Simulation of fatigue crack growth with a cyclic cohesive

Article published in International Journal of Fracture (20 14) 188:23–45The final publication is available at http://link.springer.com, DOI: 10.1007/s10704-014-9942-8

Simulation of fatigue crack growth with a cyclic cohesivezone model

Stephan Roth · Geralf Hutter · Meinhard Kuna

Received: 12 December 2013 / Accepted: 20 March 2014 / Published online: 08 April 2014

Abstract Fatigue crack growth is simulated for anelastic solid with a cyclic cohesive zone model (CZM).Material degradation and thus separation follows fromthe current damage state, which represents the amountof maximum transferable traction across the cohesivezone. The traction-separation relation proposed in thecyclic CZM includes non-linear paths for both un- andreloading. This allows a smooth transition from re-versible to damaged state. The exponential traction-separation envelope is controlled by two shape param-eters. Moreover, a lower bound for damage evolutionis introduced by a local damage dependent endurancelimit, which enters the damage evolution equation. Thecyclic CZM is applied to mode I fatigue crack growthunderKI-controlled external loading conditions. Theinfluences of the model parameters with respect tostatic failure loadK0, threshold load∆Kth and Paris pa-rametersm, C are investigated. The study reveals thatthe proposed endurance limit formulation is well suitedto control the ratio∆Kth/K0 independent ofm andC.An identification procedure is suggested to identify thecohesive parameters with the help of Wohler diagramsand fatigue crack growth rate curves.

Keywords Cyclic cohesive zone model· Fatiguecrack growth· Damage mechanics· Boundary layermodel

S. Roth· G. Hutter· M. KunaInstitute of Mechanics and Fluid Dynamics,TU Bergakademie Freiberg, 09596 Freiberg, GermanyE-mail: [email protected]

1 Introduction

The cohesive zone model (CZM) was developed byDugdale(1960) andBarenblatt(1962). Its basic ideais to concentrate the entire fracture process to a thincohesive zone. Inside this zone, the material behaviouris described by a local constitutive law, which relatesthe traction transferred across the cohesive zone to thedisplacement jump also denoted as separation. Consid-ering a crack tip as sketched in Fig.1, the stress sin-gularity expected from linear elastic fracture mechan-ics is replaced by a more realistic distribution alongthe cohesive zone. Damage initiation is determined byreaching a strength parameter called cohesive strength,t0. Below t0, i.e. ahead of the cohesive zone, the con-stitutive law captures reversibility. Due to the involvedtransition from reversible to damaged state, no explicitfracture criterion is needed. Instead, local failure is in-dicated by the vanishing local strength. The presenceof an incipient crack is not necessary. Thus, CZMs al-low to simulate crack initiation, crack propagation andfinal fracture in a unique manner.

Within the classical finite element (FE) framework,the CZM is implemented as cohesive zone element.This requires in those regions of the structure, wherecracks might occur, to prepare the mesh with cohesivezone elements. Nevertheless, the CZM is well appli-cable and widely accepted in order to simulate crackpropagation along symmetry lines in specimens, at in-terfaces and for delamination problems. Beyond that,the extended finite element method (XFEM) allows the

2 St. Roth et al., Int J Fracture (2014) 188:23–45, DOI: 10.1007/s10704-014-9942-8

δ

δ0

x

t0

t

cohesive zone

Fig. 1 Sketch of a cohesive zone ahead of a crack tip (tractiont,separationδ , cohesive strengtht0, corresponding separationδ0)

combination of CZMs and arbitrary crack propagation(Xu and Yuan, 2009a).

Separation processes under monotonic loadingconditions are covered by monotonic CZMs. The for-mulation of monotonic CZMs, underlying model as-sumptions as well as applications are well docu-mented, e.g. byBrocks et al(2003). In most of themodels, the cohesive laws are described by a scalartraction-separation relation and its curve. While the co-hesive strength and the fracture energy density char-acterise the maximum traction and the area underthe traction-separation curve, respectively, the shapeof the curve remains as a free model assumption.There are, amongst others, bilinear, trapezoidal, poly-nomial and exponential laws reported in the literature.For numerical reasons, continuously differentiable ap-proaches (polynomial, exponential) are preferred. Aversatile and quite often used cohesive law is the ex-ponential one proposed byXu and Needleman(1993).It is motivated by atomistic considerations and al-lows a closed form formulation in a thermodynam-ical framework by introducing a cohesive potentialfunction (e.g.Bouvard et al(2009); Ortiz and Pandolfi(1999); Roychowdhury et al(2002)). Modificationsconcerning the shape of the traction-separation curveare suggested e.g. byKroon and Faleskog(2005) orGoyal et al(2004); Lucas et al(2008), where the “brit-tleness” of the material is controlled by a shape param-eter. However, these models lack a closed form cohe-sive potential.

A further model assumption addresses the un-and reloading behaviour. For undamaged states, un-

and reloading is restricted to the reversible branch(linear or not) of the traction-separation curve. Af-ter damage initiation, the literature offers differ-ent approaches of the unloading behaviour with re-spect to linearity and destination of the unloadingpaths. While unloading is assumed towards the ori-gin in (Bouvard et al, 2009; Camacho and Ortiz, 1996;Goyal et al, 2004; Lucas et al, 2008; Nguyen et al,2001; Xu and Yuan, 2009a; Yang et al, 2001), whichis associated with quasi-brittle material behaviour,some residual separation after unloading is admittedin (Roe and Siegmund, 2003; Scheider and Mosler,2011; Xu and Yuan, 2009b). Linear unloading pathsare assumed (Bouvard et al, 2009; Nguyen et al, 2001;Roe and Siegmund, 2003; Scheider and Mosler, 2011;Xu and Yuan, 2009a,b; Yang et al, 2001) as well asnon-linear ones (Goyal et al, 2004; Lucas et al, 2008).Considering the cohesive zone as the area where mi-cro defects evolve, residual separation may result fromsurface mismatches at closing. Plasticity-induced sur-face mismatches (crack closure) can be incorporatedincluding plastic matrix material in the surrounding ofthe cohesive zone. Concerning the question of the lin-earity of unloading paths, the onset of damage at theapex of traction-separation curves should be regarded.In case of linear unloading, the unloading behaviourchanges abruptly when coming from the non-linear re-versible branch of the traction-separation curve to thesoftening one. Even an infinitesimal step over this bor-der results in a finite hysteresis with associated dissipa-tion, which could be hardly justified physically by aninfinite small change of the damage state. This effect isdemonstrated in (Liu et al, 2010; Roe and Siegmund,2003).

Cyclic CZMs provide the capability to simulatesubcritical crack growth. In these models, irreversibledamage accumulation is controlled by an explicit dam-age evolution equation where an endurance limit canbe incorporated (Bouvard et al, 2009; Nguyen et al,2001; Roe and Siegmund, 2003; Xu and Yuan, 2009b;Yang et al, 2001). While in monotonic CZMs the dam-age state is uniquely defined by the maximum separa-tion attained during the loading history, cyclic CZMsneed a more general damage variable. In the literature,stiffness-type (Nguyen et al, 2001), separation-type(Yang et al, 2001) and micromechanically motivateddamage variables (Roe and Siegmund, 2003) are sug-

Simulation of fatigue crack growth with a cyclic cohesive zone model 3

gested. For visualisation purpose,Ortiz and Pandolfi(1999) propose an energetically motivated conversionof a separation-type damage variable into the range be-tween zero and one.

In contrast to monotonic CZMs, cyclic CZMs al-low to specify an endurance limit to indicate a cyclicload level below which there is no damage initia-tion, see (Roe and Siegmund, 2003; Siegmund, 2004;Xu and Yuan, 2009a,b). Therefore, additional materialparameters are introduced in the damage evolutionequation of the cited models mostly with the help ofHeaviside step functions.

Fatigue crack growth (FCG) and related overloadeffects are investigated byBouvard et al (2009);Nguyen et al (2001); Roe and Siegmund(2003);Siegmund (2004), and Xu and Yuan (2009b). Thethree stages of FCG rate curves has been reproduced:near threshold region, Paris line and static failure.While the static failure load without any doubt de-pends on the fracture energy density of the cohesivemodel, both Paris exponent and fatigue threshold loadare found to depend on the endurance limit parameter.Except for these general statements, there are nofurther studies that clarify the relation between thecharacteristics of FCG rate curves and the proper-ties of CZMs. Moreover, although the shape of thecohesive law is investigated severalfold (e.g. in thecontext of crack growth resistance of elastic-plasticsolids (Hutter et al, 2011; Tvergaard and Hutchinson,1992)), its influence on the simulation of FCG has, tothe authors’ knowledge, not been studied before. Thisalso applies to the assumptions with respect to theunloading behaviour.

The present publication gives a contribution tothese questions. A versatile cyclic CZM is presented.Monotonic and cyclic properties of the CZM are dis-cussed. The applicability of the model to cyclic ho-mogeneous stress-based loading and FCG under modeI loading conditions is demonstrated. In an extensiveparametric study the influences of the parameters ofthe cyclic CZM with respect to the FCG rate curves areinvestigated systematically. Finally, a parameter iden-tification strategy is proposed.

2 Monotonic Cohesive Zone Model

The key feature of a CZM is the traction-separation-law (TSL). It relates the traction vector,ti, to the sep-aration vector,δi, accounting for the current damagestate. Both vectors consist of one normal and two tan-gential components indicated by the coordinate in-dicesi = n, r, s. In consistency with the thermodynam-ical framework proposed e.g. in (Bouvard et al, 2009;Roychowdhury et al, 2002), a cohesive potential,Γ , isintroduced in order to obtain the traction coordinatesby partial differentiation with respect to the separationcoordinates,

tit0

=∂ Γ

∂(

δiδ0

) , i = n, r, s . (1)

The cohesive potential is a measure of the normalisedstored reversible energy and captures the specific un-loading behaviour, both considered below. Introducingthe effective normalised quantities effective traction,τ ,and effective separation,λ ,

τ =1t0

√

t2n + t2

r + t2s , (2)

λ =1δ0

√

〈δn〉2+δ 2r +δ 2

s , (3)

as well as a damage parameter,D, the vectorial TSLreduces to a scalar relationτ(λ ,D). The normalisingparameterst0 andδ0 are the cohesive strength and thecorresponding separation, respectively, see Fig.1. TheMacAulay brackets

〈δn〉=12

[

δn+|δn|]

=

δn , ∀ δn ≥ 0

0 , ∀ δn < 0(4)

are introduced to ensure positive contributions of nor-mal separation. The response of the model in thecompression case is addressed below. Under mono-tonic loading conditions, the TSL describes an en-velope in the normalisedλ -τ-space bounding all ad-missible states (traction-separation envelope, TSE).Here, a generalised exponential approach based onXu and Needleman(1993) is used by introducing twoshape parameters,ε ≥ 1 andω ≥ 1,

τTSE=

λ exp(1−λ ) , ∀ λ < 1

1−[

1−[

λ exp(1−λ )]ε]ω

, ∀ λ ≥ 1. (5)


Some TSEsτTSE(λ ) are depicted in Fig.2. The ascend-ing branch of the TSE before reaching the cohesivestrength refers to reversible material behaviour (sub-script “rev”) and does not depend onε andω . At theapex, damage initiates and develops with increasingdisplacement driven loadλ . Since the maximum trans-ferable traction degrades with increasing damage state,the descending branch of the TSE is called damage lo-cus (subscript “DL”). The shape parametersε andωonly influence the shape of the damage locus. Whilethe latter defines the plateau width of the TSE,ε affectsthe slope of its softening branch. This is demonstratedin Fig.2, too. The area under the TSE represents thenormalised fracture energy density,Γ0, dissipated dur-ing the formation of new surfaces (cohesive fractureenergy or work of separation per unit area normalisedby t0δ0). In Fig.2, its value is kept constant by sizingε in dependence ofω . The normalised fracture energydensity can be decomposed into a reversible part,Γrev,and the area under the damage locus,ΓDL,

Γ0 = Γrev+ΓDL . (6)

While Γrev represents a constant value,

Γrev=

1∫

λ=0

τTSE(λ )dλ

=

1∫

λ=0

λ exp(1−λ )dλ = e−2≈ 0.71828

(7)

with Euler’s number e,ΓDL depends onε andω ,

ΓDL(ε , ω) =

∞∫

λ=1

τTSE(λ )dλ . (8)

Numerical solutions of Eq. (6) are depicted in Fig.3.Note that the five parameters of the modelt0, δ0, Γ0, ε ,andω are not independent of each other. For the spe-cial parameter setε = 1, ω = 1 andΓ0 = e, the modelof Xu and Needleman(1993) is recovered.

As stated above, under monotonic loading condi-tions damage evolution takes place exclusively on thedamage locus. Below this curve, i.e. in unloading andreloading situations, reversible material behaviour isassumed. For these reasons, the position on the dam-age locus can be seen as a measure of the currentdamage state defined by the shape of the TSE and ei-ther the current effective traction or separation, respec-tively. Here, the latter variable is chosen since it allows

0

0.2

0.4

0.6

0.8

1

1 2 4 6 8

τ TS

E

λ

ε = {1, 1.569, 2.46,3.185, 4.95}

ω = {1, 2, 5, 10, 50}

D = 1, D = 0

D → ∞, D → 1

Γ0

Fig. 2 Exponential TSEs with constant normalised fracture en-ergy densityΓ0 and variation of the shape parametersε andω

0

2

4

6

8

10

12

0 1 2 3 4 5

Γ 0

ε

ω = {1, 2, 5, 10, 50}

Γ0(ε , ω = const.)

Γ0 = e

Γrev = e−2

Fig. 3 Normalised fracture energy densityΓ0 in dependence ofthe shape parametersε andω; dashed line: reversible partΓrev =e−2; dot-dashed line:Γ0 = e, compare Fig.2

for an unambiguous description. In the following, thedamage state is defined by the separation-type “funda-mental damage variable” 1≤ D ≤ ∞. Moreover, in or-der to interpret the damage state as an effective surfacedensity of microdefects (Lemaitre, 1996), the damagevariable 0≤ D ≤ 1 known from damage mechanics isintroduced. The conversion ofD into D is described inSect.4. Nevertheless, the set of equations of the CZMis formulated in terms ofD for sake of reducing com-putational costs. Note that the conversion is more orless a matter of postprocessing and does not affect thedamage state.

Besides the shape of the TSE, the cohesive po-tential is mainly determined by the unloading be-haviour. Here we restrict to unloading towards the ori-gin without any remanent separation. For undamaged


states, un- and reloading paths follow the non-linearreversible branch of the TSE. In order to establish asmooth transition from reversible to damaged states, anon-linear approach is needed, which guarantees littlechanges of unloading behaviour in consequence of lit-tle changes of the damage state, as discussed in Sect.1.Beyond that, the (non-)linearity of the unloading pathsinfluences the amount of reversible energy, stored inthe cohesive zone at a certain damage state. This isexpressed in terms of the area under the path and theparticular value of the cohesive zone potential, respec-tively. Besides the requirement that the unloading pathand the reversible branch of the TSE coincide for theundamaged stateD = 1, all paths have to be mono-tonically increasing. Furthermore, they are bounded bythe TSE. A cohesive zone potential covering all theseproperties is found as

Γ (λ , D) =Gκ

F

[

e− [1+λF]exp(1−λF)]

, (9)

wherein the functionsF(D) andG(D) provide the spe-cific shape of the TSE and the damage dependent un-loading behaviour in terms of the fundamental damagevariableD,

F(D) =

1 , ∀ D < 1

− 1D

W

(

−1e

G1−κ)

, ∀ D ≥ 1, (10)

G(D) =

1 , ∀ D < 1

τTSE(D) , ∀ D ≥ 1. (11)

The further parameterκ enters to describe the non-linearity of the unloading behaviour. The Lambertfunction W(x) (also known as product logarithm) isdefined implicitly by

x = W(x)exp(

W(x))

(12)

and can be evaluated numerically. The use of W in-dicates, thatF is more or less an inverse function ofEq. (5). Several unloading paths for different damagestates andε = 1, ω = 1, κ = 0 are depicted in Fig.4.Note that with increasing damage the non-linearitydecreases. The dependence ofF on the fundamentaldamage variableD is depicted in Fig.5 for some com-binations of the shape parametersε , ω andκ = 0. Theinfluence ofκ is illustrated in Fig.6. As shown there,the parameterκ ≤ 1 determines the non-linearity of

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

τ

λ

D = 11.5

2

2.5

3

45

6 D → ∞

Fig. 4 TSE (ε = 1, ω = 1) and unloading paths (κ =0) from the damage locus at various damage statesD ={1, 1.1, 1.5, 2, 2.5, 3, 4, 5, 6}

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

F

D

ε = {1, 1.569, 2.46,3.185, 4.95}

ω = {1, 2, 5,10, 50}

Fig. 5 FunctionsF(D) for κ = 0 and various combinations ofεandω with Γ0 = e= const.

the unloading paths and thus the amount of stored en-ergy and dissipation, respectively. With decreasingκthe non-linearity decreases. Forκ = 1 the unloadingpaths intersect the damage locus horizontally leadingto the largest cohesive zone potential and the lowestdissipation.

Evaluating Eqs. (1) and (9), the normal and tangen-tial coordinates of the traction vector result from partialdifferentiation ofΓ with respect to the coordinates ofthe separation vector,

tnt0

=〈δn〉δ0

GκF exp(1−λF) , (13)

tr/s

t0=

δr/s

δ0GκF exp(1−λF) . (14)


0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

τ

λ

D = 2.5

κ = {−1, 0,0.5, 1}

Fig. 6 TSE (ε = 1, ω = 1) and unloading paths from the damagelocus (D = 2.5) for variousκ

With the help of Eq. (2), the relation between the effec-tive quantities is obtained as

τ(λ , D) = λGκF exp(1−λF) , (15)

plotted in Fig.4. Both branches of the TSE are recov-ered by Eq. (15),

τTSE(λ ) =

τ(λ , D = 1) , ∀ λ < 1

τ(λ , D = λ ) , ∀ λ ≥ 1, (16)

which emphasises the key role of the cohesive poten-tial. Note that the monotonic CZM is uniquely definedby Eqs. (1)–(3) and Eq. (9). Comparably to the expo-nential CZM presented here, a bilinear one is deducibleby replacing Eq. (9) with

Γ bilinear(λ , D) =λ 2 (λ f − D)

2D(λ f −1)(17)

whereλ f denotes the normalised effective separationat failure.

In order to complete the monotonic CZM, the com-pression case is considered. Negative normal separa-tionsδn < 0 are penalised by a contact formulation bysteeply rising the stiffness,

tnt0

= µe

[

1−exp

(

−GκFµ

δn

δ0

)

]

. (18)

For reasons of simplicity, friction under con-tact conditions has been neglected, compare(Roe and Siegmund, 2003). The penalisation fac-tor is chosen to beµ = 0.001 throughout all analysespresented in the following.

3 Cyclic Cohesive Zone Model

In the monotonic CZM presented above damage evo-lution is restricted to the damage locus. Under cyclicloading with constant separation amplitude the modelwould therefore allow an infinite number of repetitionswithout any accumulation of damage. In order to sim-ulate fatigue, the model has to be augmented by a dam-age evolution equation covering the following features:

i) Un- and reloading paths do not coincide forD > 1.While damage keeps constant during unloading, itmust evolve during subsequent reloading.

ii) The maximum transferable traction decreases withincreasing damage.

iii) The damage locus still forms an envelope of all ad-missible states. There is damage evolution belowthe damage locus. Once the damage locus has beenleft, it is never reached again at constant ampli-tude displacement driven loading. This would re-quire sufficient increase of the amplitude. So, thedamage locus remains the upper bound of damageevolution.

iv) The monotonic CZM forms a special case of thecyclic CZM.

v) Comparable to the cohesive strength for the undam-aged state, there is a local damage dependent en-durance limit providing a lower bound for damageevolution.

The behaviour of such a cyclic CZM is qualitativelydepicted in Fig.7. The damage evolution equationmay depend on the damage state, the current sepa-ration, its rate, and the shape of the damage locus,˙D = ˙D(D, λ , λ , . . . ). Since damage evolution is an ir-reversible process,D is non-negative,D ≥ 0. Further-more, a smooth transition from unloading to reloadingis desirable. Therefore, damage evolution is assumedto scale with the distance to the damage locus. Whiledamage evolution tends to the monotonic CZM withincreasing load, it diminishes in the vicinity of the ori-gin. In order to implement this, the ratio between thecurrent separation and the fundamental damage vari-able seems to be an appropriate measure. It ranges be-tween zero at the origin and unity at the damage lo-cus. A one-parametric power-law approach is chosenas scaling function, which leads to the following dam-


τe

λ e

τ

λ

D = 1

D → ∞

E

III

III

Fig. 7 Initial(I), un(II)- and reloading (III) paths described bythecyclic CZM including an endurance limit (E) (dashed line: dam-age locus, monotonic CZM)

age evolution equation,

˙D =

[

λD

]ρ〈λ〉 ≥ 0 . (19)

The damage exponentρ controls the influence of thecurrent state, the MacAulay brackets ensure the posi-tiveness of˙D, and restrict damage evolution to loadingconditions only.

Equation (19) features no endurance limit yet. Inorder to modify Eq. (19) to include an endurance limit,let us consider again the monotonic CZM. Here, thecohesive strengtht0 and the respective separationδ0

define the endurance limit for the initial undamagedstate. Belowt0, there is reversible material behaviourdescribed by the ascending branch of the TSE. Oncedamage has initiated, the respective endurance limitdecreases according to the damage state at the dam-age locus. Note that it is the immanent characteristicof monotonic CZMs that the endurance limit does onlychange as a consequence of an increasing cyclic loadlevel or due to single overloads.

In contrast, within a cyclic CZM a lower endurancelimit exists that indicates a cyclic load level belowwhich there is no damage initiation. The endurancelimit enters the damage evolution equation via a Heav-iside step function, which prevents damage initiationand accumulation below the endurance limit. The ini-tial endurance limitτe

0 is introduced as a further ma-terial parameter. Once damage has initiated, the argu-mentation with respect to the monotonic CZM leadsto the necessity of a suitable approach to describe the

damage dependency of the endurance limit. Moreover,the postulated smoothness of the model at damage ini-tiation used above to motivate the non-linear unloadingbehaviour, ends up in the assumption that a local statedependent endurance limit must exist, as well. Start-ing at the initial endurance limit, indicated by the ini-tial endurance limitτe

0 at the TSE, the expected en-durance limit vanishes forD → ∞. In the following,the endurance limit is assumed to depend solely on thedamage state.

The endurance limit can either be expressed interms of effective tractionτe or effective separationλ e,see Fig.7. Again, a simple power-law approach is as-sumed scaling the damage locus (compare Eq. (5)),

τe(D) = τe0Gα (20)

with the initial endurance limitτe0 and the endurance

exponentα . The dependency ofτe with respect to thenew model parameterα ≥ 1 is illustrated in Fig.8.The respective endurance separation is found withEqs. (15) and (20) from

τe(D) = λ eGκF exp(1−λ eF) (21)

as

λ e =− 1F

W

(

−τe0

eGα−κ

)

. (22)

Respective curvesλ e(D) are depicted in Fig.8, too.Note that the endurance separation behaves non-monotonic during damage evolution. The enduranceexponentα determines the relation between the maxi-mum transferable traction at the specific damage stateand the minimum load to continue damage evolution.This consideration directly leads to the endurance lo-cusτe(λ e), which envelops all endurance states com-parably to the damage locus. It is found by eliminatingD in Eq. (20) with the help of Eq. (22). A parameterplot is depicted in Fig.9 (parameter 1≤ D ≤ ∞). Eachof the depicted curves forms a lower bound of damageevolution. All states to the left of the endurance locusare endurable. Damage may only evolve between en-durance locus and damage locus. Forα = 1 andτe

0 = 1the endurance locus and the damage locus coincide.This describes the special case of the monotonic CZMwhere damage evolution is restricted to the damage lo-cus.

The endurance limit is embedded in the damageevolution equation Eq. (19) by multiplying a Heaviside


0

0.5

1

1.5

0 2 4 6

τe ,λ

e

D

α = {1.0, 1.5,2.0, 5.0}

τe0

Fig. 8 Endurance limitτe(D) (solid) and endurance separationλ e(D) (dashed) for various endurance exponentsα andτe

0 = 0.8,ε = 1, ω = 1, κ = 0

τe0

1

τe

λ e

α = {1, 1.1, 1.2,1.5, 2, 5, 10}

τe0 = 0.8

Fig. 9 Endurance locusτe(λ e) for various endurance exponentsα andτe

0 = 0.8, ε = 1, ω = 1, κ = 0 (solid lines); dashed line:TSE with damage locus

step function, H(λ −λ e), which restricts damage evo-lution to cases ofλ > λ e,

˙D =

[

λD

]ρ〈λ〉 H(λ −λ e) . (23)

This evolution equation allows an analytic integrationonce the evaluation of the MacAulay brackets revealsλ ≥ 0 and the appliedλ exceeds the current enduranceseparation,λ > λ e(D). When starting atD = D1, thefundamental damageD2 reached after loading fromλ1

to λ2 is calculated by

D2 =ρ+1√

λ ρ+12 + Dρ+1

1 −λ ρ+1res (24)

λ1 λ e = λres λ2

τ

λ

D1

D2

Fig. 10 Initial monotonic loading from the origin toλ = D1 at thedamage locus, subsequent unloading with constant damage downto λ1 and reloading with damage evolution untilλ = λ2, D = D2;damage evolution starts at the endurance locus,λres= λ e(D1)

with λres being the effective separation at which dam-age accumulation is resumed,

λres= max(

λ1, λ e(D1))

. (25)

Here, the index “1” indicates known quantities at thebeginning of the loading process and the index “2”at the end, respectively. By means of Eq. (25), thestarting point of damage evolution is shifted to theendurance separation. This situation is illustrated inFig.10. Herein, an unloading step from the damagelocus down toλ1 precedes the considered load step.Note, since dλ e ≤ dλ does not apply in the wholerange ofD and for all parameter combinations,λ ≥λ e(D2) has to be checked after analytical integrationof Eq. (23). In case ofλ < λ e(D2), D2 is recalculatedwith the inverse of Eq. (22), D2 = D(λ e = λ2), i.e. bymapping on the endurance locus.

Note that within the cyclic CZM, the initiationvalue of the fundamental damage variable is shiftedfrom one toλ e(D = 1). For D < 1, damage accumu-lates according to Eq. (23) but without any effect onneither the endurance limit nor the un- and reloadingpath, compare Eqs. (10) and (11).

4 Energetic Considerations

In damage mechanics, damage states are commonlycharacterised by a damage variableD that ranges be-tween zero and one indicating the undamaged stateand material failure, respectively. Since the presented


cyclic CZM is formulated in terms of the fundamentaldamage variableD, an appropriate conversionD(D)

is needed. This relation distributesD at the damagelocus.Ortiz and Pandolfi(1999) propose an energet-ically motivated approach. According to this, dam-age is defined as the ratio between the cohesive po-tential evaluated for the maximum attained separationand the fracture energy density. In the view of the au-thors, the application of this approach to the presentCZM is problematic because of two reasons. Firstly,in this definition damage is not limited to the dam-age locus but rather distributed at the whole TSE in-cluding the reversible branch. Secondly, the mean-ing of the measure seems questionable. While in theCZM of Ortiz and Pandolfi(1999) the cohesive poten-tial amounts the work done in a monotonic loadingprocess starting at the origin up to the current maxi-mum separation (by following the TSE), the cohesivepotential in the present CZM is interpreted as a mea-sure of the reversible or stored energy. Both quantitiesare not directly related to damage. Since damage ac-cumulation is an irreversible and dissipative process,the ratio between the dissipation densityD and thefracture energy densityD/Γ0 (both normalised) is sug-gested as a more convenient damage measure. Unfor-tunately, under cyclic loading conditions the amountof dissipated energy becomes history dependent andmaterial failure does not necessarily occur atD = Γ0.However, both conditions are fulfilled under mono-tonic loading conditions, which leads to the modifieddefinition

D :=Dmon

Γ0. (26)

Thus, the damage state after arbitrary loading history isassociated with a monotonic loading process that leadsto the same damage state (i.e. the sameD) by dissipat-ing Dmon. Moreover, the maximum load of the com-parable monotonic processλ exactly equals the fun-damental damage for which the conversion intoD issearched for. Consequently, the relation

D(D) =Dmon(λ = D)

Γ0(27)

is to be evaluated. The respective distribution ofD atthe damage locus is depicted in Fig.11.

In order to determineD , an arbitrary reloading pro-cess including damage evolution is considered first

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

τ

λ

D = 00.1

0.3

0.5

0.7

0.9D → 1

D

Γ0−D

Fig. 11 Distribution of damage variable 0≤ D ≤ 1 at the dam-age locus (ε = 1, ω = 1) and respective unloading paths (κ = 0);highlighted areas refer toD=0.5 and cover the same area demon-stratingD = D/Γ0 = 0.5

(λ1 → λ2, ˙D > 0, depicted in Fig.12). The area un-der the loading path betweenλ1 andλ2 covers the nor-malised total work,W tot

1→2, done in this step, compareFig.12: A-C-D-B. Furthermore, the cohesive potentialΓ (D, λ ) introduced in Sect.3 describes the normalisedstored energy density of a damaged state at a given de-formation. This amount of energy could be recoveredby total unloading. Inλ -τ-space,Γ is identified bythe area under the unloading path, Eq. (15). Regardingthe loading process considered, the material point pos-sesses the stored energiesE rev

1 = Γ (D1, λ1) (Fig.12:O-A-B) andE rev

2 = Γ (D2, λ2) (Fig.12: O-C-D) at ini-tial and final states, respectively. According to Fig.12:E-D-B, the energy dissipated during the loading step isfound as the difference between the sum of stored en-ergy at the beginning and total work on the one hand,and the stored energy at the end on the other hand,

D = E rev1 +W tot

1→2−E rev2 . (28)

While the cohesive potential is defined analytically byEq. (9), there is no closed solution available for the in-tegral over the loading path,

W tot1→2 =

λ2∫

λ=λ1

τ(

λ , F(

D(λ ))

, G(

D(λ ))

)

dλ . (29)

Instead, Eq. (29) is solved numerically. Note that dis-sipation is intrinsically associated with damage evo-lution. In contrast, for un- and reloading processes atconstant damage between points E and D, the total


O λ1 λ2

τ

λ

D1

D2

A

B

C

D

E

Fig. 12 Unloading at constant damage (D = D1) and reloading(λ = λ1 . . .λ2) with damage evolutionD = D1 . . . D2; energeticquantities:E rev

1 → O-A-B, E rev2 → O-C-D,W tot

1→2 → A-C-D-B, D

→ E-D-B

work equals the difference of the respective reversibleenergies. Pure changes in the direction of the tractionand separation vector at constant effective quantities(and thus unchanged damage) are assumed to be re-versible.

The dissipation densityDmon of a process loadedmonotonically from the origin up toλ = λ reduces to

Dmon=

λ∫

λ=0

τTSE(λ )dλ −Γ (D = λ , λ = λ ) , (30)

wherein Eqs. (5), (28) and (29) are used. The combina-tion of Eq. (27) with Eq. (30) evaluated forλ = D nowenables the conversion fromD to D. Due to the embed-ded integral again a closed form solution is not avail-able. That is whyD(D) is fitted with a two-parametricapproach,

D =

1−exp

(

1− Dβ

)

γ

. (31)

Because the shape of the damage locus in Eq. (5) de-pends onε and ω , and theκ-dependent unloadingpath, the distribution of the damage variable,D, andthus both of the fit parametersβ andγ in Eq. (31), de-pend onε , ω andκ . Some numerically obtained resultsfor β (ε) andγ(ε) in the range of 0.1≤ ε ≤ 10 are de-picted in Fig.13 for selected combinations ofω andκ . Obviously,ω andκ do not affectβ . The latter de-creases with increasing shape parameterε . In contrast,γ increases with increasingε , ω or κ , respectively.

0

2

4

6

8

10

12

14

0 2 4 6 8 10

β,γ

ε

β (ε , ω, κ)

ω = {1, 2, 5}γ(ε , ω, κ)

κ = {0, 1}

κ = {0, 1}

κ = {0, 1}

Fig. 13 Fit parametersβ (solid) andγ in dependence ofε =0. . .10,ω = {1, 2, 5} andκ = {0(dot-dashed), 1(dashed)}

5 Model Response to Cyclic HomogeneousStress-based Loading

In order to investigate the response of the cyclic CZMto cyclic homogeneous stress-based loading, a largenumber of simulations was performed by varying themean stress level and stress amplitude in the ranges of0≤ σm/t0 ≤ 0.95 and 0< σa/t0 ≤ 1, respectively. Thiscovers load ratios in the range of−1 ≤ R < 1. Com-parable to (Roe and Siegmund, 2003), normal loading(superscript “N”) as well as shear loading (superscript“S”) was considered. The parameters of the cyclicCZM were chosen asε = 1, ω = 1, κ = 0, ρ = 1,τe

0 = 0.25 andα = 2. The predicted number of cy-cles to failure,Nf , at a certain mean stress level areplotted in a semi-logarithmic Wohler diagram (S-N di-agram) shown in Fig.14. The expected continuouslydecreasing S-N curves are obtained as known from ex-perimental investigations, e.g. in (Suresh, 1998). Withincreasing mean stress, the number of cycles to fail-ure decrease at a constant load amplitude. Forσm = 0,σa(Nf = 1)/t0 = 1 applies while the endurance loadamplitude is found asσa(Nf = ∞)/t0 = τe

0 = 0.25 asexpected. Noticeable differences between normal andshear loading are only found for low mean stress lev-els. For higher mean stresses the curves for normal andshear loading can even not be distinguished in Fig.14.Respective constant-life plots for normal and shearloading are depicted in the Haigh diagram in Fig.15for Nf = {1, 2, 5, 10, 20, 50, 100, 200, 500}. Betweenthe upper and lower bounds,σa(Nf =1)/t0= 1−σm/t0andσa(Nf = ∞)/t0 = κ −σm/t0, each of the constant-


life plots shows a characteristic shape. However, thecurves differ from the Goodman relation where it is as-sumed that all curves are linearly leading to the point(1,0) in Fig.15. The curves are dominated by a re-gion defined by a mean stress sensitivity (afterSchutz(1967), M =

[

σa(R =−1)−σa(R = 0)]

/σm(R = 0))of M = 1. From Haigh diagrams we know that this cor-responds to an exclusive dependence on the upper loadlevel of constant amplitude cyclic loading. Regardingthe cyclic CZM, the deviation from the Goodman rela-tion is caused by the following model assumptions:

i) Under normal loading conditions there is no dam-age evolution in the compression case.

ii) Unloading is assumed towards the origin.iii) Damage evolution requires a deformation or stress

level beyond the local endurance limits,λ e or τe,respectively.

The first assumption applies only for normal loading,which causes the deviations of the corresponding S-Nand constant-life curves for shear and normal loadingin Fig.14 and Fig.15. In contrast, the assumption iii)leads, independently of the loading direction, to limitcurvesσa/t0 = σm/t0± τe

0, which are both depicted inFig.15 (dot-dashed). Below the lower limit curve inthe bottom right region, the constant-life curves canbe approximated conservatively by a linear relation-ship of Goodman type. Thus, these curves are virtu-ally bilinear. Such bilinear S-N curves are proposed in(Haibach, 1989) to account for the mean stress influ-ence of notched specimens. Nevertheless, Fig.15 re-veals that the present type of cyclic CZM is not ableto predict convex constant-life curves (Goodman, Ger-ber, . . . ) due to its basic model assumptions. Note thatthis applies in the normal loading case even withoutthe local endurance limit approach. Moreover, follow-ing (Smith, 1942) such concave constant-life curvesare associated with brittle metals.

In contrast, the cyclic CZM proposed in(Xu and Yuan, 2009b) captures the linear Good-man relation. It bases on the model presented byRoe and Siegmund(2003) wherein unloading isassumed to be of elastic-plastic type leading to aresidual separation (compare assumption ii). Thisenables non-negative separation in the compressioncase, i.e. no penetration at negative normal tractions.Thus, in this situation damage accumulates dependingon the negative load ratio reducing the mean stress

0.0

0.2

0.4

0.6

0.8

1.0

1E+0 1E+1 1E+2 1E+3 1E+4

σN/S

a/t

0

Nf

σN/Smt0

= {0, 0.1. . .0.9}

Fig. 14 Wohler diagram: S-N curves for normal (superscript“N”, solid) and shear (superscript “S”, dashed) loading (τe

0 =

0.25, α = 2, ρ = 1) for mean stress levelsσN/Sm /t0 =

{0, 0.1, . . . 0.9}

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

σN/S

a/t

0

σN/Sm /t0

Nf = {1, 2, 5, 10,20, 50, 100,200, 500}

Fig. 15 Haigh diagram: Constant-life plots for normal (super-script “N”, solid) and shear (superscript “S”, dashed) loading(τe

0 = 0.25, α = 2, ρ = 1); dot-dashed: endurance limit curve:σa/t0 = τe

0 −σm/t0 and bounds of region with mean stress sen-sitivity M = 1: σa/t0 = σm/t0 ± τe

0 (upper bound only for shearloading)

sensitivities toM < 1 (compare assumption i). Fur-thermore, in (Xu and Yuan, 2009b) the Heavisidefunction, which is associated with the endurance limit,is not evaluated (compare assumption iii). Under theseconditions, the linear Goodman relation is predicted.However, respecting the endurance limit leads toconstant-life curves comparable to the presented onesof our model due to assumption iii). For that reason,the results presented above qualitatively match thosein (Roe and Siegmund, 2003) despite the differentunloading behaviour.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

σS a/t

0

σNa /t0

Nf

R =−1

Fig. 16 Biaxial fatigue envelopes for Nf ={1, 2, 5, 10, 20, 50, 100, 200, 500} with load ratio R = −1;parameters: τe

0 = 0.25, α = 2, ρ = 1; dashed: circle(σN

a )2 + (σNa )2 = (τe

0t0)2 for limit case N = ∞; dot-dashed:reference circle

In order to compare both models also with re-spect to mixed-mode loading, simulations of homo-geneous biaxial fatigue tests were performed. As pro-posed in (Roe and Siegmund, 2003), in phase normaland shear loading was applied simultaneously with aconstant load angle,φ = tan−1(σS

a/σNa ) and a load ra-

tio of R =−1 leading to zero mean stresses,σN/Sm = 0.

The resulting biaxial fatigue envelopes are depicted inFig.16. ForNf = 1 andNf = ∞ the envelopes are quar-ter circles with radii ofσa/t0 = 1 andσa/t0 = τe

0, re-spectively. In between, the circular shape gets lost dueto the minor damage accumulation under normal load-ing in the compression case as discussed above. Thenumber of load cycles to failure in pure normal loadingexceeds that of pure shear loading at constant load am-plitudes. This applies to combined loading, too. In thiscase, the envelopes in Fig.16deviate from the circularshape looking “compressed” in direction of the shearcomponentσS

a . These results differ from those pre-sented in (Roe and Siegmund, 2003). As stated there,the resulting load amplitude for combined loading ex-ceeds the load amplitudes for both pure shear or nor-mal loading. This effect may possibly arise due to thedifferent model assumptions concerning the unloadingbehaviour.

6 Mode I Fatigue Crack Growth

This section addresses the application of the cyclicCZM to mode I fatigue crack growth (FCG) underKI-controlled external loading conditions.

6.1 Boundary Layer Model

The crack is modelled with a boundary layer formu-lation. Details on the finite element implementationare summarised in (Roth and Kuna, 2013). The geom-etry of the boundary layer is defined by the radius ofthe outer rim,Ro = 108 δ0 (Fig.17). The outer rim ofthe half model is meshed with 20-30 finite elements.The matrix material is assumed to be isotropic and lin-ear elastic with Young’s modulusE = 100t0 and Pois-son’s ratioν = 0.3. At the ligament, cohesive zone el-ements are applied. A fine meshed region in the vicin-ity of the crack tip with a width ofLa ≪ Ro allowsthe evaluation of FCG under constant (far-field) load-ing conditions. For lower load levels, a fine meshedregion of widthLa = 100δ0 with a cohesive zone el-ement size ofLf = 0.01δ0 is chosen. In order to savecomputational costs, increased values ofLf = 0.05δ0

andLa= 1000δ0 are applied at higher load levels. Theboundary layer is loaded byKI-controlled displace-ment boundary conditions according to the first termof the Williams series expansion of the far field so-lution in linear fracture mechanics withKI being thestress intensity factor of opening mode I. The appliedcyclic load ∆KI is determined in terms of load ratioR = Kmin

I /KmaxI and maximum loadKmax

I ,

∆KI = KmaxI −Kmin

I = KmaxI [1−R] . (32)

Under plane strain conditions, the energy release rate,J, and the corresponding cyclic load read

J =1−ν2

EK2

I , (33)

∆J =1−ν2

E

[

KmaxI

]2[

1−R2]

. (34)

Note that forR = 0, ∆J and∆KI can be directly con-verted into each other by use of Eq. (33). The referenceload in normalised form is associated with the fractureenergy densityJ0 = t0δ0Γ0 leading with Eq. (33) to

K0

t0√

δ0=

√

Et0

Γ0

1−ν2 . (35)


Lf

x, ∆ay

Ro

La

x, r, ∆ay

ϕ

ur/ϕ (KI)

a)

b)

Fig. 17 a) Boundary layer model withKI-controlled displace-ment boundary conditions; b) detailed mesh within the finemeshed region, 0≤ x ≤ La (taken fromRoth and Kuna(2013))

The finite element mesh of the boundary layerhalf model and the boundary conditions are depictedin Fig.17. The height of the cohesive zone elementsis just for visualisation purpose and has no influ-ence on the numerical analyses. The presented cyclicCZM was implemented as FORTRAN subroutine us-ing the ABAQUS UEL interface. For detailed infor-mation concerning the finite element formulation seee.g. (Ortiz and Pandolfi, 1999). Within the FE calcu-lations, the ABAQUS UAMP and URDFIL interfaceswere used, too. The latter allows the controlling of timeincrementation, output generation, as well as the sim-ulation abort as described detailed in (Roth and Kuna,2013).

6.2 Fatigue Crack Growth

Fatigue crack growth rate curves cover three charac-teristic stages with increasing load: the near-thresholdstage (I), the Paris region (II), and static failure(III). Within the Paris region a power-law is pre-sumed to relate FCG rates to the applied cyclic load(Paris and Erdogan, 1963),

d(

∆aδ0

)

dN=C

[

∆KI

K0

]m

. (36)

Here, two parameters are introduced: the Paris expo-nent,m, describing the slope of the straight FCG rateline in the double-log plot, and the (normalised) Pariscoefficient,C, which defines its vertical position.

In the following, cyclic load ranges are investi-gated, bounded by the assumed threshold load as wellas the static failure load. Static failure occurs if the ap-plied energy release rateJ equals the fracture energydensityJ0 andKmax

I = K0, respectively. Thus, the ref-erence load forms the upper bound ofKmax

I . The min-imum value ofKmax

I that still leads to damage accu-mulation depends on the endurance properties. As de-scribed above, damage initiation is characterised by thepoint λ = λ e(D = 1), τ = τe

0 at the TSE and thus bythe initial endurance limit. The cohesive potential atthis point (evaluated using Eq. (9)) describes the corre-sponding normalised energy density to be applied fordamage initiation. For this reason,Γinit is presumed toby a threshold value

Γinit(τe0) = Γ

(

D = 1, λ = λ e(D = 1))

. (37)

With Eq. (33) the respective stress intensity factor isfound as

Kinit

t0√

δ0=

√

Et0

Γinit

1−ν2 . (38)

Note that these initiation quantities do neither dependon the shape parametersε , ω , the unloading parameterκ , nor the endurance exponentα .

About 30 FCG analyses were performed to coverthe resulting load rangeKinit(τe

0) ≤ KmaxI < K0 with

fixed parametersε , ω , τe0, α , κ , ρ , andR. Depend-

ing on the load level, each analysis took computingtimes of at least one day up to a week using four CPUs.The parametric study presented in the following wasperformed with an HPC cluster and extensive parallelcomputing. Further details on the reduction of the com-putational costs are found in (Roth and Kuna, 2013).

The postprocessing of the analyses requires a mea-sure to define the crack extension∆a. The increment ofthe crack extension within one load cycle, d∆a, deter-mines the FCG rate. Regarding the centre of the bound-ary layer as the initial crack tip position, one suitableformulation defines the current crack tip position at thelocation where no tractions are transferred any longerand D = 1, respectively (Siegmund, 2004). Becauseof the asymptotic behaviour of the present exponentialcyclic CZM, this seems not to be appropriate. Instead,


00.2

0.40.6

0.8

1

0 20 40 60 80 100

D

x/δ0

N

d∆aδ0

a)

0

0.2

0.40.60.8

1

0 20 40 60 80 100

τ

x/δ0

N

d∆aδ0

b)

Fig. 18 Profiles at the ligament with increasing number of cycles,N, and constant crack extension increment d∆a: a) damage dis-tributions and b) respective effective traction distributions (takenfrom Roth and Kuna(2013))

another damage based definition is applied, here. Thecrack extension is determined by the evaluation of theintegral of the damage variable over the whole liga-ment,

∆a =

∞∫

x=0

D(x)dx . (39)

In Fig.18a) several damage profilesD(x) of subse-quent load cycles are presented. Each profile consistsof two parts, a transient region, 0≤ D < 1, of con-stant width and a fully damaged region,D ≈ 1, whosewidth increases with d∆a = const. per load cycle in-dicating steady-state FCG. Since the damage processis situated at the transient region, it is known as activecohesive zone (Siegmund, 2004). Note that its widthdepends on the particular definition ofD (see Sect.4).Other definitions of∆a (e.g. depending on the locationof the maximum effective traction, see (Hutter, 2013))are possible but not considered here since they haveno effect on the steady-state FCG rate d∆a. Profiles ofdistributions of effective tractions at the ligamentτ(x)are shown in Fig.18b).

In Fig.19 stabilised FCG curves for increasingcyclic load levels are depicted. Independent of the loadlevel, the cracks propagate with constant rates aftersome load cycles. Thereby, the crack extension in-

0

200

400

600

800

1000

0 5 10 15 20 25 30

∆a/δ

0

N

∆KI

Fig. 19 Fatigue crack extension,∆a/δ0, in dependence of thenumber of load cyclesN for increasing cyclic load∆KI (takenfrom Roth and Kuna(2013))

crements increase with higher load levels leading tomonotonically increasing FCG rate curves.

In order to demonstrate the near-threshold be-haviour, FCG curves at low load levels are depictedin Fig.20. It shows that the crack growth retards be-fore a stationary state is reached. The correspondingdependency of the crack extension rate d

(

∆a/δ0)

/dNwith respect to the crack extension is plotted in Fig.21.In the literature, such retardation is attributed to theshort crack effect. Following e.g.Radaj and Vormwald(2007), small cracks retard when they approach mi-crostructural obstacles like grain boundaries or inclu-sions. Besides these general remarks, the ability of thecyclic CZM to predict FCG of microstructurally smallcracks is to be addressed elsewhere. Furthermore, self-arresting cracks are found in Figs.20 and21 for load-levels below a threshold valueKth. In order to explainthis effect and to formulate an applicable criterion forfatigue crack propagation, theJ-integral proposed byRice(1968) is considered,

J =−t0δ0

∞∫

x=0

τ(

λ (x), D(x)) dλ(x)

dxdx . (40)

It equals the applied energy release rate of the far field,provided that the matrix material is elastic. Note thatthe effective tractionτ depends on the current damagestate and thus on the history experienced by the par-ticular material point. Under monotonic loading con-ditions, the fundamental damage variableD equals theeffective separationλ . The integrand in Eq. (40) thenbecomes a total differential and the integration variable


can be changed fromx to λ . This allows the evaluationof Eq. (40) as the area under the traction-separation-relation at the crack tip,

Jmon= t0δ0

λt∫

λ=0

τTSE(λ) dλ (41)

with λt = λ (x = 0) being the opening at the initialcrack tip. In cyclic loading situations,D is an inde-pendent state variable representing the loading his-tory which has to be taken into consideration. Thus,Eq. (40) cannot be simplified to Eq. (41) but has to beevaluated as a line integral along the cohesive zone. Itcan be rewritten as

Jcyc= t0δ0

Γ |x=0+

∞∫

x=0

λ (x)∫

λ=0

∂ τ∂ D

dDdx

dλ dx

, (42)

wherein the first term inside the brackets describe thestored reversible energy at the initial crack tip. Oncethe initial crack tip has failed in the sense of the co-hesive model, i.e.D(x = 0) → 1, this term vanishes.Under constant amplitude loading conditions, steady-state FCG is reached andJcyc is uniquely defined by the(constant) damage profilesD(x) or D(x), respectively.Even more, note that the separation at the crack tip thenis not determined by the current value of the far fieldenergy release rate (i.e.J) but rather by its completehistory. This effect is known as “crack tip shielding” inthe context of plasticity, see (Ritchie, 1988). In fact, thedifferences between Eqs. (41) and (42) arise from thedependence of the cohesive potentialΓ on the damagestate in terms of the fundamental damage variableD.

In general, Eq. (40) can be interpreted as thenormalised area under parametric traction-separation-plots derived fromτ-profiles andλ -profiles at the lig-ament (parameterx). The D-profile is taken into ac-count indirectly via Eq. (15). Consequently, the presentcyclic CZM offers the possibility to predict an arrestof the damaged zone provided that theτ-profiles andλ -profiles are completely covered by the respectiveendurance quantity profiles, i.e. whenever the wholeligament is marked as endurable. In these situations,further damage evolution requires an increase of theapplied cyclic load. This effect is demonstrated inFig.22 where cyclic traction-separation curves are de-picted for increasing cyclic load levels∆KI . In this

0

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100

∆a/δ

0

N

KmaxI

KmaxI < Kth

Fig. 20 FCG curves in the near-threshold region: maximum loadKmax

I < Kth causes crack arrest (non-propagating cracks)

1E-5

1E-4

1E-3

1E-2

1E-2 1E-1

d(∆a

/δ0)/

dN

∆a/δ0

KmaxI

KmaxI < Kth

Fig. 21 FCG behaviour in the near-threshold region: crack ex-tension rate vs. crack extension in dependence of the applied loadlevel; for Kmax

I < Kth cracks arrest

context, cyclic traction-separation curves are paramet-ric traction-separation-plots generated with the help ofprofiles which have been evaluated at maximum ap-plied load and steady-state FCG conditions. As can beseen, theτ-λ -curves approach the damage locus for∆KI → K0. On the other side, the lower limit curverepresenting the threshold value for FCG adhere to theendurance locus. It is found that the cyclic traction-separation curves have to exceed this limit curve toestablish steady-state FCG. Otherwise, the area at theligament where the local endurance limit is not passedshields the damage zone from further evolution. Thedamage zone and thus the micro-crack arrest.

These considerations demonstrate, that both thecyclic traction-separation-curves in theλ -τ-diagramand the respective enclosed area characterise the FCG


0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

τ

λ

∆KI

Fig. 22 Steady-state cyclic traction-separation relation in depen-dence of applied load∆KI; dashed: damage locus; dot-dashed:endurance locus

behaviour. This leads to the following criterion for fa-tigue crack propagation,

Jmax> Jth = t0δ0Γth(τe0, α) , (43)

whereupon cracks propagate whenever the maximumapplied loadJmax exceeds a critical value that dependson the endurance properties of the cyclic CZM. Thiscriterion is valid at least for stress ratios 0≤ R ≤ 1.An equivalent expression in terms of stress intensityfactors follows from Eq. (33):

KmaxI > Kth(τe

0, α) , (44)

with

Kth

t0√

δ0=

√

Et0

Γth

1−ν2 . (45)

This criterion is used to distinguish propagating andself-arresting cracks in Fig.20 and Fig.21. It is worthnoting that the extracted threshold valueKth differsfrom Kinit(τe

0). While the latter defines the load wheredamage initiates, which can be seen as a microstruc-tural threshold value, the former belongs to macro-scopic crack propagation. In contrast toKinit(τe

0), Kth

depends also on the endurance exponentα , which de-fines the shape of the endurance locus.

Summarising the above, there are three energeticquantities affecting the FCG behaviour. Firstly, thedamage initiation threshold (Γinit, Jinit, Kinit) deter-mines the minimum load level to cause damage accu-mulation. Secondly, the lower bound of FCG is definedby the FCG threshold (Γth, Jth, Kth). Last, the static fail-ure load (Γ0, J0, K0) forms an upper bound of FCG. The

τe0

τ

λ

Γinit

Γth

α

Γ0

Fig. 23 Schematic of the energetic quantities (normalised en-ergy densities) characterising FCG behaviour: damage initiationthresholdΓinit(τe

0), FCG thresholdΓth(τe0, α), and static failure

loadΓ0(ε , ω) with Γinit < Γth < Γ0 (solid: damage locus, dashed:endurance locus)

respective energetic quantities,Γinit < Γth < Γ0, are de-picted in Fig.23.

6.3 Parametric Study

The influence of the load ratioR and of the pa-rameters of the cyclic CZM on the FCG rate curves(d(∆a/δ0)/dN vs.∆KI/K0) is investigated. In particu-lar, the effect of different damage loci, endurance locias well as the non-linearity of the unloading behaviouris of interest.

Load Ratio: Fatigue crack growth rate curves for in-creasing load ratios are shown in Fig.24. The regionsI, II (Paris line) and III are recovered. Although theshapes of the curves are not affected by the load ratio,a characteristic horizontal shift of the curves to lowercyclic loads is seen. In the literature, this behaviouris known from crack closure effects. In Fig.25 thesame data are plotted in dependence of the maximumload, Kmax

I using Eq. (32). In this diagram, thresholdand static failure loads of different values ofR coin-cide. Furthermore, there is only a slight decrease ofthe crack extension rate with increasing load ratio. Ac-cording to these results, the influence of the load ratioon the∆KI-curve is well described by the linear rela-tionship Eq. (32).

Shape Parameters: The shape of the TSE and thus ofthe damage locus is characterised by a set of param-


1E-2

1E+0

1E+2

0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

R

Fig. 24 FCG rate in dependence of the normalised applied cyclicload: variation of the load ratioR = {0.0, 0.2, 0.4, 0.6, 0.8} (pa-rametersε = 1, ω = 1, τe

0 = 0.9, α = 5, κ = 0, ρ = 1)

1E-2

1E+0

1E+2

0.5 1.0

d(∆a

/δ0)/

dN

KmaxI /K0

staticfailure

threshold

R

Fig. 25 FCG rate in dependence of the normalised applied maxi-mum load: variation of the load ratioR = {0.0, 0.2, 0.4, 0.6, 0.8}(parametersε = 1, ω = 1, τe

0 = 0.9, α = 5, κ = 0, ρ = 1)

eters:ε , ω , andΓ0 whereupon only two of them areindependent of each other, see Fig.3. For increasingεwith ω = const. the normalised fracture energy den-sity Γ0 decreases. Thus, the relation between constantthreshold load and static failure load increases. Thisis noticeable in the curves depicted in Fig.26. Fur-thermore the slopes of the Paris regions, described bythe Paris exponentm, increases, too, which is a con-sequence of the aforementioned effect. Note that theresults of the inverse parameter variation, i.e. varyingω with fixed ε , show a similar relationship. In Fig.27,the influence of the shape of the damage locus is il-lustrated while keeping the normalised fracture energyconstant. Both shape parameters were chosen accord-ing to Fig.3. Additionally, the endurance exponent,α ,

1E-2

1E+0

1E+2

0.5 1.0

0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

ε

staticfailure

Fig. 26 FCG rate in dependence of the normalised applied cyclicload: variation of the shape parameterε = {0.2, 0.5, 1, 2, 5} (pa-rametersω = 1, α = 5, τe

0 = 0.9, ρ = 1, κ = 0, R = 0)

1E-2

1E+0

1E+2

0.5 1.0

0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

staticfailure

ε , ω, α

J0 = const.

threshold

Fig. 27 FCG rate in dependence of the normalised applied cyclicload: variation of the damage locus,ε = {1, 1.569, 2.46, 3.185},ω = {1, 2, 5, 10}, at constant fracture energy densityJ0 = t0δ0Γ0

and constant threshold∆Kth, α = {5, 22, 12800, 1.95×109} (pa-rametersτe

0 = 0.9, ρ = 1, κ = 0, R = 0)

was adjusted to guarantee a constant thresholdKth. Theresulting FCG rate curves are nearly identical. Thisconfirms the presumption that the shape of the TSEdoes not affect the FCG behaviour significantly for agiven ratio∆Kth/K0.

Endurance Parameters: The influence of the en-durance parameters is investigated by varyingτe

0 andα separately. Both sets of FCG rate curves, depicted inFig.28 and Fig.29, feature a pronounced but counter-directional dependency ofKth with respect to the var-ied parameters. The threshold value increases with in-creasing initial endurance limitτe

0, and decreasing en-durance exponentα . The static failure load remains


1E-2

1E+0

1E+2

0.1 0.5 1.0

0.01 0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

staticfailure

τe0

Fig. 28 FCG rate in dependence of the normalised appliedcyclic load: variation of the initial endurance limitτe

0 ={0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} (parametersε = 1, ω = 1,α = 5, ρ = 1, κ = 0, R = 0)

unchanged due to the constant fracture energy density.Regarding the Paris region, the same argumentationapplies as mentioned above. These results are in-linewith (Roe and Siegmund, 2003). FCG rate curves witha constant threshold load are given in Fig.30. For thispurpose, combinations ofα and τe

0 are identified it-eratively that yield the same ratio∆Kth/K0. The re-sults reveal increasing FCG rates for decreasingτe

0 andaccordingly increasedα . Since this effect is primarilyconcentrated to the near-threshold region, a moderatedecrease of the Paris exponentm is deducible and thusan interrelationship betweenm and the short crack ef-fect explained above. Hence, in contrast to the resultsconcerning the shape of the damage locus, the shapeof the endurance locus significantly affects the FCGbehaviour.

Damage Evolution: The evolution of the damage pa-rameter is controlled by the damage exponentρ . Lowvalues ofρ imply a high damage rate. With increasingρ the damage rate decreases, see Eq. (23). Hence, a lowparameter range of 0.01≤ ρ ≤ 2 was chosen withinthe parametric study to save computational costs. Both,endurance locus and damage locus remain unchangedresulting in constant values of threshold and static fail-ure load. The FCG rate curves are depicted in Fig.31.As expected, there is a pronounced dependency of theFCG rate with respect toρ . The higher the damage ex-ponent is, the lower is the FCG rate. This relation isvalid for the whole load range. Again discussing thisin terms of the Paris power law, a correlation ofρ with

1E-2

1E+0

1E+2

0.5 1.0

0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

staticfailure

α

Fig. 29 FCG rate in dependence of the normalised applied cyclicload: variation of the endurance exponentα = {1, 2, 5, 10} (pa-rametersε = 1, ω = 1, τe

0 = 0.9, ρ = 1, κ = 0, R = 0)

1E-2

1E+0

1E+2

0.5 1.0

0.5 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

staticfailure

α τe0

threshold

Fig. 30 FCG rate in dependence of the normalised applied cyclicload: variation of the endurance locus,α = {1.5, 2.0, 2.8}, τe

0 ={0.7, 0.8, 0.9}, at constant threshold∆Kth (parametersε = 1,ω = 1, ρ = 1, κ = 0, R = 0)

both Paris parameters is obvious. With increasingρ theParis coefficientC decreases whereas the Paris expo-nentm increases, see Fig.33.

Non-linear Unloading: As the last feature of the cyclicCZM, we address the non-linearity of the unloadingpaths controlled by the parameterκ . With −1 ≤ κ ≤0.99 the largest and most reasonable range is covered.Again, to ensure a constant FCG threshold, the en-durance exponentα is adjusted. The behaviour of theFCG rate curves in Fig.32resembles the one in Fig.30where the endurance parameters are varied. Here, theParis exponent increases with increasingκ and de-creasingα , too. Unfortunately, the effect of both pa-rameters are not dividable. Nevertheless, the general


1E-2

1E+0

1E+2

0.5 1.0

0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

staticfailure

ρ

threshold

Fig. 31 FCG rate in dependence of the normalised ap-plied cyclic load: variation of the damage exponentρ ={0.01, 0.05, 0.1, 0.5, 1, 2} (parametersε = 1, ω = 1, τe

0 = 0.9,α = 5, κ = 0, R = 0)

1E-2

1E+0

1E+2

0.5 1.0

0.1 1.0

d(∆a

/δ0)/

dN

∆KI/K0

∆J/ [t0δ0Γ0]

staticfailure

κ α

threshold

Fig. 32 FCG rate in dependence of the normalised applied cyclicload: variation of the unloading parameterκ = {−1, 0, 0.99} atconstant threshold∆Kth, α = {6, 5, 2.3} (parametersε = 1, ω =1, τe

0 = 0.9, ρ = 1, R = 0)

influence of the non-linearity of the unloading pathscan be deduced from the energetic considerations inSect.4. As can be seen from Fig.4 and Fig.11, κ con-trols the amounts of stored energy and dissipation, re-spectively. For a given damage level, high values ofκcorrespond with little dissipation. Since the incrementsof crack extension correlate with the dissipated energy,the FCG rates decrease with increasingκ .

7 Discussion

In this section an identification procedure for the pa-rameters of the cyclic CZM is proposed. The set of pa-rameters is summarised in Tab.1. It consists of nine

independent model parameterst0, δ0, Γ0, ε , κ , τe0, α , ρ

andµ . The remaining shape parameterω is uniquelydefined byΓ0 andε although no closed form solutioncould be given. The values ofω are obtained numer-ically or graphically with the help of Fig.13. How-ever, the FE analyses reveal that the shape of the TSEdoes not have a significant influence on the FCG be-haviour. This finding confirms statements whereuponthe shape of the TSE is no primary feature of a CZM(Tvergaard and Hutchinson, 1992) at least if the ma-trix material is not plastic. According to our investiga-tion, this applies with respect to the considered mode IFCG. Thus, within the parameter identification proce-dure, the shape parameters are set toε = 1 andω = 1leading to the exponential TSE ofXu and Needleman(1993). In the case of plastic matrix material, the shapeof the TSE is supposed to become important, compare(Hutter, 2013; Hutter et al, 2011). As discussed in thesections above, analytical preliminary considerationspredict specific influences of each of the remaining pa-rameters on the FCG behaviour. By means of the para-metric study, these predictions were reproduced. Be-side the qualitative statements, now the specific effectsof the parameters can be quantified. Note that the twoparametersβ andγ associated with the definition of thedamage variable are provided numerically, see Fig.13.

The sequential identification procedure starts withthe evaluation of S-N curves generated from uniaxialtensile test results withR = 0. As discussed in Sect.5and illustrated in Fig.14, the cohesive strengtht0 aswell as the initial endurance limitτe

0 can be identified.While the first is the maximum load under monotonicloading conditions (life timeNf = 1), the latter resultsfrom the respective endurable load level. Details onthe identification of the cohesive strength are found in(Schwalbe et al, 2013).

The remaining parameters are identified with thehelp of FCG experiments also performed with a loadratio of R = 0. First the damage initiation stage isregarded. This is done by microstructural investiga-tions after cyclic loading at load levels below the FCGthreshold. The load level where the evolution of mi-cro defects sets in determinesJinit and, with Eqs. (37)and (38), δ0 is calculated. Sinceδ0 describes the inter-nal length of the cohesive model, its identification on amicrostructural basis seems appropriate.


Table 1 Parameters of the cyclic CZM

parameter description value / identification

t0 cohesive strength S-N curves: maximum load levelδ0 cohesive zone length parameter FCG: damage initiation load(microstructural observation)Γ0 normalised fracture energy density FCG rate curves: staticfailure loadτe

0 initial endurance limit S-N curves: endurable load levelε shape parameter of damage locusε = 1ω shape parameter of damage locusω = 1κ unloading parameter FCG rate curves: Paris exponentmα endurance exponent FCG rate curves: threshold value∆Kth, Paris parametersm, Cρ damage exponent FCG rate curves: Paris parametersm, Cµ penalty parameter µ = 0.001β parameter for damage definition numerical identification, see Fig.13γ parameter for damage definition numerical identification, see Fig.13

The normalised fracture energy density,Γ0, is ob-tained from the asymptotic behaviour of the FCG ratecurves approaching static failure and Eq. (35). Here theexperimental challenge is to ensure small scale yield-ing condition in order to apply theK-concept of linearelastic fracture mechanics. In (Schwalbe et al, 2013)other direct identification methods based on monotonicloading are presented.

The threshold load of FCG rate curves is usedto identify the endurance exponentα . Therefore, theshape of the endurance locus is adjusted with the ini-tial endurance limitτe

0 kept constant. Finally, the Parisregion of FCG rate curves is interpreted. The damageexponentρ and the unloading parameterκ are identi-fied by non-linear optimisation. Note that the unload-ing parameter is basically a monotonic feature of a co-hesive model. That is why only minor effects on thecyclic behaviour are expected.

The parametric study in Sect.6.3 reveals that theFCG threshold and the Paris parameters cannot be setindependently of each other. They are coupled by theshape of the endurance locus via the endurance expo-nentα . The presented identification procedure so farfocuses on the identification of the fracture energy andthe FCG threshold. In Fig.33 the resulting Paris expo-nents of all FCG rate curves of the parametric studyare compared with material data taken fromFleck et al(1994). The reciprocal Paris exponents 1/m are plot-ted versus the toughness ratios∆Kth/K0. In this form,the diagram accounts for all relevant fatigue proper-ties discussed above. As shown there, the results of theparametric study cover an area between engineering al-loys (m ≈ 3. . .5) and engineering ceramics (m > 10).

The particular influences of the model parameters withrespect to∆Kth, K0 andm are reproduced as discussedabove. Although the specific sets of parameters havebeen chosen more or less arbitrary, the results demon-strate the meaning of the parameters and how theyshould be varied to enter a certain material area. Notethat so far only linear elastic matrix material was as-sumed. For metals, the model gives no satisfying re-sults because plasticity is not considered. Plastic dis-sipation significantly contributes to the static fracturetoughness (Tvergaard and Hutchinson, 1992), whichshifts the curves in Fig.33 to lower ratios∆Kth/K0

(left). Furthermore, pure plasticity is associated withm = 2 (crack blunting,Anderson(2005)). This is sup-posed to shift the curves towards lowerm (upwards).In order to apply the model also to metals, the iden-tification of the parameters of the cyclic CZM shouldconcentrate on the near-threshold and the Paris regionof a FCG rate curve. Static failure becomes part of theplastic matrix material modelling. Based on these pre-liminary considerations, it is proposed to fit the FCGregions mentioned by adjustingα , ρ , κ andΓ0, simul-taneously. The application of this modelling and theidentification concept is to be addressed elsewhere.

Further objections concern the fairly rough formu-lations of the normalised effective quantitiesλ andτ , Eqs. (2) and (3). In the present study, only mode Iloading is considered although the cyclic CZM cap-tures also mixed mode loading. Since the underlyingassumptions of the model are analogous for tangentialand normal directions, a successful validation with thehelp of appropriate experiments is expected.


0

0.1

0.2

0.3

0.1 1

50

10

5

4

31/

m m

∆Kth/K0

engineeringalloys

carbonsteels

Alalloys

Nialloys

elastomers

polymerfoams

engineeringceramics

engineeringpolymers

toolsteels

ε = {0.2. . .5}, see Fig.26τe

0 = {0.2. . .0.9}, see Fig.28α = {2. . .10}, see Fig.29constant threshold, see Fig.30ρ = {0.01. . .2}, see Fig.31

Fig. 33 Relation between the reciprocal Paris exponent 1/m and the toughness ratio∆Kth/K0: material contours taken fromFleck et al(1994) compared with numerically obtained results using the cyclic CZM

8 Conclusion

In this study, a new cyclic cohesive zone model (CZM)is developed. Based on fundamental considerations,the cyclic CZM is equipped with a number of neces-sary constituents: the shape of the traction-separationenvelope (TSE), an assumption concerning the unload-ing behaviour, the choice of scalar effective quantitiesand an appropriate damage variable as well as its evo-lution equation. The latter also comprises an endurancelimit approach, which is proposed as a damage statedependent field variable in the present cohesive model.Within the mathematical description, an exponentialTSE and simple power-law approaches for both en-durance limit and damage evolution equation are cho-sen. This model yields a completely smooth traction-separation relation under all loading conditions.

Finite element (FE) analyses are performed usingthis cyclic CZM. 1 With simulations of uniaxial andbiaxial homogeneous fatigue tests, S-N curves andconstant-life plots are generated. The fatigue crackgrowth (FCG) behaviour is investigated using a bound-ary layer model of a crack underKI-controlled plainstrain loading condition. By means of FE calculations,the influence of the parameters of the cyclic CZM onthe FCG behaviour could be analysed. The shape ofthe damage locus is found to have no significant in-fluence. Instead, the FCG rate curves are characterisedby three energetic quantities. A microscopic threshold,i.e. the damage initiation stage, is controlled byΓinit,which depends on the initial endurance limit. In con-trast, the macroscopic FCG thresholdΓth depends onthe shape of the endurance locus. Between micro andmacro threshold, the model predicts (self-arresting)short crack growth. Last, the area under the TSE equals

1 The respective ABAQUS UEL subroutine and visualisation utili-ties (Roth et al, 2012) are available on request from the authors.


the normalised fracture energy densityΓ0 describingthe static failure load in FCG rate curves.

The characteristic sigmoidal shape of FCG ratecurves in log-log plots is predicted. The Paris regionwas found to depend, again, on the endurance formu-lation as well as on the non-linearity of the unload-ing behaviour. Thus, the presented results justify theintroduction of the state dependent endurance limit.The sensitivity study has revealed the physical mean-ing of the parameters of the CZM especially those ofthe cyclic formulation.

With the help of these results, finally, a potentialparameter identification procedure is proposed. Thedevelopment of a reliable concept requires further in-vestigations in collaboration with experimentalists.

The capabilities of the cyclic CZM to predict over-load and sequence effects as well as non-linear damageaccumulation should be mentioned. A respective studyis to be published elsewhere. The same holds for theapplication of the model with regard to the simulationof short crack effects and transient crack formation.

Acknowledgements This work was performed within the Clus-ter of Excellence “Structure Design of Novel High-PerformanceMaterials via Atomic Design and Defect Engineering (ADDE)”that is financially supported by the European Union (Europeanregional development fund) and by the Ministry of Science andArt of Saxony (SMWK).

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Simulation of fatigue crack growth with a cyclic cohesive