a phase field modeling approach of cyclic fatigue crack growthspeciﬁcally, s is driven by a proper...

Int J Fract (2020) 225:89–100https://doi.org/10.1007/s10704-020-00468-w

ORIGINAL PAPER

A phase field modeling approach of cyclic fatigue crack growth

Christoph Schreiber · Charlotte Kuhn ·Ralf Müller · Tarek Zohdi

Received: 24 July 2019 / Accepted: 30 June 2020 / Published online: 17 July 2020© The Author(s) 2020

Abstract Phase field modeling of fracture has beenin the focus of research for over a decade now. Thefield has gained attention properly due to its benefitingfeatures for the numerical simulations even for com-plex crack problems. The frameworkwas so far appliedto quasi static and dynamic fracture for brittle as wellas for ductile materials with isotropic and also withanisotropic fracture resistance. However, fracture dueto cyclic mechanical fatigue, which is a very impor-tant phenomenon regarding a safe, durable and alsoeconomical design of structures, is considered onlyrecently in terms of phase field modeling.While in firstphase field models the material’s fracture toughnessbecomes degraded to simulate fatigue crack growth, wepresent an alternative method within this work, wherethe driving force for the fatigue mechanism increasesdue to cyclic loading. This new contribution is gov-erned by the evolution of fatigue damage, which canbe approximated by a linear law, namely the Miner’srule, for damage accumulation. The proposed model isable to predict nucleation as well as growth of a fatiguecrack. Furthermore, by an assessment of crack growthrates obtained from several numerical simulations by

C. Schreiber (B) · R. MüllerTechnische Universität Kaiserslautern, Kaiserslautern,Germanye-mail: [email protected]

C. KuhnUniversity of Stuttgart, Stuttgart, Germany

T. ZohdiUniversity of California, Berkeley, Berkeley, CA, USA

a conventional approach for the description of fatiguecrack growth, it is shown that the presented model isable to predict realistic behavior.

Keywords Phase field · Cyclic fatigue · Crackgrowth · Finite elements

1 Introduction

The field of cyclic mechanical fatigue is one of themost important branches of post and current researchin engineering. Catastrophic failures of different typesof technical structures like trains, airplanes, bridges,turbine rotors, pressure vessels and others, occurredas consequence of the fatigue of materials. Accord-ingly, this phenomenon must be kept in the focus ofpresent studies. From a material science perspective,the fatigue mechanism principles are mostly under-stood and explained in several textbooks by e.g. Dowl-ing (2013), Schijve (2009), Haibach (2006). The totalfatigue life of a structure is generally divided into twophases. The crack nucleation phase describes the periodin which a microcrack is initiated as a consequenceof irreversible cyclic slip in grains with proper orien-tation and the growth of this microcrack to an orderwhere it is visible with the naked eye or at least by uti-lizing a low rate magnifying glass. The second periodcontains the propagation of the macrocrack until finalfailure. For both of these periods different phenomeno-logical prediction frameworks exist. The total life time

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90 C. Schreiber et al.

approach is mainly based on so-called Wöhler curvesor S-N curves, respectively. These curves assign a cer-tain number of bearable load cycles to a certain loadamplitude. The particular cycle numbers may serve asinput for damage accumulation laws, where the lin-ear accumulation hypothesis referred to as the Minerrule (Miner 1945) is widely applied. The most pop-ular method to describe macrocrack growth is Paris’law (Paris and Erdogan 1963). Within this method thecyclic crack growth rate da/dN is described as a func-tion of the cyclic stress intensity factor�K , which basi-cally incorporates effects of external load, actual cracklength and geometry. It was found that, at least withinthe region of sub critical crack extension, this functionplotted on a double logarithmic scale is a straight line.Accordingly the crack growth rate is approximated bya power law. Extensions of this law were proposed toinclude effects like mean stress or stress ratio (see e.g.Dowling 2013). Paris’ law is also the basis for the mostwidely used numerical tool for fatigue crack growthsimulations namely NASGRO (Forman et al. 2005).

Essential studies attempt to model cyclic fatiguefailure using the framework of Continuum DamageMechanics (CDM), which was comprehensively out-lined by Lemaitre (1992) to describe the evolution of adamage variable, were presented by e.g.Marigo (1985)or Chow and Wei (1991). However, within these mod-els, the fatigue life is modeled only for the period untilcrack nucleation. A numerical model for the simula-tion of fatigue crack growth was introduced by Fishand Oskay (2005). The damage evolution within thiswork is modeled by the growth of the void volumefraction, which is governed by extensions of Gurson’sfailure model. To obtain a crack or its sharp interfaces,respectively, finite elements are deleted once a criticalvoid volume fraction is reached.

As an alternative to conventional crack growth simu-lation techniques, phase field modeling for fracture hasgained attention within the past decade. This methodprovides a very useful tool for the numerical simula-tion of problems dealing with sharp interfaces, as itdoes not require procedures for mesh disconnection,element deletion, and re-meshing for an explicit mod-eling of the interfaces. The phase at a certain locationis incorporated by introducing additional degrees offreedom. In the context of fracture, the framework wasapplied to dynamic fracture (see e.g. Schlüter et al.2014), and quasi static brittle fracture in e.g. Kuhnand Müller (2010); Borden et al. (2014); Miehe et al.

(2010). Extensions of thesemodel exist for ductile frac-ture (e.g. Kuhn et al. 2016; Borden et al. 2016) and alsoeffects like anisotropic fracture resistance were stud-ied by e.g. Teichtmeister et al. (2017), Schreiber et al.(2017), Hakim and Karma (2009). Models for fatiguedamage were so far presented by Alessi et al. (2017)and Seiler et al. (2018). In these models the phase fieldis driven by degrading the energy release rate once adamage parameter increases.

In this work, the fatigue mechanisms occurring in amaterial are interpreted as additional driving force con-tribution, contribution that originates from cyclicallyaccumulated deformation work. The formulation is anextension of the model proposed by Kuhn and Müller(2010) for brittle fracture and as a linear elastic mate-rial model is considered within our work, the range ofvalidity in terms of the number of cycles to failure NFmay be restricted to high cycle fatigue, which coversNF ≥ 1000 for many metallic materials. The drivingforce for the phase field variable s(x, t) incorporates anadditional energy contribution accounting for fatiguedamage accumulation due to an increasing number ofload cycles.

2 Phase field model for fatigue fracture

2.1 The regularized model for brittle fracture

The core idea of a phase field fracture model is to intro-duce an additional field variable to represent cracks.A continuous transition of this field variable s(x, t)within the interval [0, 1] approximates the crack faces.The crack field is 1 as long as the material remainsundamaged and drops once fracturing occurs, wherethe lower limit s = 0 represents cracks. The phasefield model from Kuhn andMüller (2010) uses the reg-ularized formulation of a variational model for brit-tle fracture (Bourdin et al. 2000). As an extension ofGriffith’s theory, which states that a crack propagatesonce the energy release rate reaches a certain criticalvalue, it is postulated that the displacement field u andthe fracture field s locally minimize the total energy Ewithin a loaded body.

E(∇u, s, ∇s) =∫

�

ψ(∇u, s, ∇s) dV

=∫

�

[(g(s) + η)W (∇u) + �(s,∇s)] dV .(1)

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A phase field modeling approach of cyclic fatigue crack growth 91

In Eq. (1) W is the elastic strain energy density, whichis multiplied by a degradation function g(s) to accountfor the stiffness reduction in cracked regions. The sur-face energy density contribution � accounts for theenergy required for the formation of a crack surface.The parameter η shall be chosen 0 < η � 1 to avoidnumerical problems within the static solution scheme.The elastic strain energy density

W = 12ε(u)TCε(u) (2)

can be formulated in terms of the infinitesimal straintensor

ε(u) = 12

(∇u + (∇u)T

), (3)

where ∇u is the spacial gradient of the displacements.In Eq. (2) the Voigt notation for symmetric tensors isused for the linearized strain tensor ε and the stiffnesstensor C . The second energy contribution in Eq. (1) isthe fracture energy density

�(s,∇s) = Gc(

(1 − s)24�

+ �|∇s|2)

, (4)

which besides a local part, further consists of a non-local part, characterized by the spacial gradient of s.The two contributions are multiplied by the criticalenergy release rate Gc. The length scale � controls thewidth of the transition zone between undamaged andbroken material. This crack surface contribution reg-ularizes Griffith’s theory, which proposes the propor-tionality of the fracture energy and the crack surface.Assuming a hyperelastic material, the stresses can bederived from the potential ψ by

σ = ∂ψ∂ε

= (g(s) + η)Cε. (5)

Different types of degradation functions g(s) are pro-posed in the literature (Bourdin et al. 2000; Kuhn et al.2015; Borden et al. 2016). However, this function hasto fulfill several criteria as g(0) = 0, g(1) = 1 andalso g′(0) = 0. These requirements are satisfied by thequadratic function

gb(s) = s2, (6)

which is used in the original regularized formulationproposed by Bourdin et al. (2000). Discussions abouta quadratic and a cubic degradation function are givenin Kuhn et al. (2015). These functions were shown toshift the stress level at which crack initiation occurs tohigher stresses and also to conserve the linear elasticstress strain behavior until crack initiation. A mixedformulation for the degradation function

gβ(s) = β(s3 − s2) + 3s2 − 2s3 (7)

was proposed by Borden et al. (2016). Notice, that forβ = 2, gβ becomes gb and for β → 0, gβ becomes thecubic degradation function introduced by Kuhn et al.(2015). Accordingly, the model behavior is effected byvarying the parameter β.

A generalized Ginzburg-Landau equation (Gurtin1996) is used to obtain the time evolution of the phasefield s via

ṡ = −M{

∂ψ

∂s− ∇ ·

(∂ψ

∂∇s)}

︸︷︷︸δψ/δs

= −M2

{g′(s)εTCε − Gc

(4��s + 1 − s

�

)} (8)

where � is the Laplace operator. The scalar mobilityparameter M in Eq. (8) acts as a viscous control param-eter to approach the stationary limit, which is specifiedby δψ

δs = 0. The limit case M → ∞ approximatesquasi static conditions.

2.2 Phase field formulation for cyclic fatigue

To provide a correct understanding of the modeldescribed in the following, it is important to commenton the interpretation of the phase field variable s. Inparticular the transition zone between the two phasesbroken and intact (0 < s < 1) requires special consid-eration. A possible interpretation is to connect s to thefraction of micro voids within a representative volumeelement, as it was proposed byVoyiadjis andMozaffari(2013). Here, s is brought in direct connection with thedamage variable from the CDM framework (Lemaitre1992). On the other hand, the phase field variable inBourdin’s original model (Bourdin et al. 2000) did nothave a physical interpretation at all, but the scalar phasefield s was rather incorporated in the energy formula-

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tion of a fractured body to enable a regularization of thecrack energy. According to this explanation, the mean-ing of s results from the phase field philosophy itself,which is to approximate sharp interfaces by a smoothtransition of an order parameter for numerical reasons.In the phase field model presented within this work,we follow this second interpretation. The term damagerefers to fatigue damage, which will be applied to drivethe phase field, but it is not explicitly connected to it.Specifically, s is driven by a proper estimate for thefatigue damage increment dDf occurring in the mate-rial within a certain cycle interval.

2.2.1 Phase field fracture model for fatigue failure

As explained in Sect. 2.1, a sequence of local mini-mizers of the total internal energy, depending on dis-placement and fracture fields, has to be found in orderto obtain the correct crack pattern, or in other wordsinformation for crack initiation, kinking or branch-ing, respectively. Suppose a sample is exposed to amonotonically increasing load. In that case the solu-tion reveals an evolution of the crack field, since atsome point the surface part of Eq. (1) will increaseas it becomes energetically more favorable to form acrack, which means to decrease s, than to allow formore strain energy. Within the regime of cyclic fatiguethe load maximums are rather small to very small com-pared to quasi static fracture loads. Presuming such asmall load and a constant Gc, the solution will not yielda decline of s in case of the classical model (Kuhn andMüller 2010), as energy consumed by irreversible pro-cesses associated with an increasing state of fatigueis not taken into account. Accordingly, crack growthwould be too costly with respect to the total internalenergy. A modification of the total energy formulationhas been proposed by Alessi et al. (2017) in order toenable crack growth driven by cyclic fatigue. Themod-ification basically applies to the crack energy

∫�

� dV ,where the fracture resistance Gc is degraded by a func-tion of a plastic strain variable, which is accumulatedover an increasing number of load cycles.

Within this work we present an alternative approachfor a regularized formulation of the total energy

E f(∇u, s,∇s, Df) = E + Eac

=∫

�

[(g(s) + η)W + �] dV +∫

�

h(s)P(Df) dV,

(9)

where an additional energy density contribution P isintroduced to account for the sum of additional driv-ing forces associated with the mechanism of cyclicmechanical fatigue. The function h(s) degrades P ina similar way as W is degraded by g(s) once a crackpropagates. The energy contribution Eac is governedby a piece-wise defined function

P(Df) ={0 for Df < Dfcq(Df − Dfc)b for Df ≥ Dfc, (10)

with the threshold value Dfc. According to this defini-tion the amount of additional energy to enter Eq. (9)remains zero until a certain critical value Dfc ofthe accumulated damage Df is reached and increasesrapidly (governed by the model parameters q and b)after this limit is exceeded. The actual accumulatedfatigue damage is in general described as

Df = D0 + dDf(σa, σ̄ , R, ε̇, D0), (11)

which is the sum of the previous damage state D0 andthe damage increment dDf as a function of the ampli-tude σa , the mean value of the stress σ̄ , the load ratio R,the stain rate ε̇, and the damage history or the previousdamage state, respectively.Themain task in this contextis to find an accurate formula for the increment dDf.Equation (11) indicates that, generally, several effectshave to be considered for an estimate of the damageincrement. However, the choice has to be made alsoin consideration of computational effort, which a cer-tain model may cause. Straightforward approaches forthe fatigue lifetime estimation of a component, whichare commonly used and were also incorporated withinapproaches for numerical fatigue estimations (see e.g.Bograd et al. 2008) are Miner’s rule (Miner 1945) andthe fatigue damage model proposed by Chaboche andLesne (1988). The main difference between these twoapproaches is that Miner’s rule does not consider theeffect of the current state of damage D0 on the incre-ment, which is formulated as

dDfi =1

NFidN , (12)

with the number of bearable load cycles NFi and thecycle increment dN . Considering Eq. (12) it becomesclear that the damage accumulation is linear andaccordingly the influence of load sequence on the

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Fig. 1 Schematic illustration of a S-N line

fatigue life is not cough. Within this work we primar-ily want to give a principal description of the proposedmodification of the phase field model. Hence, only theMiner rule was so far incorporated for the evaluationof the damage increment, as it provides an easy andvery stable way of implementation. However, it shallbe outlined at this point, that other approaches, as e.g.the model for fatigue damage from Chaboche, couldalso be incorporated within the presented framework.

The particular critical cycle number NFi can beobtained by an appropriate S-N curve. These curves(see Fig. 1) are plots, obtained within fatigue exper-iments, of the stress magnitude for cyclic loadingdepending on the corresponding number of bearableload cycles NF . A straight line is obtained using a dou-ble logarithmic scale.

With Eq. (12) and an appropriate S-N curve, the totalinternal Energy E f of a component that may containcracks caused by quasi static or by cyclic loading isthen

E f =∫

�

(g(s) + η)12εTCε dV

+∫

�

Gc(

(1 − s)24�

+ �|∇s|2)

dV

+∫

�

h(s)q

〈D0 + dN

nD

(σA

AD

)k− Dc

〉bdV,

(13)

with the driving force quantity σA, which is relatedto a stress amplitude. This variable will be explainedwithin the next section. Other quantities introduced inEq. (13) are the fatigue limit AD/2, the knee point cyclenumber nD and the slope factor k of the S-N curve. The

〈·〉n denote Macauley brackets with the definition

〈·〉n ={0 for (·) ≤ 0(·)n for (·) > 0. (14)

Considering the energy in Eq. (13) the stresses become

σ = (g(s) + η)Cε︸︷︷︸σ e

+ h(s)qb〈Df − Dfc〉(b−1) ∂Df∂ε︸︷︷︸

σ ac

.

(15)

The first term in Eq. (15) is the usual degraded stresstensor and the second term may be considered an addi-tional stress component accounting for accumulatedmicro stresses as consequence of the fatigue mecha-nism.

2.2.2 Cycle resolved simulation scheme

Depending on the magnitude of the load cycles, thefatigue mechanism may require a very large number ofcycles before consequences on a macro level like cracknucleation or extension can be observed. Accordingly,an efficient scheme is required to integrate large num-bers of cycleswith respect to a characteristic cycle num-ber, otherwise simulation times cannot be kept withina reasonable limit. Even for conventional fatigue lifeapproaches certain amounts of similar cycles are sum-marized to blocks, where the connection of all theseblocks represents thewhole loadhistory called load col-lective. Fish and Yu (2002) proposed a scheme called“cycle jump”, where the damage at the end of a simula-tion step including a certain number of cycles D(i+�Ni )is calculated via

D(i+�Ni ) = Di + �Ni · �Di . (16)

The damage at the previous step is Di and the particular�Di values are obtained by an explicit Euler method.Within this simulation scheme the block size summa-rizing several cycles, �Ni is chosen adaptive.The time-cycle transfer scheme presented within thiswork is quite similar. Suppose a constant block size ornumber of cycles per time �N

�t , respectively, leads to acertain evolution of damage. By utilizing Eq. (12), thedamage of the actual time is

Di = Di−1 + dDfMiner(�ti ). (17)

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The phase field evolution Eq. (8) can be transferredto the cycle domain using the chain rule. Accordingly,the evolution of the phase field can be reformulated interms of cycles

ds

dN= − M̂

{g′(s)1

2εTCε + h′(s)P(Df)

−Gc(2��s − s − 1

2�

)},

(18)

with M̂ denoted the mobility in the cycle domain.The damage will be updated within every simulationstep according to the change caused by a chosen cyclenumber. To ensure good convergence even for a rapidincrease of P , adaptive step size adjustment must beimplemented.

3 Numerical implementation

For the evaluation of the enhancement of this phasefield model for fatigue failure the quadratic degrada-tion function from Eq. (6) is chosen for g(s) and h(s).If volume forces are neglected within the mechanicalequilibrium and the modified evolution equation forthe phase field Eq. (18) is utilized, the governing set ofdifferential equations for the unknown displacementsu and the order parameter s is given by[∇· 00 1

M̂ddN + 2(W + P) + Gc2� − Gc2��

][σ(u)s

]=

[0Gc2�

](19)

where ∇· is used as divergence operator. Multiplyingboth equations with the respective virtual quantities δuand δs and integration by parts, the particular weakforms are obtained

0 = −∫

�

σ T · δε dV +∫

∂�t

δuT · t∗ dS (20)

with the prescribed traction vector t∗ on ∂�t and

0 = −∫

�

δs M̂−1 dsdN

dV

−∫

�

δs

(sεTCε + 2P(Df) + Gc s − 1

2�

)dV

−∫

�

2Gc�∇sT∇δs dV,(21)

where the homogeneous Neumann boundary condition∇s · n = 0 at ∂� is used. The isoparametric conceptis used in order to obtain a discrete description of thefield quantities u and s bymeans of the shape functionsHI for the nodes I = 1, . . . , n with

u(x) =n∑

I=1HI (x)ûI (22)

and

s(x) =n∑

I=1HI (x)ŝI , (23)

where the denotation ˆ(·) indicates nodal values. Byintroducing the differential operator matrices for a twodimensional setting

BuI (x) =⎡⎣H(x)I,x 00 H(x)I,yH(x)I,y H(x)I,x

⎤⎦ (24)

and

BsI =[H(x)I,xH(x)I,y

], (25)

where (·),v indicates partial derivatives ∂(·)∂v , the valuesfor ε and ∇s can be computed from

ε(x) =n∑

I=1BuI (x)ûI (26)

and

∇s(x) =n∑

I=1BsI (x)ŝI , (27)

where the Voigt notation is used. An analog discretiza-tion may be applied for the virtual quantities. Accord-ingly the contributions to the vector of the internalforces at node I become

FuI =∫

�

(BuI

)Tσ dV (28)

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and

FsI =∫

�

HI M̂−1 ds

dNdV

+∫

�

HI

(sεTCε + 2sP(Df) + Gc s − 1

2�

)dV

+∫

�

2Gc�(BsI

)T ∇s dV .(29)

In the following,we denote byσA(ε) any representativecomponent of the stress tensor or property of the stresstensor, chosen as the driving quantity for the fatiguemechanism. Thiswork aims at investigating fundamen-tals of the integration of fatigue failure within a phasefield formulation in terms of mode-I fracture. Accord-ingly, the normal stress component in loading direction

σA = 1TCε, (30)

with 1 = (0, 1, 0)T is considered as appropriate stressmeasure for cyclic tensile loading in vertical direc-tion. In order to find a solution U∗ = (u∗, s∗, ṡ∗)Tof the nonlinear system, the Newton-Raphson schemeis applied. The particular components of the currentstiffness matrix are found as

K uuI J =∂FuI∂uJ

=∫

�

BuI (s2 + η)CBuJ

+ s2BuI ( f1 + f2)C1 [C1]T BuJ dV, (31)K suI J =

∂FsI∂uJ

=∫

�

HI2s (Cε + f3 · C1)T BuJ dV,(32)

K usI J =∂FuI∂sJ

=∫

�

(BuI

)T 2sHJ (Cε + f3 · C1) dV,(33)

K ssI J =∂FsI∂sJ

=∫

�

HI HJ

(εTCε + 2P + Gc

2�

)

+ 2Gc�(BsI

)T BsJ dV (34)with

f1 = ∂2P(Di−1 + �Di )

∂D2i·(kσ (k−1)AnD AkD

dN

)2, (35)

f2 = ∂P(Di−1 + �Di )∂Di

· k(k − 1)σ(k−2)A

nD AkDdN , (36)

Fig. 2 Set up for phase field simulation of C(T)-specimen

f3 = ∂P(Di−1 + �Di )∂Di

· kσ(k−1)A

nD AkDdN (37)

and the damping matrix is zero, except for the compo-nent

DsṡI J =∂FsI∂s,N

=∫

�

HI HJ M̂−1 dV . (38)

Using the implicit Euler method for the transient prob-lem is sufficient as the number of cycles is directly pro-portional to time. However, automatic step size adjust-ment is applied to ensure a stable computation whenthe phase field decreases due to rapid increase of thefatigue damage.

4 Examples and validation

For an experimental characterization of the fatiguecrack growth behavior of materials the so-called com-pact tension (CT) specimen is, among others, fre-quently used. Accordingly, this specimen geometrywas also chosen for the conducted simulations withinthis study, whose results are presented in the following.The definition of the geometry of the C(T)-specimen isgiven in the ASTM E 399 standard (ASTM 2009). Forthe simulations the software FEAP 8.4 was used. Themodel was implemented into a user element routinefor a 4-node quadrilateral element. Figure 2 shows the

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Fig. 3 Schematic illustration of the approximation of cyclicloading within simulation

setup for the mode-I simulations. For the simulations,force was used as control variable. The force trans-mission by a bolt was approximated by a distributedsinusoidal load across the upper semi-circle of the bolthole. The pulsating load is approximated within thetransient simulations by means of a ramp function,where each simulation step represents a certain incre-ment of load cycles (seeFig. 3).As only constant ampli-tude loading was considered the applied force was keptconstant after the maximum load was reached. For allnumerical experiments a material with Young’s mod-ulus E = 2.1 · 105 MPa, Poisson’s ratio ν = 0.3 andcritical energy release rate Gc = 2.33 · 103 J/m2 wasconsidered. Further, the parameters AD = 200MPaand k = 5 were used.

4.1 Crack initiation and model characteristics

Themain feature of the original phase field formulationfor brittle fracture is a crack pattern that locally mini-mizes the total energy. The solution is allowed to yieldcracks and cracks will appear once their formation isbeneficial with respect to the total energy. However, aspreviously explained, the modified phase field modelfor fatigue fracture has an additionalmechanism,whichincludes a fatigue related driving force. The plots inFig. 4 show the evaluation of the stress contributionsσ e and σ ac of Eq. (15) in vertical direction on a line,which extends in horizontal direction from the notchtip of the C(T)-specimen. Furthermore, the effectivestress in vertical direction σA is indicated. For simplic-ity all stresses are normalized by the threshold stressAD . A corresponding contour plot of the phase fields within a square section (edge length approximately1mm) around the notch tip is indicated in every illus-tration. Note that for this simulation the initial s field

was 1 over the whole specimen. In other words, thespecimen was not pre-cracked and the crack initiationperiod will be illustrated in the following. The plot inFig. 4a shows the situation after the initial ramp up,i.e. complete load applied. The phase field remains 1indicating intact material everywhere. The stress at thenotch tip is slightly above AD . This state stays unal-tered until a number of approximately 1.4 · 105 cycleswas applied (Fig. 4b). At this point an increase of theσ ac contribution can be recognized, which indicatesthat the damage at the notch tip has grown to a rele-vant amount such that the energy density contributionP significantly affects the energy functional. Conse-quently the phase field decreases and the stresses aredegraded. After another 0.3 · 105 cycles the phase fieldat the notch tip is almost zero (Fig. 4c) andσA increases,while σ ey decreases due to the degradation. This obser-vation illustrates that σA rather than σ ey is a suitabledriving force, as otherwise the fatigue damage wouldstop to grow. A fatigue crack was initiated after a cyclenumber of approximately 2.0·105 (Fig. 4d). Both stresscontributions are zero at the crack tip as they are fullydegraded. Note that for the subsequent evolution ofthe energy density contribution P a higher stress isobserved as driving force as for the crack nucleation.This will certainly lead to a smaller number of cyclesuntil the next crack growth increment occurs, comparedto the interval for crack initiation. This behavior is inagreementwith experimental findings (see e.g.Haibach2006).

In order to assess the crack growth behavior, anothersimulation was set up and ran until a fatigue crack witha total length of 15mm was generated. Figure 5 showsthe results obtained from this simulation by means ofdifferent illustrations. The plot on the right shows thecrack evolution with respect to the number of appliedload cycles. For selected N-a states, contour plots, rep-resenting the phase field variable s over the specimen,are indicated among the line plot. The crack length ismeasured from the tip of the V-notch to the actual cracktip (maximum x-coordinate with s = 0). A nonlinearfunction of the crack length with respect to the numberof applied load cycles is obtained from the simulation.The steep gradient at the end of the simulation indicateshigh crack growth rates. This behavior is qualitativelyin line with experimental findings (see e.g. Dowling2013). An illustration of the particular energy contri-butions of the total energy (Eq. (9)) with respect to thenumber of load cycles for the same simulation is shown

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(a) (b)

(c) (d)

Fig. 4 Evaluation of stress contributions on a horizontal straight line at the crack tip after a 0 cycles, b 1.4 · 105 cycles, c 1.66 · 105cycles, and d 2.0 · 105 cycles

in the left diagram of Fig. 5. Note that for this simula-tion the node at the crack tip was initially set to s = 0.This explains the small deviation of EΓ from zero atthe beginning. However, this evaluation provides goodinsight into the mechanism of the proposed phase fieldmodel. The onset of crack growth is indicated in theplot by the first increase of the crack energy contribu-tion E� . It may be recognized, that the energies Eac

representing the fatigue energy and EW representingstrain energy are small compared to E� . Accordinglyfor the illustration these contributions were factorizedby 10. The fatigue energy Eac increases as consequenceof the rising fatigue damage induced by cyclic loading.Accordingly, s decreases, which yields an increase ofE� and a decrease of Eac to compensate the additionalcrack energy andminimize the total energy. The energycontribution Eac increases subsequentlywithin anotherarea according to the actual crack pattern and the pro-cess starts again. The strain energy contribution EW isalmost constant for the larger part of the simulation and

increases with higher rates only after approximately2.5 · 105 cycles due to higher stresses with increasingcrack length.

4.2 Evaluation of crack growth rates

The growth behavior of a macro crack, which isexposed to cyclic loads is still mainly described by the arelation between the growth rate da/dN and the cyclicstress intensity factor�K , which represents the proper-ties of crack length, loading conditions, and geometry.This law, denoted as Paris’ law, was first presented inParis and Erdogan (1963) and is given by

da

dN= C (�K )m , (39)

were C and m are material dependent parameters. In adouble logarithmic diagram one obtains the power law

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Fig. 5 Illustration of crack growth data from phase field simulation as a plot of the different energy contributions (left) and as a N-aplot (right)

of Eq. (39) as a straight line. Within common exper-imental procedures for the identification of the crackgrowth behavior, a discrete form of the growth rate

(da

dN

)p

≈ ap − ap−1Np − Np−1 (40)

is utilized with p denoting the p-th data point. Forthe approximation of the �K values associated to theC(T)-specimen we stuck to Srawley (1976), who rec-ommends:

�K = �FB

√L

2 + aL(1 − aL

)3/2 ·[0.886 + 4.64 ( aL

)2 − 13.32 ( aL)3

+14.72 ( aL)4 − 5.69 ( aL

)5],

(41)

where the actual crack length a is considered as thehorizontal distance from the center of the holes to thecrack tip (see Fig. 2). According to this, for anotherassessment of the proposed phase field model severalsimulations with different maximum force values forthe introduced mode-I test of the C(T)-specimen were

performed. A fatigue crack was generated within everysimulation and driven to amaximum length of approxi-mately 6mm. Figure 6 summarizes the results of thesesimulations. Within a double logarithmic plot of thecrack growth rates with respect to the stress inten-sity factor all the data points lie close to a straightline. This straight line, which confirms Eq. (39) with

C = 2.8 ·10−7 mm/cycle(MPa

√m)m andm = 2.5 was estimated

by means of a least square fit and reveals a coefficientof determination of 0.996. The associated N-a plots arepresented in Fig. 6a. According to the different levelsof the maximum load for each simulation, large differ-ences are obtained for the number of cycles required togenerate a certain crack length. However, all samplesfit to the Paris parameters of Fig. 6a, which is indeed avery important property regarding a comparison withexperimental findings.

Concluding remarks

A phase field model, which is able to cover thephenomenon of cyclic mechanical fatigue, has beenpresented as enhancement of the model for brittle

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(a) (b)

Fig. 6 Evaluated crack growth data from four different phase field simulations as a N-a plot and b double logarithmic diagram of crackgrowth rates vs. ΔK factors

fracture from Kuhn and Müller (2010). The driv-ing force for the fatigue mechanism was realized bya new energy contribution accounting for the irre-versibility of mechanical fatigue. Numerical simu-lations show that our model generally yields rea-sonable results in terms of fatigue crack nucleationand growth and therefore describes the phenomenon.Simulated crack growth rates confirm the Paris law,which represents experimental findings. However, asfatigue of materials is a rather complex field, whichis effected by many quantities like load ratio, loadsequence, strain rate, temperature, environment and notforgetting the class of the material, further enhance-ment of the current phase field model has to bedone in order to accomplish a comprehensive formula-tion.

Acknowledgements Open Access funding provided by Pro-jekt DEAL. Funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation)-2524083 85-IRTG 2057

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https://doi.org/10.1002/pamm.201800207https://doi.org/10.1002/pamm.201800207

A phase field modeling approach of cyclic fatigue crack growthAbstract1 Introduction2 Phase field model for fatigue fracture2.1 The regularized model for brittle fracture2.2 Phase field formulation for cyclic fatigue2.2.1 Phase field fracture model for fatigue failure2.2.2 Cycle resolved simulation scheme

3 Numerical implementation4 Examples and validation4.1 Crack initiation and model characteristics4.2 Evaluation of crack growth rates

Concluding remarksAcknowledgementsReferences

a phase field modeling approach of cyclic fatigue crack growthspeciﬁcally, s is driven by a proper...

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