the fourth term to avoid the singularity associatedwith higher order terms for a tending to unity. The critical value ofbiaxiality ratio, ac, which indicates the onset of the saturation ofeffective triaxiality can be calculated using Eq. ?? to be

ac ¼ffiffiffi3p

Hsat � 1ffiffiffi3p

Hsat þ 1: ðA:5Þ

Based on the parametric study in [19], the authors conclude thatinitiation of ductile fracture at biaxiality ratios, a < ac, is controlledby parameter C and thus, can be determined from a low constrainttest such as A-notch tension test. The other parameter, ratio S/C isassociated with fracture at high constraint. These parameters areshown to be stress-state independent, and therefore, can be treatedas material constants.

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