inﬂuence of the loading path on fatigue crack growth under...

Int J FractDOI 10.1007/s10704-009-9396-6

ORIGINAL PAPER

Influence of the loading path on fatigue crack growth undermixed-mode loading

V. Doquet · M. Abbadi · Q. H. Bui · A. Pons

Received: 24 April 2009 / Accepted: 2 September 2009© Springer Science+Business Media B.V. 2009

Abstract Fatigue crack growth tests were performedunder various mixed-mode loading paths, on maragingsteel. The effective loading paths were computed byfinite element simulations, in which asperity-inducedcrack closure and friction were modelled. Applicationof fatigue criteria for tension or shear-dominated fail-ure after elastic–plastic computations of stresses andstrains, ahead of the crack tip, yielded predictions ofthe crack paths, assuming that the crack would propa-gate in the direction which maximises its growth rate.This approach appears successful in most cases con-sidered herein.

Keywords Mixed-mode · Non-proportionalloading · Fatigue crack · Bifurcation

1 Introduction

Under non-proportional cyclic loading, the stress orstrain ranges were shown not to be sufficient to modelthe multiaxial cyclic behaviour of metals (Benallaland Marquis 1987; Doquet and Pineau 1990; Feaugasand Clavel 1997) and parameters describing the load-ing path had to be introduced into constitutive equa-tions to capture extra-hardening or softening effects

V. Doquet (B) · M. Abbadi · Q. H. Bui · A. PonsLaboratoire de Mécanique des Solides, EcolePolytechnique, CNRS, Palaiseau cedex 91128, Francee-mail: [email protected]: http://www.lms.polytechnique.fr

(Benallal and Marquis 1987). Concerning fatigue crackgrowth under non-proportional mixed-mode, a sim-ilar question arises: are �KI, �KII (or �KIII) andR ratios sufficient to predict crack paths and growthrates? Does the loading path have an intrinsic influ-ence, or can all this influence be captured throughappropriate corrections for closure and friction effectson stress intensity factors, that is, by the use of�Keffective

I ,�KeffectiveII ,�Keffective

III ?This problem is complex and no clear answer

emerges from the literature on non-proportional mixed-mode I and II (Gao et al. 1983; Hourlier et al. 1985;Wong et al. 1996, 2000; Planck and Kuhn 1999; Yu andAbel 2000; Doquet and Pommier 2004) or mode I andIII (Feng et al. 2006). The latter concludes: “with identi-cal loading magnitudes in the axial and torsional direc-tions, the crack growth and crack profiles are stronglydependent on the loading path. The observed resultscreate a great challenge to the crack growth theoriesfor general loading conditions”.

At least three types of interactions between mode Iand mode II can be distinguished.

1.1 Kinematic interactions between the crack faces

Mode I reduces crack faces contact and friction andthus tends to increase �Keffective

II (Yu and Abel 1999),whereas static or intermittent mode II produces a rel-ative shift of the crack faces thus increasing asper-ity-induced closure and reducing �Keffective

I (Stanzlet al. 1989; Dahlin and Olsson 2003). But, on the other

123

V. Doquet et al.

hand, pure mode II loading on a rough crack gener-ates a cyclic mode I, because gliding asperities wedgethe crack open. Several analytical models (Tong et al.1995; Yu and Abel 1999; Bian et al. 2006) were devel-oped to describe the latter effect, but the assumptionof rigid asperities leads to overestimate the induced KI

and to underestimate �KeffectiveII . Elastic–plastic finite

element simulations taking into account the contact andfriction of crack face asperities can be used to overcomethis limitation, as shown below.

On the other hand, shear-mode loading superim-posed on mode I may have an indirect influence onplasticity-induced closure, because it modifies the plas-tic flow at the crack tip.

1.2 Plastic interactions

Plastic interactions concern plastic flow ahead of thecrack tip. An example is given on Fig. 1a. Consideringthe yield surface and plastic flow of the material aheadof the crack tip, a static tensile stress along the ligamentdue to applied mode I should lower the yield stress inshear and thus enhance cyclic shear-mode plasticity.This is confirmed by the finite element computationsof the shear strain range ahead of the crack tip, shownon Fig. 1b. It should also promote tensile plastic flowand ratchetting, at least for a transient period (i.e. untilthe center of the yield surface has been shifted to pointE by kinematic hardening). This may cause an intrinsicacceleration of mode II crack growth, that is: not dueto an increase in�Keffective

II . Similarly, an applied shearstress associated to present or past mode II loading canenhance tensile plastic flow compared to pure mode I.Similar effects were also shown to occur due to resid-ual tensile or shear stresses, in sequential mixed-modeloading (Doquet and Pommier 2004). The latter effectmight thus also increase plasticity-induced closure.

A possible influence of non-proportional loadingahead of the crack tip on the constitutive equations ofthe material—extra cyclic hardening in some single-phase alloys (Benallal and Marquis 1987; Doquet andPineau 1990), or extra cyclic softening in some tita-nium alloys (Feaugas and Clavel 1997)—also has to bekept in mind.

1.3 Damage interactions

Damage interactions concern fatigue damage aheadof the crack tip. Gao et al. (1983) argue that since

C

Shear stress(a)

Tensile stress

Loading pathwith a static KI

E

Elastic domain of materialahead of the crack tip

BD

Loading pathin cyclic mode II

(b)

2,0E-02

2,5E-02

3,0E-02

pure mode IIwith with with

KI= 5KI=10KI=15

5 0E 03

1,0E-02

1,5E-02

shea

r st

rain

ran

ge0,0E+00

5,0E-03

-180 -135 -90 -45 0 45 90 135 180

angle (degrees)

A

Fig. 1 Illustrations of “plastic interactions” between mode I andmode II. a Schematic loading path, initial yield domain (VonMises criterion) and plastic flow for a small volume element ata crack tip loaded either in reversed mode II or reversed mode II+ static mode I. The shift of the yield surface in the direction ofplastic flow due to kinematic hardening at point C and D has notbeen represented, for sake of clarity. b Angular distribution ofthe shear strain range 20µm ahead of the crack tip, computed byFEM, for reversed mode II, �KII = 20MPa

√m, with various

static KI, in maraging steel

dislocation glide and damage induced by non-propor-tional mixed-mode loading ahead of a crack tip aredistributed over a wide range of slip planes orientation,whereas it is concentrated, under proportional loading,the plastic strain in the maximum shear direction willbe lower in the first case, for equivalent�KI,�KII, sothat the crack growth rate would be intrinsically lowerunder non-proportional loading.

However, it is generally accepted that shear-initiateddecohesion along a slip band in fatigue is triggered byan opening stress and that the presence of an openingstress can—to a certain degree—compensate a reduc-tion in shear stress or strain range.

Findley (1957) took this effect into account in acrack initiation criterion where the damage function:

βFind = �τ + kσn,max (1)

123

Influence of the loading path on fatigue crack growth under mixed-mode loading

includes the peak opening stress σn,max, computedalong the facet which undergoes the maximum shearstress range, �τ . The constant k determines the“weight” of the tensile stress for the material of inter-est. If mode II fatigue crack growth—observed when�Keffective

II is above a material-dependent threshold—isdue to shear-initiated fatigue fracture of grains locatedin the crack plane ahead of the tip, then, according toEq. 1, this fracture should occur earlier when an open-ing stress due mode I loading is present. and lead tofaster crack growth, if no bifurcation occurs.

The relative importance of these three types of inter-actions is likely to depend on the microstructure, andmechanical properties of the material, as well as onthe loading conditions: R ratio, frequency and envi-ronment. A coarse microstructure, a high friction coef-ficient, or a small or negative mode I R ratio shouldpromote kinematic interactions, while the importanceof plastic interactions should depend on the yield stressand type of hardening i.e. isotropic, linear or non lin-ear kinematic hardening. The importance of damageinteractions should be higher in materials which exhibita substantial difference in their fatigue resistance inpush–pull and torsion (which results in a high value ofthe coefficient k in Eq. 1).

The present investigation is based on crack growthtests under various mixed-mode loading paths on mar-aging steel and observation of crack paths and growthrates. It also includes elastic–plastic finite element sim-ulations in which asperity-induced closure and frictionare modeled. It provides some elements about thesethree types of interactions. Moreover, application offatigue criteria for tension or shear-dominated fail-ure after elastic–plastic computations of stresses andstrains cycles, ahead of the crack tip—assuming thatthe crack propagates in the direction which maximisesits growth rate—yields predictions of the crack pathsand growth rates that correspond well to experimentalobservations.

2 Experiments

2.1 Procedures

The material investigated is a maraging steel (Z2NKD18-8-5), for which kinetic data concerning mode IIfatigue crack growth are available from a previousstudy (Pinna and Doquet 1999). It has a very high

yield stress (Rp0.2 ≈ 1720 MPa ) but very low harden-ing capacity (Rm/Rp0.2 ≈ 1.03) and limited ductility(around 8%). Tubular specimens (10.8 and 9 mm outerand inner diameters) were used for push–pull, reversedtorsion and reversed torsion plus static tension tests, inorder to fit constitutive equations and two fatigue crite-ria: one for shear-dominated failure—which occurredsystematically in torsion—(Findley’s criterion (Find-ley 1957)) and one for tension-dominated failure whichoccurred in push–pull (Smith, Watson and Topper’s cri-terion (Smith et al. 1970). This preliminary part of thestudy is reported in Appendix A.

Some tubular specimens had a circular hole (370µmin diameter) from which a 1 to 1.5 mm-long transverseprecrack was grown in mode I. Square shaped goldmicrogrids with a 4µm pitch were laid on some spec-imens, along the precrack plane. The grids were usedto measure crack face relative opening and sliding dis-placements by digital image correlation, using sequen-tial SEM images taken during load cycles applied in situby a tension–torsion machine (Bertolino and Doquet2009).

The pre-cracked specimens were submitted to ten-sion and torsion cyclic loadings, following variousloading paths (Table 1). These loadings included var-ious blocks with constant nominal �KI and �KII orconstant static KI or KII. Loading E and F, which bothappear as truncated ellipses in KI − KII plane, cor-respond to 90◦ out-of-phase tension and shear, but inthe first case, loading is fully reversed, while in thesecond case, only shear-mode loading is reversed, butR=0 for mode I. Loading G, represented as an inclinedtruncated ellipse in KI − KII plane corresponds to 30◦out-of-phase push–pull and torsion.

2.2 Nominal and effective loading paths

KnominalI ,Knominal

II were computed for a crack emanat-ing from a hole in a infinite plate, but corrections wereapplied to take into account the influence of the curva-ture of the tube wall, which, according to Erdogan andRatwani (1972) increases as the crack grows.

Two-dimensional plane stress finite element simu-lations of applied loadings were performed with roughcrack faces (Fig. 2), taking into account the contactand friction of the asperities, according to Coulomb’slaw. The Cast3 M code was used, with nearly 40,000four nodes quadrangular and three nodes triangular

123

V. Doquet et al.

Table 1 Loading parameters and bifurcation angles

Loading path ref Loading parameters Measured bifurcation angle Predicted bifurcation angle

KI

KII

KI

A �KII = 20 MPa√

m, KI = 7.5 MPa√

m 0 0

�KII = 20 MPa√

m,KI = 10 MPa√

m 0 0

�KII = 20 MPa√

m,KI = 13 MPa√

m 0 0

KI

KII

I

B �KI = 15 MPa√

m,KII = 10 MPa√

m < 4◦ 0

KI

KIIC1 � KI = 5.6 MPa

√m,� KII = 20 MPa

√m 0 0

� KI = 11 MPa√

m,� KII = 20 MPa√

m 0 0

� KI = 16.4 MPa√


m 0 0

� KI = 21 MPa√


m 0 0

� KI = 26 MPa√

m ,� KII = 20 MPa√

m < 4◦ 0

KI

KIIC2 � KI = 16 MPa

√m,� KII = 30 MPa

√m 0 0

� KI = 23.7 MPa√


m 0

� KI = 32 MPa√


m <4◦ 0

K

KI

KII D � KI = 10 MPa√

m ,� KII = 20 MPa√

m 0◦ for 80µm then 30◦ 0

KII

KI

E � KI = 7.5 MPa√


m 0 0

� KI = 10 MPa√


m 0 0

K

KII

KI

F � KI = 7.8 MPa√

m,�KII = 20 MPa√

m 0 0

�KI = 10.6 MPa√


m 0 0

�KI = 16 MPa√


m 50 0

KI

KIIG �KI = 5.3 MPa

√m,�KII = 10.6 MPa

√m 38 54

123


Fig. 2 Finite elementmodel for the computationof effective loading paths.a Mesh and boundaryconditions. b, c Deformedmesh for two levels ofapplied shear-mode loading.Asperities wedge the crackopen

Fig. 3 Outcome of finiteelement simulations of arough crack loaded in modeII. a Effective loading pathfor R= −1,b asperity-induced �KI as afunction of �Knominal

II , forR= −1,c �Keffective

II /�KnominalII as a

function of R,d asperity-inducedKImax and �KI as afunction of R

IIno

min

al

(c)

am

)

(a)10

5

IIef

fect

ive /

KI 0.6

1

0.4

0.2IIef

fect

ive (M

P

0

-5

)R

K

-1 -0.5 0 0.5 10

KIeffective (MPa m)

0 0.5 1 1.5 2-10

(MPa

m) 2.5

2

1.56

8

10

KImax

DKItive(M

Pam

KImaxKI

KK

Ieffe

ctiv

e 1

0.5

0 2

4

ectiv

e ,K

Ieffe

c

KIInominal (MPa m)

0 5 10 15 20 250

R

KIm

axef

f

-1 -0.5 0 0.5 1

(d)(b)

0.8

elements. A sinus profile with amplitude h—rangingfrom 5 to 20µm—and period p—around 180µm—wasassumed, which is simplified, but reasonable comparedto the observed precrack profiles.

The effective stress intensity factors, KeffectiveI ,

KeffectiveII , were computed at each time step (by analysis

of crack faces displacements). Plasticity-induced clo-sure—probably limited in such a high-strength steel—was not taken into account.

Simulations of pure mode II loading were performedfor a rough crack, with h = 5, 10 and 15µm and fric-tion coefficients, µ, between 0 and 1, for �Knominal

IIranging from 5 to 25 MPa

√m and R ratios from −1

to 0.7. Figure 3a shows an example of computed effec-tive loading path—which appears non-proportional—with some asperity-induced mode I, rising linearly with�Knominal

II , for a given crack roughness profile (Fig. 3b).The computed effective loading path is rather similarto that obtained with an analytical model—assumingrigid asperities—by Yu and Abel (1999).�Keffective

II /�KnominalII was found to be independent

of �KnominalII in the range simulated, but decreasing

with R, when the latter is positive, as shown on Fig. 3c.This curve also is similar to that obtained by Yu andAbel and consistent with some experimental observa-tions on Al alloys, by Planck and Kuhn (1999) who

123

V. Doquet et al.

Fig. 4 Influence of crackroughness and frictioncoefficient on (a) �Keffective

Iand (b) �Keffective

II inreversed mode II

(a) (b)1

0.8

0.6KIIn

om

inal

h = 5 micronsh = 10 micronsh = 15 microns

0.2

e /K

IIno

min

al

h= 5 micronsh=10 micronsh=15 microns

0.4

0.2

0

KIIef

fect

ive /0.1

KIef

fect

iv

friction coefficient

0 0.2 0.4 0.6 0.8 1

friction coefficient

0 0.2 0.4 0.6 0.8 10.0

Fig. 5 Computedevolutions of effective stressintensity factors: (a) KI forloading B and (b)KI and KII for loading D

10

(MP

am

) (b)

8

10

m)

(a)

ectiv

e

0

5

20 60 100 140 180

e, K

IIeffe

ctiv

e KI

2

4

6

ffec

tive

(MP

a KII = 5KII = 10KII = 15

KIef

fe

-10

-5

KIef

fect

iv

KIILoading D

00 50 100 150

KIe

Time step (s) Time step (s)

Loading B

Fig. 6 Mutual influence ofmode I and mode II on theeffective fraction of �K ofthe other mode (h= 10µm,p= 180µm) (a)�KII = 20 MPa

√m and

(b) �KI = 10 MPa√

m

inal

(a) (b)1.2

1min

al1.2

90° out-of-phasemixed-mode + static mode Iiv

e /K

IIno

m

0.8

0.6ct

ive /

KIn

o1

0.8

0.6

0 2 4 6 8 10 12 14 16

reversed mixed-modesequenceMode II + static mode IK

IIeffe

ct

0 5 10 15 20 25 30

mixed-mode + static mode I

reversed mixed-mode or 90° out of phase

mode I + static mode II

0.4

0.2

0

KIef

fe

0.4

0.2

0

KInominal (MPa m) KII

nominal (MPa m)

report no coplanar crack propagation for R values �0.6, independently from�Knominal

II ,while for lower R-ratios, mode II crack growth was observed, when�KII

was high enough.The plateau on this curve, as well as on the curve

showing the asperity-induced KI (Fig. 3d), for Rbetween −1 and 0 is also consistent with the absenceof any measurable difference in mode II crack growthkinetics for R=0 or R=−1 reported by Otsuka et al.(1987) on Al alloys or Pinna and Doquet (1999) onmaraging steel. Note that for R > 0, the maximuminduced KI is predicted to increase substantially, butthe amplitude �KI to decrease.

The higher the friction coefficient, µ, the smallerthe asperity-induced �KI and �Keffective

II (Fig. 4). Asit could be expected, Mode I induced by asperities ispredicted to increase and �Keffective

II to decrease withthe height h of the asperities.

Figure 5 shows the evolutions in time of effectivestress intensity factors computed for loading B (cyclicmode I plus static mode II ) and D (in-phase mixed-mode plus static mode I), with h= 10µm and fric-tion coefficient µ = 1. Important asperity-inducedclosure—increasing with KII—was found for cyclicmode I plus static mode II. Closure effects were foundless important for mixed-mode plus static mode I andabsent for all other investigated loading paths. Fig-ure 6 shows the mutual influence of each mode on theeffective stress intensity factor of the other mode for�KI = 10 MPa

√m and K�KII = 20 MPa

√m.

A static mode I was found to reduce or even suppresscrack faces interactions and to increase�Keffective

II . Forh= 10µm and µ = 1,KI = 5 MPa

√m is predicted to

be sufficient for�KeffectiveII to reach�Knominal

II . This isconsistent with the measured load-sliding displacementloops measured in the SEM, at various distance behind

123


Fig. 7 Nominal KII versussliding displacementmeasured by SEM digitalimage correlation at variousdistance behind the crack tipfor (a) pure reversed modeII and (b) reversed mode II+ static KI = 7.5 MPa

√m

1212(a) (b)

MPa

m)

0

4

8

x = 3 micronsx= 10 micronsx = 25 micronsx = 71 microns

K effective0

4

8x = 3 micronsx = 10 micronsx = 25 micronsx = 45 micronsx = 120 micronsx = 192 microns(M

Pam

)

K effective

KII

nom

inal

(

-8

-4

x = 109 micronsx = 181 microns

KIIeffective

-8

-4

KII

nom

inal

KIIeffective

Sliding displacement ( m)

-12-12

Sliding displacement ( m)

-2.5 -1.5 -0.5 0.5 1.5 2.5 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

the crack tip, using digital image correlation (Fig. 7).Figure 7a, obtained in pure mode II, shows verticalsegments—denoting a variation in KII without slidingof the crack locked by asperities and friction —aftereach reversal. The effective part of the cycle—i.e. thepart during which sliding occurs—can be deduced fromthese loops and �Keffective

II evaluated as 15 MPa√

m.By contrast, the loops of Fig. 7b, obtained with thesame �Knominal

II = 21 MPa√

m but with a superim-posed static KI = 7.5 MPa

√m do not exhibit such

vertical segments, which indicates that �KeffectiveII =

�KnominalII .

In-phase push–pull and reversed shear (R= −1) wasfound to enhance crack faces interactions and to reduce�Keffective

II , compared to pure mode II (�KeffectiveII =

11.4 MPa√

m for �KII = 20 MPa√

m and �KI =10 MPa

√m).

By contrast, out-of-phase push–pull and shear wasalso found to reduce crack faces interactions andto increase �Keffective

II compared to in-phase push–pull or pure mode II, but to a lesser extent thana static KI. Figure 8 shows the computed evolu-tion of �Keffective

II /�KnominalII and effective R ratio

for mode II (that is: R = KeffectiveIImax /Keffective

IImin ) withthe phase angle for �KII = 20 MPa

√m,�KI =

5 or 10 MPa√

m, h = 10µm and µ = 1. As the phaseangle increases from 0 to 90◦, the tensile part of thecycle provides less and less assistance to the forwardslip of crack faces, but the compressive stage providesless and less resistance to their backward slip. Thisresults in an increase of�Keffective

II and a decrease of theeffective mode II R ratio. By contrast, �Keffective

I ≈�Knominal

I for any phase angle.Very limited influence of mode I on �Keffective

II wasfound here for sequential loading, but simulations per-formed for �KI = 15 MPa

√m�KII = 20 MPa

√m

for a much softer ferritic-pearlitic steel in a previous

1

DKI=10

DKI=5

/K

IInom

inal

KI=10MPa m

KI= 5MPa m

0.8

0.6

0.4

phase angle (degrees)K

IIeffe

ctiv

e

0.2

0

DKI=10

DKI 5

(b)

KI=10MPa m

0

-0.2

DKI=5KI= 5MPa m-0.4

-0.6

-0.8

1

0 30 60 90

0 30 60 90

phase angle (degrees)

effe

ctiv

e R

fo

r m

od

e II

(a)

Fig. 8 Computed evolution of (a)�KeffectiveII /�Knominal

II and (b)effective R ratio for mode II with the phase angle, for out-of-phasepush–pull and reversed torsion, for �KII = 20 MPa

√m, h =

10µm, p = 180µm

study (Pinna and Doquet 1999) have shown that forsuch loading, the residual crack tip opening increasesfrom cycle to cycle. This effect, which would prob-ably be observed in maraging steel at higher �KI islikely to reduce crack faces interference and increase�Keffective

II .Note that for similar nominal loading ranges,�KI =

10 MPa√

m and �KII = 20 MPa√

m, variations inthe loading path lead to variations in �Keffective

I from7.9 (path B) to 10 MPa

√m (path E or F) and in

�KeffectiveII from 11.4 (path D) to 20 MPa

√m (path

A)! Such large differences in effective loading arelikely to produce differences in crack paths and growthrates.

123

V. Doquet et al.

2.3 Observations

Table 1 summarises the observations, concerning thecrack paths. Pictures of crack paths after loading C1, Dand F are shown on Fig. 9.

Coplanar growth over more than 700µm wasobserved for loading A—reversed mode II, with�KII = 20 MPa

√m and an increasing static KI. No

deceleration was observed, contrary to what would hap-pen in pure mode II, for the same�Knominal

II (Pinna andDoquet 1999). This is partly due to a higher�Keffective

II ,since it was shown above that KI = 7.5 MPa

√m

is enough to suppress crack face interference. Butthis acceleration of shear-mode crack growth by astatic mode I cannot be reduced to a KI-inducedincrease in �Keffective

II since compared to the growthrate for pure mode II with �Keffective

II = 20 MPa√

mdeduced from an interpolation of Pinna’s data (Pinnaand Doquet 1999), the growth rate measured dur-ing the first block, with the smallest KI is alreadyfive times higher and even increases slightly withKI : 8.2510−7, 8.6810−7 and 8.7410−7 respectively,for KI = 7.5, 10 and 13 MPa

√m.

This effect can thus be attributed to a large extent toplastic and damage interaction ahead of the crack tip.

For loading B—cyclic mode I plus static mode II—the crack deviation did not exceed 4◦ and the growthrate was not significantly different from the growth ratemeasured for the same�KI during precracking in puremode I.

Crack growth was nearly coplanar for sequentialloadings C1and C2. The slight deflections observedduring the final blocks of these tests seem not to bedue to the applied mixed-mode, but to incipient shear-lips, due to the relatively high �KI. Figure 10 showsthe evolution of measured coplanar crack growth rateswith �KI for test C1and C2. For such sequential load-ing, it may seem reasonable to predict the coplanarcrack growth rate by a simple sum:

da

d N(sequence) ≈ da

d N(�K effective

I )

+ da

d N(�K effective

I I ) (2)

Those predictions—based on the values of �KeffectiveI

and �KeffectiveII computed for sequential loading with

h= 10µ m and µ = 1 and the experimental mode Iand mode II Paris laws—are plotted for comparisonon Fig. 10. Equation 2 appears to underestimate the

measured crack growth rates by a factor of 3 to 5.5. Asimilar synergetic effect was reported in Wong et al.(1996, 2000) and Doquet and Pommier (2004) for railsteel.

Loading D—in phase mixed-mode plus static modeI—produced 80µm coplanar crack growth, followedby bifurcation at 30◦.

Coplanar crack growth was observed for path E (90◦out-of-phase, but R=0 for mode I) and during thefirst and second block of fully reversed, 90◦ out-of-phase loading F. Bifurcation at 50◦ occurred during thethird block. For loading F, a large quantity of frettingdebris appeared along the rough precrack, but muchless along the smoother coplanar crack part, grown inmixed-mode. Much less fretting debris was formed forpath E and the fracture surfaces were not mated, con-trary to the previous case.

Loading G—fully reversed, 30◦ out-of-phase push–pull and torsion—led to bifurcation at 38◦. The crackwas covered by fretting debris that had to be removedto allow observations.

These fretting debris are thus specially abundant formixed-mode loadings with a compressive stage whileshearing is being applied.

3 Analysis

Elastic–plastic FE simulations of applied loadingswere performed for rough crack faces, using constitu-tive equations with isotropic and non-linear kinematichardening fitted to measured stress–strain curves (seeappendix B). Findley’s damage function was computedahead of the crack tip and averaged along the radialsegment of length l (l = 20µm in the computationsreported below) for which the shear stress range wasmaximum, whereas for the computation and averagingof Smith-Watson-Topper’s damage function, two possi-ble choices of critical plane were considered: the planesalong which either the peak normal stress, σn,max, orthe normal stress range �σn were maximum. Thesetwo cases, which can yield very different predictions,as shown below, will be denoted by SWTa and SWTb.These two potential growth directions were consideredby Hourlier et al. (1985) and Dahlin and Olsson (2003).The former concluded that the direction of maximum�σn was more suitable for alloys showing a limitedinfluence of the R ratio on their mode I kinetics, which isthe case of the maraging steel investigated here (da/dN

123


Fig. 9 Crack paths for (a) sequential mixed-mode loading (C1) (b) in-phase mixed-mode + static mode I (D) (c) fully reversed 90◦out-of-phase mixed-mode (F)

Fig. 10 Evolution ofcoplanar crack growth rateswith �KI in sequentialloading for (a)�KII = 20 MPa

√m (test

C1) and (b)�KII = 20 MPa

√m (test

C2). Predictions based on asimple sum of crack growthrates in mode I and II areplotted for comparison

8,E-07

6,E-07

4,E-07

2,E-07

0,E+00

le)

1,5E-06(a) (b)

dN

(m/c

yc

6,0E-07

9,0E-07

1,2E-06

dN

(m

/cyc

le)

sequentiasequential

superposition

sequentiasequential

superposition

5 10 15 20 25 30

da/

m)

0,0E+00

3,0E-07

15 20 25 30 35

da/

K MPa mKI (MPa I ( )

is at maximum 2.5 times higher for R=0.7 than forR=0, for a given �KI). The latter concluded that thedirection of maximum σn,max should be preferred forhigh-strength metals with limited ductility, which isalso the case of the maraging steel investigated here!

The analysis of the crack path for cyclic mode I + sta-tic mode II (test B) is very useful to solve that dilemma.For these loading conditions, the peak σn,max occurs at62◦ while the maximum�σn is at 0◦, whatever the sta-tic mode is. Since no bifurcation occurred, the crackfollowed the maximum �σn plane. Thus SWTb crite-rion was used for tension-dominated failure and Find-ley’s criterion, for shear-dominated failure.

The potential numbers of cycles to failure N f (l) arecomputed, using the analytical relations between dam-age functions and fatigue lives, fitted to experimen-tal data on smooth specimens (see Appendix A). Thecrack is supposed to grow in the direction where fail-ure—be it tension or shear-dominated—occurs first or,in other words, in the direction where its growth rate ismaximum. It is consistent with experimental observa-tions (Fig. 11) suggesting transient simultaneous copla-nar growth and branch crack growth until the slow-est crack gets arrested, due to increasing shielding bythe other crack growing faster. It generalises the ideaof Hourlier et al. (1985) who promoted it, under the

123

V. Doquet et al.

Fig. 11 Zoom on the crack tips after loading D (a) and G (b)suggesting transient simultaneous coplanar growth and branchcrack growth

restrictive assumption that mode I would necessary pre-vail. The potential growth rates in corresponding direc-tions are then estimated as:

da

d N≈ l

N f (�)(3)

3.1 Prediction of crack path and growth rate underfully reversed mode II loading

Figure 12 compares the potential crack growth ratein fully reversed mode II predicted by Findley’s cri-terion and that of a branch crack, in mode I, accord-ing to SWTb for various amplitudes. These computa-tions were performed for a smooth, frictionless crack.Coplanar growth is predicted for�Keffective

II higher than15 MPa

√m, while bifurcation is predicted below this

threshold value, which is consistent with the observa-tions and measurements by Pinna and Doquet (1999).Most popular bifurcation criteria (based on the maxi-mum tangential stress, minimum strain energy density,minimum radius of the plastic zone and many others)

N (m

/cyc

le)

branch crack, mode I

main crack mode II

10-6

10-7

10-8

10 100

da/

d

main crack, mode II

fit of experimental Mode II data

KIIeffective (MPa m)

10-9

10-10

Fig. 12 Potential crack growth rate in reversed mode II pre-dicted by Findley’s criterion and that of a branch crack, in modeI, according to SWTb

would not predict this transition in the crack growthmode as �KII increases and would predict bifurca-tion whatever the loading range. The predicted modeII crack growth rate is correct above 25 MPa

√m, but

a bit too small below that value.The computations of �Keffective

II for reversed modeII on a rough crack reported on Fig. 4 show that�Keffective

II decreases with h and µ, so that for �KII =20 MPa

√m, coplanar growth is predicted for h=5

and 10µm, whatever the friction coefficient, and forh=20µm, only if µ is smaller than 0.44, while bifur-cation is predicted above this value. Also, �Keffective

IIis predicted to decrease for positive R ratios (Fig. 3c),so that for �KII = 20 MPa

√m, coplanar growth is

predicted from R = −1 to 0 and bifurcation if R > 0.Slight changes in crack roughness or tribological

conditions are thus likely to change the crack paths.Mode II crack growth is favoured by smooth crack face,a low friction coefficient and reversed loading.

3.2 Prediction of crack paths in non-proportionalmixed-mode

The predicted crack paths for mixed-mode loadings Ato G reported in Table 1 were obtained for h = 10µmand µ = 1

3.2.1 Reversed mode II plus static mode I

For loading A, coplanar crack growth is predicted, inaccordance with the experiment. Figure 13 comparesthe predicted and measured coplanar crack growth ratesfor various static KI, normalized by the growth ratein pure mode II, for �Keffective

II = 20 MPa√

m. Theorder of magnitude of KI-induced acceleration is wellpredicted, but tends to be exagerated, as KI increases.

123


8

4

6

aliz

ed d

a/d

N

predicted

0

2

no

rma predicted

observed

0 5 10 15

Static KI (MPa m)

Fig. 13 Comparison of predicted and measured influence of astatic mode I on the coplanar crack growth rate, for�Keffective

II =20 MPa

√m

Increasing KII

10-9

da/

dN

(m

/cyc

le)

with static mode II

pure mode I

10-10

10-11

101 100

KIeffective(MPa m)

10-12

Fig. 14 Influence of a static mode II on the coplanar crackgrowth rate in Mode I, as predicted by SWTb criterion

3.2.2 Cyclic mode I (R=0) plus static mode II

As explained above, the SWTb criterion predicts copla-nar growth in that case, in accordance with our obser-vations during test B. The coplanar crack growth rate ispredicted to decrease as the static KII increases, due toasperity-induced decrease of �Keffective

I (see Fig. 6a).However, as shown on Fig. 14, on which the computedgrowth rate is plotted versus �Keffective

I , the decelera-tion is not as important as the decrease in �Keffective

Iwould suggest: an intrinsic acceleration of crack growthby the static mode II is predicted. This is again an exam-ple of plastic interactions: an increase in the �εn termof the SWT criterion, caused by the static shear stressalong the ligament associated with KII.

It is in accordance with experimental results ofStanzl et al. (1989) who report, for 13Cr steel, mode Icrack growth acceleration by a static mode II, for rel-atively short cracks (i.e. with few asperities) or afterremoval of asperities by a fine saw-cut, but decelera-tion for long cracks (many asperities, so that the KII-

9,0E-07

1,2E-06

1,5E-06

m/c

ycle

)

sequential

superposition

local approach

0,0E+00

3,0E-07

6,0E-07

da/d

N (

KI (MPa m)

15 20 25 30 35

Fig. 15 Improvement of the crack growth rate prediction by thelocal approach, for the sequential test C2, compared to a simplesuperposition (Eq. 1)

induced decrease in �KeffectiveI becomes predominant

over the intrinsic acceleration). The observations madeby You and Lee (1997) on a mild steel suggest the sameeffects: acceleration of mode I crack growth by a sta-tic mode II, if R > 0.2 (no crack face interaction) butdeceleration for lower R ratios.

3.2.3 Sequential mixed-mode loading

Coplanar growth is predicted for sequential loadingsC1 and C2, in accordance with experimental observa-tions. In addition, the local approach described above,which integrates both plastic and damage interactionsimproves the prediction of crack growth rates, as com-pared to the simple superposition of Eq. 2, as illustratedfor test C2 on Fig. 15.

3.2.4 In-phase mixed-mode plus static mode I

For test D, the potential crack growth rates predictedby Findley and SWTb criteria are quite close, so that aslight change in crack roughness changes the predictedcrack path: for h = 10µm, coplanar crack growth ispredicted, while for h = 15µm, bifurcation shouldoccur. The fact that coplanar growth was first observedover 80µm before bifurcation might be the result ofsuch an instable situation.

3.2.5 Out-of-phase mixed-mode loading

Simulations of out-of-phase push–pull and reversedtorsion were performed for �KII = 10.6 MPa

√m,

�KI = 5.3 MPa√

m and for �KII = 20 MPa√

m,�KI = 10 MPa

√m. Figure 16a and b compares the

evolutions of predicted potential tensile and shear-mode crack growth rates with the phase angle and

123

V. Doquet et al.

Fig. 16c shows the corresponding bifurcation angles(similar for both amplitudes). For the smallest loadingrange, bifurcation in the direction experiencing max-imum opening stress range is predicted, whatever thephase angle, with a bifurcation angle increasing from54◦, for in-phase loading, to 64◦ for 90 out-of-phaseloading. Note that except for in-phase loading, thebranch crack does not undergo pure mode I, since puremode I or mode II do not exist when loading is non-proportional. For the highest loading range, a transitiontowards coplanar growth is predicted for 90◦ out-of-phase loading. These predictions are consistent withthe results of test E (�KII = 20 MPa

√m,�KI =

10 MPa√

m, 90◦ out-of-phase, coplanar growth) andG (�KII = 10.6 MPa

√m,�KI = 5.3 MPa

√m, 30◦

out-of-phase, bifurcation). In the latter case, the mea-sured bifurcation angle (38◦) was however signifi-cantly smaller than predicted (54◦).

Simulations were performed for �KII = 20 MPa√m and �KI = 5, 10 or 15 MPa

√m or path E (fully

reversed), as well as for path F (R=0 for mode I). All ofthem predicted coplanar crack growth, with a growthrate increasing with �KI. The points correspondingto these computed growth rates fall near the predictedmode II kinetics on a da/dN versus �Keffective

II plot. Inthat case, mode I is just predicted to increase�Keffective

II ,but not to trigger significantly damage ahead of thecrack tip.

4 Conclusions

The “kinematic interactions” between mode I and II canbe predicted by finite element simulations taking intoaccount the contact and friction of crack face asperities.Such simulations show that for fixed nominal loadingranges, variations in the loading path induce important

Fig. 16 Evolution with thephase angle of potentialtensile and shear-modecrack growth rates, forout-of-phase push–pull andreversed torsion, computedwith h = 10µm, p=180 µmfor (a) �KI =5.3 MPa

√m,�KII =

10.6 MPa√

m or (b) �KI =10 MPa

√m,�KII =

20 MPa√

m and (c)corresponding bifurcationangles

)

(a) (b)shear-mode (Findley)10-9 10-7

/dN

(m

/cyc

le)

Shear-mode (Findley)

Tensile mode (SWTb)

a/d

N (

m/c

ycle

)

tensile mode (SWTb)

10-10 10-8


da

0 30 6 0 0 30 60 90


d

10-11 10-9

(c)

60

80

deg

rees

)

0

20

40

rcat

ion

an

gle

(d

shear-mode (Findley)

tensile mode (SWTb)

-200 30 60 90


bif

u

Fig. 17 Fit of fatiguecriteria

(a)105 (b) 105

104

s to

fai

lure

s to

fai

lure

104

103

Cyc

les

y = 2E+27x-7,622

2

Cyc

les

103

y=2.1027x-7.622 R=0.981

y = 1,672e+07 * x (̂-2,9937) R= 0,9962

102

y=1.672 107x-2.994 R=0.996 R2 = 0,9807

1 10 100 100 1000 10000102

0

123


Fig. 18 fit of constitutiveequations

100 p100

200

100simulation

200simulation

a)

0

-100

0

Str

ess

(Mp

strain

-200 -200

-0.01 0 0.01

E 192 GPa

0.323

A 1170 MPa

200

100

simulationtest

Mpa

)

C 900 MPa

0.775

5

0

-100Str

ess

(

R0 1000 MPa

Rm 730 Mpa

B 10

0.01 0.02

-200

strain

strain

100

-100

pa)

Str

ess

(M

-0.01 0-0.02

0.020.010-0.01-0.02

variations in�KeffectiveI and �Keffective

II likely to changethe crack path and growth rate.

However, linear elastic fracture mechanics param-eters like �Keffective

I and �KeffectiveII cannot represent

the complex interaction of mode I and II in terms ofcrack tip plastic flow that is responsible for an intrin-sic acceleration of mode II crack growth by a sta-tic mode I or of mode I crack growth by a staticmode II.

In addition, for non-proportional loading, pure modeI or pure mode II crack growth do not exist: what-ever the crack path, the crack is loaded in mixed-mode, so that the Paris laws for pure mode I or puremode II are not appropriate to predict the growthrates.

For these reasons, an approach based on elas-tic–plastic FE computations and local application offatigue criteria was preferred to analyze the crack pathsobserved during mixed-mode experiments on marag-ing steel. The predictions of crack paths were suc-cessful in most cases. Regarding the crack growthrates, the local approach performs better, for sequentialloadings, than a simple superposition, which does notcapture the observed synergetic interaction of mode Iand II.

Acknowledgment This study was supported by the AgenceNationale de la Recherche.

Appendix A: Fit of fatigue criteria

Tension-dominated failure which occurs in push–pullcan be predicted with Smith, Watson and Topper’s crite-rion (Smith et al. 1970) in which the damage parameteris

βSW T = �εnσn max (4)

Figure 17a shows the measured fatigue lives in push–pull or repeated tension as a function ofβSWT. An expo-nential law was fitted and used for crack growth pre-dictions.

Shear-initiated decohesion along a slip band infatigue occurs earlier when an opening stress is present.Findley (1957) took this effect into account in a crackinitiation criterion where the damage function:

βFind = �τ + kσn max (5)

incorporates the peak opening stress σn max, computedalong the facet which undergoes the maximum shearstress range, �τ . Figure 17b shows the measuredfatigue lives in reversed torsion with various static ten-sile stresses as a function ofβSWT in which k=0.2 gavethe best correlation. There again an exponential fit wasobtained.

123

V. Doquet et al.

Appendix B: Fit of constitutive equations

Constitutive equations were fitted to some stress–straincurves obtained in push–pull, for which the materialexhibits pronounced cyclic softening. A possible influ-ence of non-proportional loading was not investigated.These equations—in which f denotes the yield func-tion, J2 the second invariant of the stress deviator, p the

cumulated equivalent plastic strain,X

and R kinematic

and isotropic hardening variables and A, C, ψ,ω, B,R0,Rmax, B, materials coefficients are:

f = J2(σ − X)− R(p) (6)

d X = 2

3C A

[1 + (1 − ψ)e−ωp] dε

p− C Xdp (7)

R = R0 + (Rmax − R0)(1 − e−Bp) (8)

dp =√

2

3dε

p: dε

p(9)

Figure 18 provides some examples of experimental andfitted stress–strain curves and the corresponding valuesof the material coefficients.

References

Benallal A, Marquis D (1987) Constitutive equations for nonpro-portional cyclic elastoviscoplasticity. J Eng Mater TechnolTrans 109:326–336

Bertolino G, Doquet V (2009) Derivation of effective stressintensity factors from measured crack face displacement.Eng Fract Mech 76:1574–1588

Bian L-C, Fawaz Z, Behdinan K (2006) A mixed-mode crackgrowth model taking account of fracture surface contactand friction. Int J Fracture 139:39–58

Dahlin P, Olsson M (2003) The effect of plasticity on incipientmixed-mode fatigue crack growth. Fatigue Fract Eng MaterStruct 26:577–588

Doquet V, Pineau A (1990) Extra hardening due to cyclic non-proportional loading of an austenitic stainless steel. ScriptaMetallurgica 24:433–438

Doquet V, Pommier S (2004) Fatigue crack growth under nonproportional mixed-mode loading in ferritic-pearlitic steel.Fatigue Fract Eng Mater Struct 27:1051–1060

Erdogan F, Ratwani M (1972) A circumferential crack in a cylin-drical shell under torsion. Int J Fract Mech 8:87–95

Feaugas X, Clavel M (1997) Cyclic deformation behaviour of analpha/beta titanium alloy. 1. Micromechanisms of plasticityunder various loading paths. Acta Materialia 45:2685–2701

Feng M, Ding F, Jiang Y (2006) A study of loading path influ-ence on fatigue crack growth under combined loading. IntJ Fatigue 28:19–27

Findley WN (1957) Fatigue of metals under combination ofstresses. Trans ASME 79:1337–1348

Gao H, Brown MW, Miller KJ (1983) Fatigue crack growth undernon-proportional loading. In: Proceedings of the interna-tional symposium on fracture (ICF), Beijing 22–25 Nov1983, Science Press, China, pp 666–671

Hourlier F, d’Hondt H, Truchon M, Pineau A (1985) In: MillerKJ, Brown MW (eds) Multiaxial fatigue. ASTM STP 853.ASTM, Philadelphia, pp 228–248

Otsuka A, Mori K, Togho K (1987) Mode II fatigue crack growthin aluminium alloys. In: Current research on fatigue cracks,current Japanese materials research, vol 1. Elsevier, Lon-don, pp 149–180

Pinna C, Doquet V (1999) The preferred fatigue crack propaga-tion mode in a M250 maraging steel loaded in shear. FatigueFract Eng Mater Struct 22:173–183

Planck R, Kuhn G (1999) Fatigue crack propagation under non-proportional mixed-mode loading. Eng Fract Mech 62:203–229

Smith KN, Watson P, Topper TH (1970) A stress–strain functionfor the fatigue of metals. J Mater 5:767–778

Stanzl S, Czegley M, Mayer HR, Tschegg E (1989) Fatigue crackgrowth under combined Mode I and Mode II loading. In:Wei RP, Gangloff RP (eds) Fracture mechanics: perspec-tives and directions, ASTM STP 1020. ASTM, Philadel-phia, pp 479–496

Tong J, Yates JR, Brown MW (1995) A model for sliding modecrack closure I & II. Eng Fract Mech 52(4):599–623

Wong SL, Bold PE, Brown MW, Allen RJ (1996) A branch cri-terion for shallow angled rolling contact fatigue cracks inrails. Wear 191:45–53

Wong SL, Bold PE, Brown MW, Allen RJ (2000) Fatigue crackgrowth rates under sequential mixed-mode I and II loadingcycles. Fatigue Fract Eng Mater Struct 23:667–674

You BR, Lee SB (1997) Fatigue crack growth behaviour ofSM45C steel under cyclic mode I with superimposed sta-tic mode II loadings. Fatigue Fract Eng Mater Struct20(N◦7):1059–1074

Yu X, Abel A (1999) Modelling of crack surface interferenceunder cyclic shear loads. Fatigue Fract Eng Mater Struct22:205–213

Yu X, Abel A (2000) Fatigue crack growth in notched thin-walledtubes under non-proportional mixed-mode loads, vol IV. In:Proceedings of the 10th Int offshore and polar engineeringconference, Seattle, May 28–June 2, 2000, pp 23–28

123

inﬂuence of the loading path on fatigue crack growth under...

Documents