crack path paper

7/27/2019 Crack Path Paper

1/13

This article appeared in a journal published by Elsevier. The attached

copy is furnished to the author for internal non-commercial research

and education use, including for instruction at the authors institution

and sharing with colleagues.

Other uses, including reproduction and distribution, or selling or

licensing copies, or posting to personal, institutional or third partywebsites are prohibited.

In most cases authors are permitted to post their version of the

article (e.g. in Word or Tex form) to their personal website or

institutional repository. Authors requiring further information

regarding Elseviers archiving and manuscript policies are

encouraged to visit:

http://www.elsevier.com/authorsrights
http://www.elsevier.com/authorsrightshttp://www.elsevier.com/authorsrights


2/13

Author's personal copy

Crack path evaluation on HC and BCC microstructures under multiaxial

cyclic loading

V. Anes, L. Reis , B. Li, M. Freitas

Instituto Superior Tcnico, ICEMS & Dept. of Mechanical Engineering, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e i n f o

Article history:

Received 22 December 2012

Received in revised form 22 March 2013

Accepted 24 March 2013

Available online 4 April 2013

Keywords:

Crack path

Multiaxial fatigue

Microstructure

Fatigue life

Experimental tests

a b s t r a c t

In this paper the multiaxial loading path effect on the fatigue crack initiation, fatigue life and fracture sur-

face topology are evaluated for two different crystallographic microstructures (bcc and hc): high strengthlow-alloy 42CrMo4 steel and the extruded Mg alloy AZ31B-F, respectively.

A series of multiaxial loading paths were carried out in load control, smooth specimens were used.Experimental fatigue life and fractographic results were analyzed to depict the mechanical behavior

regarding the different microstructures.A theoretical analysis was performed with various critical plane models such as the FatemiSocie, SWT

and Liu in order to correlate the theoretical estimations with the experimental data. A new approachbased on maximum stress concentration factors is proposed to estimate the crack initiation plane, esti-

mations from this new approach were compared with the measured ones with acceptable results. To

implement this new approach a virtual micro-notch was considered using FEM. Moreover, the multiaxialloading path effect on stress concentration factors is also studied. The obtained results clearly show theeffect of the applied load conditions on local microstructures response.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

Structural failure is often caused by fatigue cracks which fre-

quently initiate and propagate in the critical regions, generallydue to complex geometrical shapes and/or multiaxial loading con-ditions. Fatigue crack initiation and crack growth orientation have

been paid growing research attentions, since it is a crucial issue foran accurate assessment of fatigue crack propagation lives and forthe final fracture modes of cracked components and structures[13]. Although there are a lot of publications regarding multiaxial

fatigue behavior of steels, there are very few studies regardingmultiaxial fatigue of magnesium alloys; Bentachfine et al. [4] stud-

ied a lithiummagnesium alloy under proportional and non-pro-portional loading paths under low-cycle and high cycle fatigueregime observing the deformation mode evolution and plasticitybehavior. It was stated that the phase shift angle in the non-pro-

portional loading paths decreases the material fatigue strength.The comparative parameter used to correlate experimental datawas the von Mises equivalent stress/strain. However with this ap-

proach the material under non-proportional loadings keeps a con-stant equivalent stress and in this way no change in the materialoccurs along each loading cycle. However, the constant change in

the direction of principal stress during the loading period due to

the phase shift increases the anisotropy on the plastic deformation

at grain level and causes, in certain cases, the decrease on fatiguelife [5]. Biaxial fatigue studies were performed by Ito and Shimam-

oto [6] with cruciform specimens made of a magnesium alloy. Itwas analyzed the fatigue crack propagation as well as the effectof microstructure on the material fatigue strength. From the biaxial

low cycle fatigue tests, it was concluded that the twinning densityevolution is strictly related with crack initiation and slip bandsformation on wrought magnesium alloys. Recently, Yu et al. [7],have also studied in-phase and out-phase behavior under strain

controlled tests on AZ61A extruded magnesium alloy using tubularspecimens. The conclusions were similar to Bentachfine et al. [4],

the presence of the phase shift angle leads to decrease the fatiguestrength compared with in-phase cases for the same equivalentstrain amplitude. At low-cycle fatigue regime, it was reported akink in the strain life curve which is a typical behavior for uniaxial

fatigue regime in magnesium alloys. Furthermore, the effect ofcompressive mean stress was evaluated, concluding that a com-pressive mean stress enhances fatigue life [8].

There are mainly three types of shear transformations beyondslip mechanisms namely deformation twinning, stress induced atmartensitic transformations and kinking. The twinning deforma-

tion occurs in hc metals deformed at ambient temperature and atbcc metals when they are deformed at lower temperatures. Twin-ning mechanism occurs when is created a boundary on the mate-rial lattice defining a symmetric region due to shear strain at

0142-1123/$ - see front matter 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ijfatigue.2013.03.014

Corresponding author. Tel.: +351 966415585; fax: +351 218417954.E-mail address: [email protected] (L. Reis).

International Journal of Fatigue 58 (2014) 102113

Contents lists available at SciVerse ScienceDirect

International Journal of Fatigue

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j f a t i g u e


3/13


atomic level. This twin boundary defines a mirror image betweendeformed and undeformed lattice grid [9,10].

At wrought Mg alloys the crack initiation is also associated withmaterial inclusions but in most cases the twinning deformationand slip bands inherent to the twinning density flow are the main

cause for the crack initiation. Crack propagation follows in generalalong the deformation twins fields [1113].

The goal of this paper is to evaluate the mechanical behavior oftwo different microstructures, bcc and hc, subjected to the same

multiaxial loading paths and point out the main differences con-cerning the multiaxial loading effect on the fatigue crack initialpath, fatigue life and fracture surface topology. In this work a crit-

ical plane study was performed comparing the agreement betweenexperimental data and the theoretical one. Critical plane modelssuch as FatemiSocie, SWT, Liu1 and Liu2 were applied and a

new approach based on the highest stress concentration factor todetermine the critical plane direction was performed.

Regarding the two different microstructures the applicability of

the studied models and the corresponding obtained results are

discussed.

2. Materials and methodology

In this work two materials were studied, one is the low-alloy

steel DIN 42CrMo4 (AISI 4140), the other one is the extruded Mgalloy AZ31B-F with 3% of aluminum and 1% zinc. The mechanicalproperties of both materials were determined by the authors fol-

lowing the standard procedures, namely following ASTM E8 andASTM E606 standards, and are presented in Table 1.

2.1. Material cyclic behavior

Structural materials have different properties regarding cyclicand monotonic regimes, for instance the cyclic yield stress can be

quite different from monotonic one, depending on the materialbehavior. Some materials can soften or harden or even maintain

similar to monotonic properties under a cyclic regime [14,15]. An-other category is point out by Lopez and Fatemi, where cyclic soft-ening at low strain amplitudes is followed by cyclic hardening athigher strain amplitudes [16].

When a cyclic softening occurs the cyclic yield stress is lowerthan the monotonic one, in this case, usually, the cyclic curve is un-der the monotonic curve for all strains. Material hardening occurs

when the cyclic regime induce material plasticity in such way thatthe cyclic yield stress appears above the monotonic yield stress, aswell as the cyclic curve appears above the monotonic curve.

Magnesium alloys tends to have a cyclic hardening behaviorbeing highly dependent on the grain refinement, purity, latticeintrinsic behavior like twinning or foundry transformation

processes [7,8,1518]. On the other hand 42CrMo4 material tendsto cyclically softening.

Identifying the cyclic material behavior is of prime importance

to accurately interpret the material local stress states. The soften-ing behavior leads to have a local stress lower than the one esti-

mated with the monotonic curve, however, if the monotoniccurve is used as reference to fatigue experiments then the cyclic to-tal strains will be greater than the monotonic ones. At cyclic hard-ening the opposite occurs, the local stresses are greater than themonotonic ones for the same total strain value. Under these

assumptions it can be concluded that fatigue models used in orderto estimate crack initiation planes and fatigue life estimations

must be implemented taking in to account the cyclic behavior ofthe material once they are based on stress and strain amplitudes[5,18,19].

Material cyclic curve can be used as a reference to fatigue load-ings in order to implement elasticplastic numeric simulations,however the fatigue cracknucleation process induces micro-notcheswhich in turn induces stress risers, with great probability of local

plasticity. Besides, the fatigue crack opening occurrence involvesplasticity in the fatigue process which leads to conclude that the

plasticity effect on the stress and strains states cannot be neglectedon the determination of the crack initiation plane [12,20,21]. In this

Table 1

Mechanical properties of the materials studied, 42CrMo4 and AZ31B-F.

42CrMo4 AZ31B-F

Microstructure type bcc hc

Poissons ratio 0.3 0.35

Density (Kg/m3) 7830 1770

Hardness (HV) 362 86

Tensile strength (MPa) 1100 290

Yield strength (MPa) 980 203

Elongation (%) 16 14

Youngs modulus (GPa) 206 45

r0

f Fatigue strength coefficient (MPa) 1154 450b Fatigue strength coefficient 0.061 0.12

e0f Fatigue ductility coefficient 0.180 0.26

c Fatigue ductility exponent 0.53 0.71

Fig. 1. 42CrMo4 monotonic and cyclic behavior.

Fig. 2. AZ31 alloy monotonic and cyclic behavior.

V. Anes et al. / International Journal of Fatigue 58 (2014) 102113 103


4/13


study the elasticplastic behavior of the material is based on theexperiments with bulk materials, and consequently it is used, as

approximation, for evaluating the local plasticity.In multiaxial fatigue analysis, generally it is considered firstly

the load or strain histories and then it is used a cyclic plasticity

model to determine the stress or strain tensors to perform a criticalplane search in order to evaluate the inherent fatigue damageparameter [5].

Fig. 1 shows the cyclic and monotonic results for the high

strength steel 42CrMo4, this material shows a cyclic softening,stating from 0.3% of strain. Fig. 2 shows the AZ31 magnesium alloymonotonic and cyclic behavior, the magnesium alloy shows a cyc-

lic hardening from 0.6% of strain. Curves plotted in Figs. 1 and 2were determined based on standards, ASTM E8 and ASTM 606requirements.

TheAZ31 alloy cyclicresponseat differentstrainhysteresis loopsshows different yield stress in tension and compression. The com-pression yield stress has a value close to the monotonic value com-

pared with the tensile one which is about 26% greater than the

monotonic yield stress, 203 MPa and 256 MPa, respectively.Currently the commercial FEM softwares regarding stress/strain

analyses do not take into account different material behaviors, i.e.different values for the yield stress at tensile and at compressionstates of a material, excepting for cast iron case [22]. In this workto overcome this issue concerning the AZ31 alloy behavior, it

was considered, as a conservative approach, the tensile branch ofthe AZ31 cyclic response.

2.2. Chaboche plasticity model

In general, plasticity models have three main parts to follow the

material response under plastic deformation. One of them is theyield function, which determines when a material yielding occurs.

The most common yield criterion is the von Mises function, wherea combination of principal stresses on the octahedral plane deter-mines the yield boundary. The second part of plasticity models isthe flow rule, which is based on constitutive equations where the

stresses and strains are computed on incremental plasticity proce-dures, where the next plastic deformation is dependent on theprior one. This rule is generally based on the Drucker postulate[23], where the plastic strains increments are normal to the yield

surface defined by the yield function. The third part is the harden-ing rule, which establishes the changes on the yield surface during

the plastic deformation [24].Chaboche plasticity model is a nonlinear kinematic hardening

model, the yield function F is given through the follow equation:

Fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 s a

:s a

r k

01

where s is the deviatoric stress, a is the back stress, and k is the yieldstress [25,26]. The kinematic hardening is governed through theback stress tensor; this model component contribution is relatedwith the yield surface translation. The back stress tensor calcula-tions are present in the following equations:

fag Xni1

faig 2

fDagj 2

3CifDe

plg cifaigDepl

1

Ci

dCidh

Dhfag 3

whereDepl is the accumulated plastic strain, h is temperature, and Ci

and ci are the Chaboche material parameters. In Eq. (3) the harden-ing modulus and back stress variation (recall term) are represented

by the first and second terms respectively. The third term is relatedwith temperature variation. In commercial software Ansys, the

Chaboche plasticity model allows to use various kinematic modelsand material constants; C1 and c1 are model inputs for one kine-matic model, however extra sets of C and c can be added. In thisstudy the C1 and c1 are determined based on the material cyclic re-sponse, present on previous subsection.

2.3. Chaboche plasticity calibration

Chaboche material parameters can be determined throughstressstrain tests under stabilized hysteresis loops. Having the

plastic strainvalues andthe inherent recallterm, which is thediffer-ence between the stress amplitude Dr/2 applied and the materialcyclic yield stress (k), for a fixed total strain, the relation between

these two values can be fitted with the Eq. (4). From a fitting proce-dure it is possible to obtain the material constants C1 and c1.

Dr2

k C1

c1

tanh c1Depl

2

4

Fig. 3 shows the data fitting curves obtained for 42CrMo4 andAZ31 materials, regarding Eq. (4). Table 2 presented the Chabochematerial constants determined for both materials.

2.4. Loading paths and specimen test

To evaluate the microstructures influence, four biaxial cyclicloading paths were selected, see Fig. 4. The first one is a pure uni-axial cyclic tension test, case PT, and the second one is a pure shear

loading, case PS. Case PP is a proportional biaxial loading and theOP case is a 90 out of phase loading case. In Fig. 5 is presentedthe specimen geometry and dimensions.

All tests were performed at ambient temperature and endedwhen the specimens were totally separated.

3. Theoretical analysis

To estimate the critical plane orientation and multiaxial fatiguelife it is often used critical plane models, where the multiaxial

Fig. 3. 42CrMo4 and AZ31Chaboche plasticity model calibration.

Table 2

Chaboche plasticity input parameters for 42CrMo4 and

AZ31B-F.

Material C1 c1

AZ31 25,000 90

42CrMo4 19,281 69

104 V. Anes et al. / International Journal of Fatigue 58 (2014) 102113


5/13


stress tensor is projected in several planes in order to determine

the one which maximizes the inherent damage parameter, i.e.the critical plane. These damage parameters are then associatedwith the material CoffinMason equation to estimate fatigue life.Next subsections present some critical plane models and a new ap-

proach to estimate the critical plane orientation.

3.1. FatemiSocie criterion

FatemiSocie [19] proposed a model which predicts that thecritical plane is the plane orientation h with the maximum FSdamage parameter:

Dc2

1 kFSrn;maxry

max

h

5

where Dc/2 is the maximum shear strain amplitude on a h plane,rn,max is the maximum normal stress on that plane, ry is the mate-rial monotonic yield strength and k is a material constant, k = 1.0 inthis case [27].

3.2. SWT criterion

Smith et al. [14] proposed a model which was later extended tomultiaxial fatigue situations in terms of maximum normal strain

by Socie and Marquis [5]. This model is based on the principalstrain range plane, and maximum stress on that plane:

maxh

rnDe12

6

where rn is the normal stress on a plane h, and De1 is the principalstrain range on that plane.

3.3. Liu criterion

Liu [28] proposed an energy method to estimate the fatigue life,based on virtual strain energy (VSE). This model considers twoparameters associated with two different modes of fatigue cracks,

a tensile failure mode (Mode I), DWI, and a shear failure mode(Mode II), DWII. Failure is expected to occur on the plane h in the

material, having the maximum VSE quantity. According to ModeI fracture, the parameter, DWI is:

DWI maxh

DrnDen DsDc 7

For Mode II fracture, the parameter, DWII is, see the following

equation:

DWII DrnDen maxhDsDc 8

Fig. 4. Loading paths: (a) case PT, (b) case PS, (c) case PP and (d) case OP.

Fig. 5. Specimen test geometry and dimensions.

Fig. 6. (a) Identification of nominal and local stress points and (b) micro-notch geometry and / angle.



6/13


whereDs and Dc are the shear stress range and shear strain range,respectively, Drn and Den are the normal stress range and normalstrain range, respectively.

3.4. New maximum kt approach regarding critical plane orientation

As early stated [5,7] loading paths have a strong influence on

the crack nucleation and initiation process, as a nucleation resultintrusion and extrusion occurs at material surface leading to createmicro-notches where plasticity mechanisms, at micro-notch root

or at surface, govern the early crack opening process.Mode I governs the crack opening process. One characteristic of

this mode is that the opening direction is perpendicular to thecracked surface.

The present methodology aims at being used mainly for well-known geometries where nominal stress can be calculated. In this

approach the main concept is based on the assumption that thecrack opening front has the biggest stress values. These valuescan be determined using numerical tools such as Ansys [29],

Abaqus [30] or even an in-house FEA software. One issue relatedwith this approach is the crack geometry to adopt on numeric sim-ulations, since the crack initiation direction is the missing variableand the crack geometry has a strong influence on the final result;

with this in mind a virtual micro-notch was implemented usingFEM, a spherical cap geometry with depth equal to half diameterwas adopted. This geometry does not have a favorable direction

to induce particular stress risers since it is equal in all directions.In this way the stress risers locations are strictly dependent onthe loading path type and stress levels.

In Fig. 6a is shown the nominal and local regions in the speci-

men test, the nominal region is sufficiently far from the localregion.

Fig. 6b shows the spherical cap (micro-notch) numerically

implemented, the spherical radius is 100 lm and spherical capdepth is also 100 lm. These values are in agreement with magne-sium alloys grain size which has around 50 lm as average value

depending on the heat treatment and grain growth.Concerning the mesh, a convergence study was conducted to

assure no influence in the number of nodes used. The control

nodes, around the cap edge, at micro-notch outer, and on specimensurface are equally separated, in order to optimize the critical

plane estimations, also an additional node at micro-notch root

(depth) is monitored. Thus, the critical plane orientation at initia-tion stage will be estimated by the highest kt value, between the

maximum axial kt and the maximum shear kt, determined atspherical cap (micro-notch) considering one period of loading pathapplied for the different loading paths. In this work the kt values, in

tension and shear, are obtained by comparing the local stresses atnotch cap for the several nodes and the nominal stresses, in tensionand shear, at a location apart from the notch cap, considering thesame direction for both situations, respectively, see Fig. 6a.

4. Results and discussion

4.1. Fatigue life analysis

Fig. 7 presents the experimental fatigue life results obtained for

both materials concerning the nominal equivalent von Misesstress. Results show that the relative damage between the loadingpaths considered in this study tends to have a different relativearrangement in the selected materials.

For the AZ31 material the PS, PT and PP loading cases almosthave the same trend line, leading to conclude that von Mises equiv-alent stress is a good candidate to perform fatigue life correlation

for this material, excepting in the situation of non-proportionalloading cases.

On 42CrMo4 material the PS loading case is the less damage and

the PT and PP SN curves are between OP and PS damage limits.Concerning Mg alloy, the OP loading case seems to present a

bilinear tendency. Due to the OP loading nature the strain energyinvolved is higher than the other loading cases on same fatigue life

region, i.e., the activation of twinning mechanisms leads to a differ-ent mechanical behavior. Moreover, regarding SN curves trends,

the damage rate between loading paths is similar for the samematerial excepting for the loading case OP where the damage rate

is bigger. In Tables 3 and 4 are presented the fatigue data shown inFig. 7.

4.2. Loading path effect on kt

In Table 5 are shown numerical results for the stress concentra-

tion factor inherent to the studied loading paths and consideringthe 42CrMo4 material with and without the Chaboche plasticity

model implemented on Ansys software, respectively.

Fig. 7. SN fatigue life considering nominal stresses for: (a) 42CrMo4 and (b) Mg AZ31B-F.



7/13


The stress analysis monitoring was performed at micro-notchroot (spherical capdepth)and at theintersectionboundarybetweenmicro-notch cap and specimen test surface. The calculus of kt(s) are

based on the shear and axial stress components at micro-notch rootand micro-notch cap boundary nodes, divided by the nominal shearand nominal axial stress components, respectively.

This approach based on stress concentration factor determina-

tion was implemented to avoid to use the concept of equivalentstress where the relation between axial and shear stress compo-

nents remains under discussion. Therefore, in this study, it wasdetermined a kt for axial loading and another one for shear loading.The nominal axial and shear stresses considered are about 30%lower than the material cyclic yielding stress, in order to ensure

no plasticity at nominal stress control point, see Fig. 6a.

Comparing the results with and without plasticity in Table 5, itcan be concluded that the results from plasticity numerical ap-

proach are lower. This result is justified through the Chabocheplasticity input, whereas the material cyclic curve instead of mono-

tonic one was applied, regarding the 42CrMo4 cyclic softeningbehavior.

Excepting the PP loading case, the highest kt concerning the ax-ial and shear components occurs at micro-notch root with and

without plasticity simulation. The average results for axial kt onthe loading cases OP and PP are greater than for shear kt, under nu-

meric plasticity simulation the axial kt results are about 25% great-er at micro-notch root and 35% at micro-notch surface. Underelastic simulation the axial kt values are 7% greater at micro-notchroot and 45 % greater for micro-notch surface.

In Table 6 are shown the mean values of axial and shear kt atmicro-notch cap surface under plasticity simulation, the greatestvalues are obtained for OP loading case and the lowest for the PPloading case. This means that for OP loading case, during one cycle,

this loading has in average, the highest stress level on micro-notchcompared with the PP loading case.

Regarding elastic and plastic simulations it is clearly shown thatthe obtained values for stress concentration factors are not onlydependent on the micro-notch geometry and nominal stress but

also on the loading path type.

4.3. Critical plane orientations

4.3.1. Critical plane modelsFig. 8 shows the critical plane results for the loading paths stud-

ied for the 42CrMo4, the AZ31 results are very similar from these

Table 3

Fatigue life for 42CrMo4 material.

Loading

case

Normal stress (MPa) Shear stress (MPa) Nf

PT 700 0 6040

600 0 19,951560 0 53,752

550 0 56,929

495 0 247,953

485 0 269,178

480 0 1,000,000

(runout)

PS 0 546 2088

0 484 11,302

0 440 70,610

0 402 159,854

0 395 315,668

0 391 1,000,000

(runout)

OP 510 294 56411

495 286 97,548

490 283 107,374

485 280 197,548475 274 316,712

465 269 618,128

450 260 1,000,000

(runout)

PP 610 352 4114520 300 27,204

495 286 48,740

470 271 97,366

465 269 109,087

445 257 239,600440 254 311,401

435 251 564,088

425 245 1,000,000

(runout)

Table 4

Fatigue life for AZ31 material.

Loading

case

Normal stress (MPa) Shear stress (MPa) Nf

PT 140 0 131,64

135 0 22,873130 0 38,102

120 0 62,352

105 0 721,573

100 0 1,000,000

(runout)

PS 0 75 88,871

0 69 128,769

0 64 227,808

0 59 388,236

0 53 1,000,000

(runout)

OP 106 61 7182

95 55 859578 45 11,986

74 43 167,525

73 42 576,336

71 41 1000,000 (runout)

PP 106 61 16,800

92 53 46,878

78 45 69,169

74 43 242,685

71 41 353,71867 39 1,000,000

(runout)

Table 5

Kt variation due to loading path at micro-notch root and surface with and without

plasticity model, maximum value in a period of time (1 cycle).

With plasticity Without plasticity

Loading path Location Kt axial Kt shear Kt axial Kt shear

PT Root 1.66 NAN 1.79 NAN

Surface 1.46 NAN 1.64 NAN

PS Root NAN 1.53 NAN 1.74

Surface NAN 1.36 NAN 1.34

OP Root 1.66 1.53 1.79 1.79

Surface 1.47 1.36 1.65 1.40

PP Root 1.34 1.15 1.67 1.78

Surface 1.54 1.36 2.14 1.60

Table 6

Kt mean values with plasticity on micro-notch surface along one loading period (1

cycle).

With plasticity model 42CrMo4

Loading path Location Kt axial Kt shear

PT Surface 0.85 NAN

PS Surface NAN 0.85

OP Surface 0.85 0.93

PP Surface 0.65 0.70



8/13


Fig. 8. Fatigue damage parameter and critical plane orientation for 42CrMo4.

Fig. 9. Critical plane estimation based on maximum kt for PT loading case with plasticity model: (a) axial and (b) shear.



9/13


ones only changing on the damage parameter values apart that thecritical plane estimations are equal for both materials.

Critical plane orientation is determined for each loading paththrough the maximum damage parameter obtained along the load-

ing period. The biaxial stress components for each critical plane

model represented in Fig. 8 were determined for the same equiva-lent von Mises stress. From the results can be concluded that SWTand Liu1 damage parameters have the same values along all projec-

tions, moreover, for the same equivalent stress, the FatemiSociedamage parameter for the OP loading case is bigger than the othercases, i.e, PT, PS and PP loading cases.

4.3.2. Maximum kt approach

In Figs. 912 are shown the results for the maximum kt ap-proach on the estimation of the crack initiation plane orientation.

On this approach the maximum value for axial and shear kt isdetermined along the loading period and assigned to the respectivemicro-notch cap interface with specimen surface node and micro-notch root node.

Linking the graph origin with a point in the respective kt curvewill give the kt magnitude for a specific direction. Linking thegraph origin to the maximum kt in axial will give the crack initia-tion plane estimated by the axial kt, the same procedure is used to

estimate the crack initiation plane using the shear kt. From thesetwo kt estimations the one with the greatest kt value will deter-

mine the crack initiation plane orientation inherent to that loadingpath.

Fig. 9 shows the kt gradient in the notch cap regarding the axial

and shear computed stress components for the PT loading case. InFig. 9a the highest kt value in axial is achieved on the nodes locatedat 0 and 180, see U referential at Fig. 6b, whereas in Fig. 9b the

highest kt shearvalues were obtained at 45. From this two resultsthe most high kt value will indicate the critical plane estimation. ForthePT loading case, Fig.9a, themaximumaxial kt value indicatesthe

criticalplaneat0, andthe highest axial kt valueoccurs at notchroot.In Fig. 10a is shown the axial kt gradient for the PS loading case,

here the maximum axial kt was found at 45, however for thisloading case the maximum kt is the shear one. In Fig. 10b, the high-est shear kt on micro-notch cap interception with specimen sur-face occurs at 45 and the highest shear kt is found on the

micro-notch root. It is expected that the crack initiation process oc-curs at maximum kt location propagating at 45 through notch

cap reaching the specimen surface at same direction.In Fig. 11a is shown the OP loading case results, the highest kt is

the axial one which suggests that the crack initiation plane occursat 0. On this loading case the kt at micro-notch root is almost

Fig. 10. Critical plane estimation based on maximum kt for PS loading case with plasticity model: (a) axial and (b) shear.

Fig. 11. Critical plane estimation based on maximum kt for OP loading case with plasticity model: (a) axial and (b) shear.



10/13


equal to the one verified at notch cap interception with specimensurface. The axial kt is greater than the shear one, see Fig. 11b, thusthe maximum kt estimation for the critical plane in this loading

path is given through the axial kt.In the PP loading case the axial kt, Fig. 12a, estimates the critical

plane on the

13

direction, moreover in this loading case the sur-face micro-notch axial kt is greater than the one verified on the mi-cro-notch root, the same was found for the shear kt i.e. the surface

results are greater than the ones verified at notch root.

4.4. Experimental fractographic analysis

In this study only early crack propagation was considered,

which occurs in a very thin layer of the surface, where the gradienteffect is relatively small. Regarding the fracture surface topology

analysis it was studied the identification of crack initiation localto measure the crack initiation angle and the different crack prop-agation zones of fracture surfaces to analyze the influence of the

different loading paths on fracture topology.

Fig. 12. Critical plane estimation based on maximum kt for PP loading case with plasticity model: (a) axial and (b) shear.

Fig. 13. Loading case PT: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.



11/13


Fig. 14. Loading case PS: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.

Fig. 15. Loading case PP: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.



12/13


Fracture surfaces and crack initiation angles for loading pathsstudied in this work are presented in Figs. 1316. These results,measured angle and fracture surface topology are representative

of the different loading path and material, respectively.

Regarding Fig. 13 and PT loading case, both fracture surfacessuggest a ductile fatigue failure mechanism with two different

zones and roughness: a fatigue zone (FZ), with smooth area andan instantaneous zone (IZ), with rough area. In the smoothest part

of the fatigue zone area is possible to identify the crack initiationlocal, as in the rough part, in the IZ, the final fracture. The rough-ness change in fracture surface indicates a different crack growthspeed proving that the failure did not happen suddenly. Fatigue life

is spent mainly in mode I crack growth and finishes with rough andcrystalline appearance. No expressive propagation marks were ob-

served in the high strength steel, however in Mg alloy were ob-served a slight river marks pointing the crack growth and theinitiation spot. Initiation angles measured in both materials were0.

Fatigue crack results for loading case pure shear PS, are pre-sented in Fig. 14. The steel specimen fracture surface shows a un-ique initiation source with an initiation plane oriented at 45; thisis a typical result for twisting loads on ductile materials [31]. It is

expected that a reduction of this angle value occurs if there areother stresses than the shear ones involved in the fatigue process,i.e., if mixed mode crack propagation is present. The FZ and IZ show

a strong granulated surface in the steel specimen, however in the

Mg specimen fracture surface is smoother. In general both fracturesurfaces are similar, the FZ have a fracture plane equally oriented

and the IZ in both cases have a similar arrangement. At Mg speci-men it is observed ratchet marks with two initiation spots growing

toward the center of the specimen, and in the IZ can be seen amaterial riffling like progression marks.

In Fig. 15 is shown the fracture surfaces results for the propor-tional loadingcase, PP. In thissituation the obtainedfracture surfacs

topology are a little different. The steel specimen fracture surfaceshows one crack origin and three distinct zones: the usual FZ andIZ zones and a surface wear region. In Mg specimen fracture surface

can be identified two crack origins and many river marks pointingout to first crack origin. The crack initiation plane angle measuredfor the steel specimen was 16 and 40 for Mg specimen.

The results for the loading case OP are shown in Fig. 16. Fracturesurfaces are also similar in this loading case, supporting the idea onthe topology convergence in both materials in high cycle regime.At Mg specimen IZ can be seen diagonal riffles starting from the

Fig. 16. Loading case OP: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.

Table 7

Critical planes measured and estimated for 42CrMo4 and AZ31B-F.

Case PT Case PS Case PP Case OP

AZ31 42CrMo4 AZ31 42CrMo4 AZ31 42CrMo4 AZ31 42CrMo4

Measured 0 0 45 45 40 16 5 0

FS 40 40 3; 87 3; 87 13 13 0 0

SWT 0 0 45 45 25 25 0 0

Liu 1 0 0 45 45 25 25 0 0

Liu 2 45 45 90; 0 90; 0 20; 70 20; 70 90; 0 90; 0

Max kt 0 0 45 45 13 13 0 0



13/13


FZ throughout the end of IZ. Diagonal riffles in instantaneous zoneindicate a biaxial loading at instant fracture time. In this case the

crack initiation plane angle measured for the steel specimen was0 and 5 for Mg specimen.

4.5. Critical plane analysis

Table 7 presents the measured crack initiation angles and alsothe critical planes estimations by the FS, SWT, Liu I and Liu II crit-

ical plane models and maximum kt criterion for the studied load-ing paths and materials.

The SWT and Liu1 models estimate well the crack initiationplane in both materials for the loading cases PT and PS. The first

principal strain has a key role on these results, however, the FSand Liu2 models have poor results for this two loading cases.

For the PP loading case, the results are satisfactory in the

42CrMo4, regarding FS and Liu2 criteria, on contrary, for the Mg al-loy, the estimated results differ from the experimental data, theminor deviation is 20 considering Liu2 model. In the OP loading

case the estimated critical plane angle by all models agree wellwith the experimental results.

From the maximum kt approach results can be concluded thatthe crack orientation planes were very well estimated for the PT

and OP loading cases, and for the PS and PP loading cases the re-sults are relatively close to the experimental trends.

These results show that maximum kt approach is a sensitive ap-proach to the stress amplitude ratio variations, being sensitive toaxial or shear predominance on a multiaxial loading, in contrastwith other approaches which are only sensitive to axial or shear

stresses.

5. Conclusions

This paper studies the influence of multiaxial loading conditionson the fatigue crack initial path, fatigue life and fracture surfacetopology on two different crystallographic microstructures. Fromthe experimental and theoretical work carried out with two mate-rials, a low-alloy steel and a Mg alloy, some remarks can be drawn:

Regarding fatigue life analysis the damage between the appliedloading paths tends to have different relative arrangement in bothmaterials.

The damage rate between loading paths is similar for the samematerial excepting the OP loading case where the damage rate ismore pronounced.

Concerning fractographic analysis in loading cases PT and PS,the fracture surface topology in both 42CrMo4 and Az31B-F spec-imens are similar and independent on the equivalent stress level.

Under multiaxial loading regime the loading path and equivalentstress level have a huge influence on the AZ31B-F surface topology.In high cycle fatigue regime the fracture surface is strongly depen-

dent on the loading path type; for the same loading path the42CrMo4 and AZ31B-F fracture surface tends to be similar.

Regarding critical plane analysis the crack initiation angle inpure axial and pure torsional loading cases do not change with

the equivalent stress level. Moreover, at uniaxial loading casesthe initiation angles do not vary for the studied materials, howeverin multiaxial loadings that is not true. The influence of the loading

path trajectories on stress concentration factors was clearly shownand the maximum kt approach proved to be sensitive to the stressamplitude ratio and local stress states, achieving very good results

with the loading conditions applied in this study.

Acknowledgements

The authors gratefully acknowledge financial support from FCT

Fundao para a Cincia e Tecnologia (Portuguese Foundation forScience and Technology), through the Project PTDC/EME-PME/

104404/2008.

References

[1] Collins J. Failure of materials in mechanical design: analysis, prediction,prevention. 605 Third Ave, New York, NY 10016, USA: John Wiley & Sons, Inc.;1993. p. 654.

[2] Dieter GE, Bacon D. Mechanical metallurgy. New York: McGraw-Hill; 1986.[3] Frost NE, Marsh KJ, Pook LP. Metal fatigue. Dover Publications; 1999.[4] Bentachfine S, Pluvinage G, Toth LS, Azari Z. Biaxial low cycle fatigue under

non-proportional loading of a magnesiumlithium alloy. Eng Fract Mech1996;54:51322.

[5] Socie DF, Marquis GB. Multiaxial fatigue. Society of Automotive Engineers;2000.

[6] Ito Y, Shimamoto A. Effect of microstructure on fatigue crack growth resistanceof magnesium alloy under biaxial stress. Key Eng Mater 2005;297:155964.

[7] Yu Q, Zhang J, Jiang Y, Li Q. Multiaxial fatigue of extruded AZ61A magnesiumalloy. Int J Fatigue 2011;33:43747.

[8] Li Q, Yu Q, Zhang J, Jiang Y. Effect of strain amplitude on tensioncompressionfatigue behavior of extruded Mg6Al1ZnA magnesium alloy. Scripta Mater2010;62:77881.

[9] Barnett MR. Twinning and the ductility of magnesium alloys: Part II.Contraction twins. Mater Sci Eng A 2007;464:816.

[10] Barnett MR. Twinning and the ductility of magnesium alloys: Part I: Tensiontwins. Mater Sci Eng A 2007;464:17.

[11] Begum S, Chen DL, Xu S, Luo AA. Low cycle fatigue properties of an extrudedAZ31 magnesium alloy. Int J Fatigue 2009;31:72635.

[12] Yang F, Yin SM, Li SX, Zhang ZF. Crack initiation mechanism of extruded AZ31magnesium alloy in the very high cycle fatigue regime. Mater Sci Eng A2008;491:1316.

[13] Mayer H, Papakyriacou M, Zettl B, Stanzl-Tschegg SE. Influence of porosity onthe fatigue limit of die cast magnesium and aluminium alloys. Int J Fatigue2003;25:24556.

[14] Smith KN, Topper TH, Watson P. A stressstrain parameter for the fatigue ofmetals (stressstrain function for metal fatigue including mean stress effect). JMater 1970;5:76778.

[15] Zberov Z, Kunz L, Lamark TT, Estrin Y, Janevcek M. Fatigue and tensilebehavior of cast, hot-rolled, and severely plastically deformed AZ31magnesium alloy. Metall Mater Trans A 2007;38:193440.

[16] Lopez Z, Fatemi A. A method of predicting cyclic stressstrain curve fromtensile properties for steels. Mater Sci Eng: A 2012;556:54050 [30 October].

[17] Knezevic M, Levinson A, Harris R, Mishra RK, Doherty RD, Kalidindi SR.Deformation twinning in AZ31: influence on strain hardening and textureevolution. Acta Mater 2010;58:623042.

[18] Polak J, Klesnil M, Lukvs P. High cycle plastic stressstrain response of metals.Mater Sci Eng 1974;15:2317.

[19] Fatemi A, Socie DF. A critical plane approach to multiaxial fatigue damageincluding out-of-phase loading. Fatigue Fract Eng Mater Struct2007;11:14965.

[20] Xia ZC, Hutchinson JW. Crack tip fields in strain gradient plasticity. J MechPhys Solids 1996;44:162148.

[21] Li H, Chandra N. Analysis of crack growth and crack-tip plasticity in ductilematerials using cohesive zone models. Int J Plast 2003;19:84982.

[22] Pietruszczak ST, Mroz Z. Finite element analysis of deformation of strain-softening materials. Int J Numer Meth Eng 2005;17:32734.

[23] Chen WF, Han DJ. Plasticity for structural engineers. J Ross Pub; 2007.[24] Chaboche JL. Constitutive equations for cyclic plasticity and cyclic

viscoplasticity. Int J Plast 1989;5:247302.[25] Chaboche JL. A review of some plasticity and viscoplasticity constitutive

theories. Int J Plast 2008;24:164293.[26] Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge University

Press; 1994.[27] Reis L, Li B, de Freitas M. Crack initiation and growth path under multiaxial

fatigue loading in structural steels. Int J Fatigue 2009;31(1112):16608[NovemberDecember].

[28] Liu KC. A method based on virtual strain-energy parameters for multiaxialfatigue life prediction, vol. 1191. Springer; 1993. p. 67.

[29] Madenci E, Guven I. The finite element method and applications in engineeringusing ANSYS. Springer; 2007.

[30] Giner E, Sukumar N, Tarancon JE, Fuenmayor FJ. An Abaqus implementation ofthe extended finite element method. Eng Fract Mech 2009;76:34768.

[31] Handbook M. Fractography, vol. 12. Ohio, USA: ASM International Metals Park;1987.


crack path paper

Documents