A multiscale approach of fatigue and shakedown for notched ...€¦ · A multiscale approach of fatigue and shakedown for notched structures G. Bertolino a, A. Constantinescu a,*,

Theoretical and Applied Fracture Mechanics 48 (2007) 140–151

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A multiscale approach of fatigue and shakedownfor notched structures

G. Bertolino a, A. Constantinescu a,*, M. Ferjani a, P. Treiber b

a Laboratoire de Mecanique des Solides – CNRS UMR 7649, Ecole Polytechnique, 91198 Palaiseau, Franceb Institut fur Materialprufung – Universitat Stuttgart, Germany

Available online 19 July 2007

Abstract

The aim of this paper is to analyse the fatigue phenomena in the presence of stress gradients. It is well-known that mostfatigue criteria fail to predict the lifetime of components in the presence of high stress concentrations or stress gradients, asit is the case in the neighbourhood of cracks, holes notches and encountered for example in riveted or threaded structures.Proposed is a numerical approach in the framework of the high cycle fatigue domain in order to give a qualitative answer.The work starts from the numerical computation of macroscopic loading corresponding to some fatigue experiments onspecimens with an inclusion of metallic grains embedded in a macroscopic matrix. The computed fields are then analysed interms of the HCF (high cycle fatigue) criterion [1], which is based on the estimation of the shakedown limit at the grainscale. The infinite lifetime prediction is based on the assumption that fatigue occurs if at least one grain fails, i.e. reachesplastic shakedown. The predictions at mesoscopic and macroscopic scales are close if the macroscopic stress distribution ishomogeneous. However in the case of the stress gradient, lifetime predicted at the macroscopic scale is underestimatedwhen compared to the predictions made at the mesoscopic scale. Another result is that the gap between microscopicand macroscopic predictions obtained from these numerical computations can roughly be estimated by a diminution ofstress of the same order of magnitude as found in the experiments and phenomenological observations.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Stress gradient; High cycle fatigue; Dang Van criterion; Polycrystalline plasticity

1. Introduction

A well-known fact in the engineering design ofstructures is that most fatigue criteria fail to predictcorrectly the lifetime of components in the presenceof high stress concentrations or stress gradients.

0167-8442/$ - see front matter � 2007 Elsevier Ltd. All rights reserved

doi:10.1016/j.tafmec.2007.05.002

* Corresponding author. Tel.: +33 1 69 33 33 30; fax: +33 1 6933 30 26.

E-mail address: [email protected](A. Constantinescu).

This is the case in the neighbourhood of cracks,holes notches, . . . as encountered for example in riv-eted or threaded structures. More precisely, it canbe stated that the local fatigue criteria based onthe hot spot method, i.e. defining the most criticalpoint in a structure such as the criterion in [1,2],generally underestimate the fatigue lifetime. Thecritical point of a structure in the neighbourhoodof a geometrical stress concentration as angles,notches, cracks, etc., stems from the elastic solu-tion of loading problem and the applied fatigue

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G. Bertolino et al. / Theoretical and Applied Fracture Mechanics 48 (2007) 140–151 141

criterion. The problem addressed in this paper is ananalysis of the physical phenomena at the smallerscales contributing to fatigue in order to give a qual-itative explanation of this apparent contradiction.

Let us first mention that various phenomenolog-ical methods have been developed to overcome thisdifficulty. The concept of a ‘‘critical distance’’ [3,4]adapts the fatigue criterion by applying it at a givendistance ahead of the notch tip. This distance iscommonly of the order of the dimension of themetal grains. An application of this method to cylin-drical notched specimen made of Ti6Al4V alloy isgiven in [3]. A similar concept, using a volumetricapproach to compute an average stress taking placein a fatigue process zone, instead of the critical dis-tance, has been proposed for example in [5,6].

Other recent discussions at the macroscopic levelof the influence of the stress gradient in bendingfatigue experiments have been presented in [7,8].The difficulties in applying fatigue criteria in theneighbourhood of stress concentrations arising infretting fatigue problems is discussed in [9].

As a consequence of this short review of the mod-els and discussions of lifetime predictions in the pres-ence of stress gradient, one can finally conclude thatthe common practice is to include the stress gradienteffect in fatigue analysis by introducing a notch fac-tor or a critical distance parameter. This effectivelycorresponds to a lowering of the stress level andcan be interpreted as the application of the fatiguecriterion over a small averaging volume, rather thanat the mathematical maximum point (hot spotmethod). This phenomenological approach providesacceptable results from the practical point of view.However a satisfactory link between the methodol-ogy and the underlying physical phenomena is stillan open problem.

This paper proposes a numerical approach in theframework of the high cycle fatigue domain in orderto give a first qualitative answer to this difficult ques-tion. The work starts from the numerical computa-tion of macroscopic loading corresponding to somefatigue experiments on specimens with an inclusionof metallic grains embedded in a macroscopicmatrix. As such the scale difference between the mac-roscopic level of the fatigue loading and the meso-scopic scale of the grains is gapped and as aconsequence the apparition of shakedown at thegrain scale can be observed and understood. If thefatigue analysis uses the criterion in [1] that is basedon the estimation of the shakedown limit at the grain

scale, then estimate can be made from the mechani-cal fields computed in the different grains if grainfailure and therefore fatigue crack initiation are apossible outcome or not. The results present the dis-tribution of mesoscopic pressure versus mesoscopicshear computed for the family of metallic grains con-sidered and the permit to estimate the fatigue crite-rion [2] in each grain. Recall that the criterion in[2] is based on the assumption that fatigue occurs ifat least one fails, i.e. reaches plastic shakedown.Therefore now it is possible to compare the fatiguepredictions for the complete set of grains with thefatigue prediction by applying directly the criterion[2] at the macroscopic scale. One can easily remarkthat predictions at mesoscopic and macroscopicscales are close if the macroscopic stress distributionis homogeneous. However in the case of the stressgradient, fatigue occurs earlier if only the macro-scopic scale is considered. Another result is thatthe gap between mesoscopic and macroscopic pre-dictions can roughly be estimated to a diminutionof stress of the same order of magnitude as in thephenomenological approached cited previously.

The paper starts with the presentation of thematerial model and of the simulated bending testin the second section. The next section recalls thebasic assumptions underlying the mesoscopic andmacroscopic fatigue criterion [1]. The forth sectiondiscusses the results of the computations in termsof fatigue predictions. The last section presents theconclusions of the work and opens some furtherresearch perspectives.

2. Shakedown theory and Dang Van HCF

Let us consider a structure under a mechanicalloading. Its fatigue lifetime will be determined fromthe mechanical fields: elastic and plastic strains,stresses, etc. computed over a cycle. The underlyinghypothesis is that the mechanical response of thestructure is stabilized, i.e. that the fields will evolveunder closed loops.

Fatigue phenomena can be characterized at threescales:

• the microscopic scale of dislocations, which are theunderlying elements of persistent slip bands andelastoplastic strains;

• the mesoscopic scale of grains, where fatigue anddamage phenomena are concentrated either atthe grain boundary or in the interior;

142 G. Bertolino et al. / Theoretical and Applied Fracture Mechanics 48 (2007) 140–151

• the macroscopic scale of the structure, at whichloads are applied and industrials design isperformed.

An inspection of the three scales during a cyclicloading would conduct to different observationsdetermined by the fatigue regime (see Fig. 1):

(i) In the high cycle fatigue regime (HCF), con-sider two fatigue domains corresponding tofinite and infinite lifetime. Physical observa-tions at the macroscopic scale show thatstructures are macroscopically in an elasticshakedown state. At the mesoscopic scale ofthe grains, it is now commonly accepted thatelastic shakedown occurs only for the infinitelifetime. The case of a finite lifetime can beassociated with a grain oriented in such away with respect to the loading, that it canonly not reach an elastic shakedown state. Itwill therefore be in plastic shakedown or rat-cheting state which will conduct to failureafter a finite number of cycles. The stress con-centration due to this mesoscopic failure willtherefore equally conduct a macroscopic crackinitiation associated with failure.

(ii) In the low cycle fatigue regime (LCF), physicalobservation at both macroscopic and meso-scopic scales show extensive existence of plas-tic strains. Moreover homogenisation theoryshows that strains and stresses at the twoscales tend to be closer to each other withincreasing plastic strain. This can be translatedsaying that the higher the applied load, thecloser the mesoscopic and the macroscopicscales will behave.

In this work consider only the high cycle fatiguedomain as defined in (i) and focus essentially on infi-

N number of cycles to failure

Low cycle fatigue High cycle fatigue

Infinite lifetime Load

on

stru

ctur

e

Finite lifetime

Fig. 1. Description of the low and high cycle fatigue domains in aWoehler diagram.

nite lifetime. Recall the main assumptions of thefatigue criterion [2] as presented in [1]. As only anisolated grain will eventually be submitted to plas-tic shakedown, this problem is reduced to theanalysis of an elastoplastic inclusion in an infiniteelastic matrix and perform the passage betweenthe macroscopic and mesoscopic fields using stan-dard homogenisation theory. The homogenisa-tion in the problem (2) that applies the hypothesis[3], i.e. equality of mesoscopic e and macroscopicstrains E:

e ¼ E ð1Þ

Under the assumption of equality of the macro-scopic and mesoscopic moduli, one obtains aftera short computation the following equality ofmacroscopic and mesoscopic stresses, R and r,respectively:

r ¼ R� 2lep ¼ Rþ q� ð2Þ

where l is the shear modulus and q* should be inter-preted as a mesoscopic residual stress field. Theknowledge of the mesoscopic stress tensor permitsnow to compute the mesoscopic shear stress s(t)and the hydrostatic pressure p(t).

The criterion in [2] states that the lifetime is infi-nite if elastic shakedown is reached at the macro-scopic and the mesoscopic scale. This is equivalentto the statement, that the mesoscopic stress tensorsatisfies the following inequality:

max½sðtÞ þ apðtÞ� 6 b ð3Þ

where the coefficients a and b are defined fromexperiments as

a ¼ t1 � f1=2

f1=3; b ¼ t1 ð4Þ

The a and b coefficients are deduced from twofatigue tests giving the fatigue limit t1 in alternatetorsion and f1 in alternate flexion.

A common way to represent the criterion in [2] isto display the stress path in a mesoscopic shear s(t)versus hydrostatic pressure p(t) diagram as schemat-ically drawn in Fig. 2. The line defined by the crite-rion is the frontier between infinite lifetime andfatigue, i.e. if a load path crosses the line, fatiguewill occur.

From a practical point of view, one disposesactually of different algorithms which permit toautomatically compute the mesoscopic shear stresss(t) and the hydrostatic pressure p(t) from the stress

Fig. 2. Schematic view of the Dang Van criterion in the {s,P}space.


results over a complete cycle of a finite elementcomputation.

3. Material and bending simulation

3.1. Model geometry

The numerical model used in the present workconsists of a domain that contains a number ofgrains which is surrounded by a homogeneousmatrix. The domain containing the grains is theintegral part of the fatigue analysis, whereas thehomogeneous matrix allows the application ofthe loadings.

The grain domain shown in Fig. 3 includes 156grains and has a rectangular geometry with sidelengths of 147 lm and 133 lm. The grain distribu-tion was microscopically observed from an austen-itic steel with an average grain size of about 10 lm.

The discussed austenitic steel consists of a face-centred cubic (FCC) crystal structure. The lattice

Fig. 3. Description of the micro-structural geometry of the graininclusion.

orientation has been observed with OrientationImaging Microscopy using Electron Back ScatteringDiffraction (EBSD). The individual grains were thenmeshed in two dimensions using linear three noddedelements.

3.2. Materials properties

The model used in the simulation consists of twodifferent material properties which are applied tothe grain containing domain and the homogeneousmatrix. The homogeneous matrix was modelledwith a simple linear elastic material law withoutplastic deformation. A more complex model isneeded to describe the mechanical behaviour ofthe grains. The fatigue properties of a material arelinked with grain plasticity and thus with the crystalstructure. Therefore, it is necessary to consider thelattice structure of the material and to know the lat-tice orientation of all the grains.

From the macroscopic point of view, one canobserve plastic deformation in a metal if the equiv-alent stress exceeds the yield limit ry of the material.On the microscopic scale, this plastic yielding isexplained by the movement of dislocations in thecrystal. This movement is induced by a force thatacts on a slip plane in the direction of the Burger’svector (slip direction). These slip planes and slipdirections depend on the lattice orientation andthe direction of external forces. It is a well-knownfact that the resulting shear stress and not the nor-mal stress induces the movement of dislocations inthe crystal and causes the plastic yielding. Theresulting shear stress s caused by a normal stress rthat acts on a slip system can be calculated withthe Schmid law given in the relation below [10–12]

s ¼ r cos j cos k ¼ rS ð5Þ

where j and k are the angles between the normalstress direction and the normal of the slip planeand slip direction, respectively. And S is defined asthe Schmid factor.

The shear force acting on a dislocation dependson the orientation of the slip system relative to thedirection of the external normal stress. Each slipsystem has a corresponding Schmid factor. Consid-ering an external normal stress, the slip system withthe maximum Schmid factor is subjected to thehighest shear stress. Local plastic yielding will occurif the shear stress s in a slip system exceeds a thresh-old values s0 and induce the movement of disloca-tions in this slip system. This threshold value is a

Table 1Slip planes and directions in a FCC crystal

Slip plane Slip directions

(111) [1�10], [�101], [01�1](1�11) [0�1�1], [10�1], [110](�1�11) [0�1�1], [101], [�1�10](�111) [01�1], [101], [�1�10]

Table 2The macroscopic material parameters: E Young modulus, mPoisson ratio, ry yield limit, rTS ultimate stress limit and H

kinematic hardening modulus

E [GPa] m ry [MPa] rTS [MPa] H [MPa]

210 0.3 800 1300 4000


material parameter and is independent of the latticeorientation or the considered slip system.

The discussed material with a face-centred cubic(FCC) crystal has four slip planes with three slipdirections per slip plane. The FCC lattice structurewith one of the twelve slip systems is shown inFig. 5. All the corresponding slip planes and slipdirections are listed in Table 1.

The material behaviour of each grain is capturedby a simplified phenomenological elastoplastic con-stitutive law with a linear kinematic hardening. It isassumed here that the applied stress resolved alongthe slip direction on the slip plane (to give the shearstress) initiates and controls the extent of plasticdeformation. Yield begins on a given slip systemwhen the shear stress on this system reaches a criti-cal value, called the critical resolved shear stress,independent of the tensile stress or any normalstress acting on the lattice plane.

For each grain g, the local yield criterion fg(rg) isobtained by the Schmid law. The individual yieldstress rc

g depends on the active slip (gliding) system s

rcg ¼ min

s

s0

Ssg

ð6Þ

The critical resolved shear stress s0, the Youngmodulus E and the macroscopic kinematic harden-ing modulus H were identified from a macroscopictensile test. The identification of the mesoscopicmaterial parameters was done using the grain inclu-sion embedded in a homogeneous elastoplastic

Fig. 4. Grain box embedded in the homogenous matrix and the appli

matrix and by adjusting the parameters so as to givea response, which when averaged over all the grainscorresponded to the macroscopic behaviour as illus-trated in Fig. 6.

The complete set of macroscopic material param-eters is presented in Table 2.

3.3. Computation of the fatigue criteria

All the FEM stress computations were performedusing the object oriented finite element toolboxCast3M [13] which includes a number of pre-processing, solving and postprocessing routines.Parts of the postprocessing procedures were per-formed using MATLAB.

The loading case discussed in this paper corre-sponds to five cycles of a pure plane strain bendingof the specimen as described in Fig. 4.

Once the stabilized stress and strain fields areobtained, the stress results of the FEM computa-tions have then been post processed in the followingsteps:

1. Determination of the shakedown state. This hasbeen checked by computation of the relative shift

of the cumulated plastic strain of the last loadingcycle when compared to the one before. The shiftis interpreted as plastic shakedown or ratchetingif it exceeds a limiting value and as elastic shake-down otherwise. The limiting value was deter-mined empirically from observations of singlegrains.

ed boundary tractions creating a stress gradient in the specimen.

Fig. 5. FCC crystal with one slip plane and its associated slipdirections.


2. Computation of the Dang Van criterion for each

slip system in each grain (slip system projection)

[2,14]. The computation of the Dang Van crite-rion for each slip system is done using the preciseknowledge of the grain orientation and the slipsystems of the grain. As such, one can preciselycompute the mesoscopic shear and pressure oneach slip system, and compute the inequality in[2] in each case in order to determine the criticalgrain and slip direction.

3. Computation of the Dang Van fatigue criterion ineach grain. The computation of the criterion[2,14] in each grain is done using the classicalalgorithms of the Dang Van criterion with themean stress field computed over each grain asan input value.

4. Computation of the Dang Van fatigue criterion for

the homogenous structure. The computation ofthe criterion in [2,14] is done in this case by using

Fig. 6. Simulated grain mean and experimental behaviours of acyclic tension test.

the classical algorithms of the criterion with thestress field computed from a homogenous elasticstructure submitted to the same load. In this caseonly the hot spot of the structure, i.e. most criti-cal point, is plotted in the diagram [2].

4. Results and discussion

4.1. Preliminary results: uniform applied load

The first step was to confirm that the number ofgrains constitutes a representative sample and thatshear obtained by the projection of the mesoscopicstress vector onto the slip systems leads to valuessimilar to the mesoscopic pressure and shearobtained with the assumptions taken in the DangVan criterion. For this purpose, the model wasloaded with a fully reversed shear load. In this case,

Fig. 7. Grain critical instant in the Dang Van plot: (a) case of areversed shear loading inducing purely elastic strains; (b) case of areversed shear loading inducing elastoplastic strains.


the stress tensors of the macroscopic and meso-scopic stresses are equal due to the absence ofhydrostatic pressure. Thus, in the case of a fullyreversed loading, the centre of the smallest hyper-sphere circumscribing the stress path is zero in allits components. The mesoscopic shear is thus equalto the amplitude of the imposed shear stress.

Fig. 7a represents the values of mesoscopic shearagainst mesoscopic pressure determined with thecriterion in [2] as well as with the slip system projec-tion. The most critical point of the stress path of thelast cycle is plotted for each grain. It can be seenthat the points of the most critical instants deter-mined with the criterion in [2] are represented bythe same point that corresponds to the homoge-neous solution.

The grain values computed with the slip systemprojection vary in their magnitudes of mesoscopicshear. This is a direct consequence of the differentlattice orientations of the metallic grains which leadto different projections for the shear stress on theslip direction. The criterion in [2] when applied tothe homogeneous solution computes the maximalmesoscopic shear under the assumption of an uni-form spatial orientation of slip directions, as it willhomogenise the solution over a large number ofgrains. Thus, under a homogeneous applied stress,both criteria provide equal points in the diagram,proving that the homogenisation over the 150 grainsfunctioned as expected.

In a next step, the shear stress was increased to arange where some of the grains were plastified.

Fig. 8. rxx and rxy components of the residua

Fig. 7b shows the values of mesoscopic shear andpressure obtained with the Dang Van criterion forthis loading. It can be observed that the range ofmesoscopic shear and pressure increases with theinitiation of yielding in several grains. This is adirect consequence of the residual stress fieldinduced by grain plasticity. The stress componentsrxx and rxy of the residual stress field of theunloaded specimen after the last load cycle are dis-played in Fig. 8. The plot of the rxy residual stressesgives an overview about the plastified grains. Thegrains are surrounded by residual stresses (see rxx

distribution) which induce the hydrostatic pressureand cause the spread of points in Fig. 7b.

In addition to the grain orientation (O1)observed with EBSD, two further calculations wereperformed with randomly (uniform distribution)generated lattice orientations (O2,O3) based onthe same grain structure. The clouds of grain criticalinstants with different lattice orientations differinsignificantly where the primary orientation (O1)contains the most critical grains. That confirms thatthe model contains a sufficient number of grains andleads to rather conservative results.

4.2. Influence of the stress gradient

As in the case of the shear loading, the first loadin the case of a bending moment induced a commonelastic shakedown state for all the grains. Themesoscopic pressure and shear computed with theDang Van criterion as well as with the slip system

l stress field after the last loading cycle.

Fig. 9. Critical instant of the stress path for each grain in theDang Van plot for a repeated bending loading.

Fig. 10. Critical instant of the stress path for each grain in theDang Van plot for different stress gradient.


projection are compared with the homogeneoussolution in Fig. 9. One can remark that the homo-geneous solution can be used as a good referencefor the grain values of the computation [2] whichare right below it. The points form a straight line,because the values of pressure and shear dependonly on the grain’s position in the grain mesh andconsequently on the imposed pressure.

However, regarding the shear in the slip systems,it can be observed that the most critical grain isplaced a certain distance from the point represent-ing the homogeneous solution. Additionally, themesoscopic shear in the slip systems of some lessstressed grains is less than 50% of the value of thehomogeneous solution.

As the difference between the two computationsis only the apparition of a stress gradient this indi-cates that the stress gradient has a non-negligibleeffect on the fatigue limit.

A more significant influence can be observed ifthe imposed stress exceeds the yield limit of somegrains located in the most stressed zone of the struc-ture. From the macroscopic point of view, the mate-rial response has to be in elastic shakedown in orderto permit the computation of the homogeneoussolution and the application of the criterion [2].Thus, the value of the applied load has to bebetween the yield limit of the weakest grain andthe macroscopic elastic shakedown limit of the ref-erence material. Fig. 10 shows the plots [2] for aload with a common maximum pressure of700 MPa applied with four different stress gradients.In order to improve the understanding of the dia-gram, only the points in the neighbourhood of the

most critical grain are plotted. A tentative line [2]of infinite lifetime is drawn through the point ofthe most critical grain belonging to the data set witha stress gradient v = 0 in order to suggest a criticalline [2].

The large range of mesoscopic pressure indicatesthe progress of plastic strain within the grains. Itcan be seen that the distance between the most crit-ical grain and the line [2] increases with an increas-ing stress gradient. Here, it seems the distance of theclouds to the suggested line [2] is increased withincreasing the stress gradient.

Let us recall that the high stress gradient willimpose a varying stress level on the population ofthe 156 grains under observation, more preciselyonly about 10–15 grains will sustain the maximalapplied stress level. Moreover each of these grainshas a particular orientation which implies that onlya small number of slip directions will enter the com-putation when the fatigue criterion is projected overthe slip systems. However when computing theDang Van criterion the underlying assumption is amaximisation over all possible planes, as slip sys-tems are supposed to be homogenously distributed.The difference displayed in Fig. 10 between slip sys-tems and homogenous computation illustrates thefact that homogenisation does not function for ahigh stress gradient.

Compare the critical instants of each grain withthe homogeneous solution. And regard it as anadditional factor the pressure, that the macroscopicapplication of the criterion in [2] is more conserva-tive than the grain values of the higher gradients,but less conservative for the lower gradients. Thisis a consequence of a distribution of plastic strain

Table 3Shakedown state of the grains for various stress gradients

Stress gradient v [MPa] 0 300 600 900Number of grains remaining elastic 28 51 86 107Number of grains in elastic shakedown 128 105 70 49Number of grains in plastic shakedown 0 0 0 0Plastified area/ total area [%] 75 66 46 32


in same grains which is smoothed out when highergradients are applied.

In order to estimate the quantity of plasticity inthe sample the total number of grains which are inelastic shakedown as well as the fraction of the plas-tified area has been analysed. The results are sum-marised in Table 3.

As one can see, the number of plastified grainsand the plastified area decreases significantly withan increasing value of the stress gradient. Fig. 11shows the plastic strains for two different stress gra-dients after the applied load cycles. One can alsoremark an increase of the maximum value of epl

xx ofmore than 50%, when the gradient is increased fromv = 0 MPa/mm to v = 900 MPa/mm.

In the previous paragraph, an important influ-ence of the stress gradient has been observed, as dis-played using the projections on the slip directions ineach grain in the plots [2]. However, up to now, theidentification of the material parameters a and b,parameters defining the fatigue limit, has not beendiscussed. The fatigue life line is defined accordingto theory by the elastic shakedown limit at thegrains scale. Use the preceding computations inorder to determine precisely the shakedown limit

Fig. 11. Plastic mesostrain for v = 0 and v = 900 MPa/mm

at the grain scale and define in this way the fatiguelimit. The natural question underlying to this proce-dure is: does the fatigue limit depend of the macro-scopically applied stress gradient?

4.3. Shakedown and fatigue limit with stress gradient

Thus, the fatigue limit was determined followingthe steps of the technique described below:

1. Apply a macroscopic load with the given stressgradient in order to attain only the yield limitof the weakest grain.

2. Compute the projection on each slip system forall grains.

3. Determine the shakedown state of each grain.4. If all grains are in elastic shakedown, increase the

apply load with fixed increment, and continuewith step 2 until to reach a plastic shakedown.

This procedure for identifying the fatigue limit,considered as an elastic shakedown threshold wasperformed for five constant stress gradients v. Theresults of these fatigue simulations are presented inFig. 12a.

In order to interpret the results, let us recall thatthe simulated fatigue tests discussed before were allperformed with a repeated load where rmin = 0. Inthe computational approximation to the fatiguelimit, only the maximum stress was varied. Regard-ing the different mean values in the calculated fati-gue limits, there is a mean stress effect besides thestress gradient effect which has to be considered in

for a common maximum stress of rxx = 700 MPa.

Fig. 12. Fatigue limit results: (a) computed fatigue limits as afunction of the stress gradient; (b) computed fatigue limits fordifferent mean stresses.

Fig. 13. Computed fatigue limits related to the common meanstress rm = 0.


the results presented before. This mean stress influ-ence was analysed with further fatigue simulationsof repeated loadings with different mean stresses.

This influence can be simply quantified in termsof the law of Gerber parabola:

rA ¼ rejrm¼0 1� rm

rTS

� �2" #

ð7Þ

where rejrm¼0 is the endurance limit for a fullyreversed loading (R = �1) and rTS is the ultimatetensile strength. This assumption was taken inaccordance with its experimental validation in uni-axial loading of ductile materials like it has beenthe case in the simulations. The endurance limit inreversed tension-compression rejrm¼0 as well as thetensile strength rTS in Eq. (7) were considered as‘‘scale parameters’’ to achieve the best fit for thecomputed values. Therefore, two further fatiguelimits in tension-compression with different meanstresses were computed such that three pairs of val-ues (see Fig. 12b) were at the disposal to obtain thevalues of rejrm¼0 and rTS which are, respectively,equal to 590 MPa and 1400 MPa. It is interestingto remark that the obtained values match the mate-rial parameters of the referenced steel 42CrMo4,

which has a material behaviour close to the one rep-resented in Fig. 3.

With the model specific parameter rTS, the stressgradient dependent fatigue limits can be related to acommon mean stress. In order to compare the com-puted data with experimental results, this meanstress was chosen to rm = 0 (reversed loading).The re–v plot of the related values is shown inFig. 13.

The linear trend is in accordance with the exper-imental fatigue results obtained on cylindrical rotat-ing bending specimens with various radii in [16].

4.4. Adaptation of the Dang Van criterion

The computed shakedown limits was used todetermine the parameters a and b of the fatigue cri-terion in [2].

The material parameter b equals the fatigue limiton reversed torsion loading. Accepting the experi-mentally justified fact that the fatigue limit for shearloads is not affected by a stress gradient effect [16],the parameter b was fixed and only the slope ofthe line [2] of infinite lifetime was varied.

However, obtaining the fatigue limit with theshakedown criterion for alternating shear load wasless simple. All the grains responded with a plasticshakedown state if the equivalent stress exceededthe local yield limit of the grains. A possible expla-nation is that the number of imposed load cycleswas not sufficiently large to impose the elasticshakedown state. Considering the experimentalresults of [15,17], the fatigue limit for alternatingshear was considered to be the shear yield limit ofthe weakest grain. As a consequence the fatigue lim-its for alternating shear is given by the followingrelation:


sTe;�1 ¼ min

grsgr

y ¼ mingr

rgryffiffiffi3p� �

¼ 1ffiffiffi3p s0

Smax

¼ 323 MPa ð8Þ

Fig. 15. Grain critical instants at the computed fatigue limit forv = 12 MPa/mm and included lines of infinite lifetime in theDang Van plot.

The slope a of the infinite lifetime line [2] can beachieved in two different ways:

(a) Using the fatigue limits of a reversed bendingassociated with the shakedown criterion. Thistechnique is not accurate, as the valuesbecause the alternate fatigue limits are froman empiric Gerber law.

(b) By drawing the line through the point [0, b]and the most critical instant of the stress pathof (i) the homogeneous solution, (ii) the mostcritical grain in plastic shakedown or (iii) themost stressed grain regarding to the values ofmesoscopic shear and pressure.

Taking into account the expected accuracy, pro-cedure (b) can be tested. The results based on thethrees different points for stress distributions withv = 300 MPa/mm and v = 1200 MPa/mm are dis-played in the Dang Van diagram shown in Figs.14 and 15. It shows the most critical instant of themesoscopic stress path for every grain. As onewould expect, the line defined by the grain withthe maximum values of mesoscopic shear and pres-sure has the smallest slope and represents conse-quently the most conservative approach. One cangenerally remark that the cloud of values computedusing the criterion [2] in each grain is largely spre-aded and does not permit to draw ay conclusion.However, the projection over the slip systems ineach grain are a consistent approach and are gen-erally well placed under the line defined by the

Fig. 14. Grain critical instants at the computed fatigue limit forv = 3 MPa/mm and included lines of infinite lifetime in the DangVan plot.

homogeneous computation as one would expect.Moreover the distance between the homogeneoussolution and the cloud of projection on the slip sys-tem is increased with increasing stress gradient if themaximal stress is constant. This is in accordancewith the fact that the mean stress over all grains willdecrease with an increasing gradient.

5. Conclusions

This work presented a numerical fatigue analysisof specimens using a simplified polycrystallinemodel at the mesoscopic/grain scale. The impactof the applied stress gradient on fatigue was shownconsidering the example of a bending experiment.

It was shown that high localization of stressescauses the failure of the homogenisation assump-tions as only few grains are exposed to very highstresses. This failure implies a fundamental changein the application of fatigue criteria in general andmore particularly of critical plane criterion [2] incases of high stress gradient. The results also pro-vide a physical explanation for critical distancesand notch factors encountered in engineeringpractice.

A finer quantitative analysis both at the meso-and macroscopic level should allow the develop-ment of a better understanding and ability to predictfatigue for stress states involving high gradientssuch as those found in notched or cracked speci-mens and bending experiments.

On the one hand side, a direct measurement ofsurface strains is actually possible using opticalstrain analysis under electronic microscopy couldbe used to check a certain number of assumptionsof the presented work.


On the other hand side, the numerical modellingitself can be also improved by using a more complexhardening model in the polycrystalline behaviour. Itis also obvious that finer structures like surfacestate, grain boundaries, inclusions and impuritieshave been completely neglected in this frame setting.

A further drawback is the usage of a simple twodimensional computation under a plane strainassumption. The three dimensional computationdoes not address a more complex material behav-iour, but demands a three dimensional measurementof the grain structure which was not available at thetime of the work. However this could be obtainedsometime in the near future.

References

[1] K. Dang Van, I.V. Papadopoulos, High-cycle metal fatigue,from theory to applications, in: K. Dang Van, I.V. Papad-opoulos (Eds.), CISM Courses and Lectures no. 392,Springer-Verlag, 1999.

[2] K. Dang Van, A high cycle fatigue criterion applied inbiaxial and triaxial out-of-phase stress conditions, in: D.L.McDowell, R. Ellis (Eds.), Advances in Multiaxial Fatigue,ASTM STP, 1191, ASTM, Philadelphia, 1993, pp. 120–130.

[3] R.A. Naik, D.B. Lanning, T. Nicholas, A.R. Kallmeyer, Acritical plane gradient approach for the prediction of notchedHCF life, Int. J. Fatigue 27 (5) (2005) 481–492.

[4] D. Taylor, Geometrical effects in fatigue: a unified theoret-ical model, Int. J. Fatigue (21) (1999) 413–420.

[5] H. Adib, G. Pluvinage, Theoretical and numerical aspects ofthe volumetric approach for the fatigue life for the notchedcomponents, Int. J. Fatigue (25) (2003) 67–76.

[6] A. Banvillet, T. Palin-Luc, S. Lasserre, A volumetric energybased high cycle multiaxial fatigue criterion, Int. J. Fatigue25 (8) (2003) 755–769.

[7] F. Morel, T. Palin-Luc, C. Froustey, Comparative studyand link between mesoscopic and energetic approaches inhigh cycle multiaxial fatigue, Int. J. Fatigue 4 (23) (2001)317–327.

[8] F. Morel, T. Palin-Luc, A non-local theory applied to high-cycle multiaxial fatigue, Fat. Fract. Eng. Mat. Struct. 25 (7)(2002) 649–665.

[9] D. Nowell, D. Dini, Stress gradient effects in fretting fatigue,Tribol. Int. 36 (2) (2003) 71–78.

[10] S. Suresh, Fatigue of Materials, second ed., CambridgeUniversity Press, Cambridge UK, 1998.

[11] G.J. Weng, A Micromechanical theory of grain-size depen-dence in metal plasticity, J. Mech. Phys. Solids 31 (3) (1983)193–203.

[12] K. Kowalczyk, W. Gambin, Model of plastic anisotropyevolution with texture-dependent yield surface, Int. J. Plast.20 (2004) 19–54.

[13] CAST3M.: An object oriented finite element toolbox, http://www-cast3m.cea.fr/.

[14] I.V. Papadopoulos, A new criterion of fatigue strength forout-of phase bending and torsion of hard metals, Int. J.Fatigue 16 (6) (1994) 377–384.

[15] R.G. Pogoretskii, G.V. Karpenko, Effect of test piece lengthon the fatigue strength of steel in air, Proc. KhimicheskayaMekhanika Materialov 1 (1965) 90–94.

[16] I.V. Papadopoulos, Invariant formulation of a gradientdependent multiaxial high-cycle fatigue criterion, Eng. Fract.Mech. 55 (1996) 513–528.

[17] G. Sines, Behavior of metals under complex static andalternating stresses, in: S. Waisman (Ed.), Metal Fatigue,McGraw-Hill Book Company, New-York, 1959, pp. 145–169.

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A multiscale approach of fatigue and shakedown for notched ...€¦ · A multiscale approach of fatigue and shakedown for notched structures G. Bertolino a, A. Constantinescu a,*,