Vol. 128 (2015) ACTA PHYSICA POLONICA A No. 4

Proceedings of the International Symposium on Physics of Materials (ISPMA13)

Quantitative Model of the Surface Relief Formation

in Cyclic Straining

J. Poláka,*

and J. Manb

aCEITEC IPM, Institute of Physics of Materials AS CR, �iºkova 22, 616 62 Brno, Czech RepublicbInstitute of Physics of Materials AS CR, �iºkova 22, 616 62 Brno, Czech Republic

The cyclic strain localization in crystalline materials subjected to cyclic loading results in speci�c dislocationarrangement in persistent slip bands consisting of alternating dislocation rich and dislocation poor volumes. Inthe present model the interaction of mobile dislocations in persistent slip band leading to the formation of pointdefects during cyclic straining is considered. Point defects are steadily produced and simultaneously annihilateddue to localized cyclic straining. The non-equilibrium point defects migrate to the matrix and result in transfer ofmatter between persistent slip band and the matrix and formation of the internal stresses both in the persistentslip band and in the matrix. Plastic relaxation of internal stresses leads to the formation of the characteristicsurface relief in the form of persistent slip markings (extrusions and intrusions). Process of point defect migrationis quantitatively described and the form of extrusion and parallel intrusions is predicted under some simplifyingassumptions. The predictions of the model are discussed and predicted surface relief is compared with selectedobservations of surface relief produced by cyclic straining.

DOI: 10.12693/APhysPolA.128.675

PACS: 62.20.me, 81.05.Bx, 61.72.Ff, 61.72.Hh, 61.72.Lk, 61.72.J�

1. Introduction

Cyclic plastic straining in crystalline materials di�erssubstantially from unidirectional plastic straining in spiteof the fact that in both cases dislocation motion playscentral role. The dislocation motion in cyclic strainingin one direction is severely limited. The repeated un-loading and alternating deformation in both directions istypical. Number of stress and strain reversals can achieveseveral tenths up to several milliards before fracture ofthe specimen or component takes place. High number ofreversal (or cycles) results in very high cumulative strainabsorbed by the material during the fatigue life in spiteof the fact that the strain introduced in each cycle is verysmall.Repeated cyclic plastic straining leads to the forma-

tion of speci�c dislocation structures di�erent from thoseformed in unidirectional cyclic straining [1]. Moreover,with increasing number of cycles in majority of crys-talline materials cyclic plastic strain ceases to be dis-tributed homogeneously and signi�cant inhomogeneitiesarise. Cyclic plastic strain localization is the substantialfeature of the cyclic loading. Inhomogeneous deforma-tion of the material results also in the inhomogeneousdistribution of dislocations in the material and disloca-tion arrangement changes during the fatigue life. Thebands of localized slip � persistent slip bands (PSBs) �arise. Localized cyclic deformation leads to the formationof the speci�c surface relief in the form of persistent slipmarkings (PSMs) which represent the nuclei for fatiguecracks.

*corresponding author; e-mail: polak@ipm.cz

Dislocation motion in the PSBs has been considered byseveral authors in the past [2�5]. Number of theories hasbeen proposed to explain the formation of the surface re-lief and initiation of the early fatigue cracks. Important�ndings represent experimental data on the resistivitymeasurements during cyclic plastic straining [6�8] reveal-ing that due to dislocation interactions point defects aresteadily produced in cyclic plastic straining. Experimen-tal �ndings lead to the �rst attempts to propose physi-cally founded mechanism leading to the formation of thepersistent slip markings and later to the initiation of fa-tigue cracks [4]. Further extension of the model based onthe production and migration of point defects in PSBsleads to the explanation of the steady extrusion and in-trusion growth [9], to the description of the quantitativegrowth of extrusions [10] and later to the generalizationand quanti�cation of the model of surface relief formationin cyclic plastic straining [11, 12].In the present contribution some experimental data on

the dislocation arrangement in the PSBs and on the sur-face relief produced in cyclic plastic straining are pre-sented. Dislocation motion and interactions within PSBsare considered and production rates of point defects areestimated. The point defect production and migrationin the speci�c dislocation structure of PSBs is quantita-tively described and predictions concerning the surfacerelief are presented.

2. Dislocation structure of PSBs and

characteristic relief of PSMs

Dislocation arrangement of cyclically deformed mate-rials is studied using transmission electron microscopy(TEM) [2, 3, 13, 14] or using electron channeling contrast(ECC) [15, 16]. Most studies were performed in copper

(675)

676 J. Polák, J. Man

single crystals [13, 14] but recently the interest is concen-trated to polycrystalline materials [17�19].Typical dislocation arrangement in fatigued copper

single crystal (fcc structure) containing individual PSBis shown in Fig. 1a. Dislocation structure in a grain offatigued polycrystalline ferritic stainless steel (bcc struc-ture) cycled with low constant plastic strain amplitudesis shown in Fig. 1b. In both cases the foil is perpendic-ular to the active slip plane and nearly parallel to theprimary Burgers vector. PSBs with the ladder-like dislo-cation structure intersect the basic matrix structure bothin copper single crystal and in a grain of ferritic stain-less steels. Ladder-like dislocation structure in three di-mensions corresponds to alternating thin dislocation-richwalls and thick dislocation-poor channels.

Fig. 1. Dislocation structure in fatigued metals (foilsare perpendicular to the to the primary slip plane (a) in-dividual PSB in copper single crystal, (b) parallel PSBsin a grain of ferritic stainless steel.

Surface relief of cyclically deformed materials also wit-nesses the localized deformation of the material. PSMsarise on the originally smooth surface due to elastic-plastic cyclic loading. Figure 2 shows PSMs on the sur-face of a grain of 316L steel cycled with low plastic strainamplitude. Both the SEM image (Fig. 2a) and AFM im-age of plastic replica of the surface (Fig. 2b) reveal thinPSMs consisting of extrusions and intrusions and the ma-jority of the surface of the grain is not disturbed by cyclicplastic loading. Plastic replica of the surface was used inorder to distinguish the intrusions.

Fig. 2. Surface relief of fatigued 316L steel (a) SEMimage of the surface with parallel PSMs, (b) AFM imageof the plastic replica of the surface.

3. Mechanisms of cyclic plastic straining in PSB

and point defect production rates

Basic mechanism of localized cyclic plastic straining inthe PSB is shown in Fig. 3. Thin dislocation rich wallsconsisting mostly of edge dislocation dipoles and multi-poles emit dislocation loops in the channels on both sides.Dislocation loops expand and reach the opposite wall,edge segment is built in the wall and most plastic strainis carried by the screw segments. Dislocation arrange-ment and dislocation interactions within the channel areschematically shown in detail in Fig. 4. Dislocations inPSB are projected in the (12̄1) plane (Fig. 4a) and in theprimary (111) plane (Fig. 4b). High plastic strain withina cycle is accommodated by the formation of the dislo-cation loops starting from the edge dislocation segmentsin the walls and by their expansion into the channels.The majority of the loops reach the neighbor wall with-out interaction with other dislocations within the channeland the edge sections become integrated in the dipolarstructure of the wall. The largest number of dislocationinteractions take place in the walls. Point defects createdin the walls are e�ectively annihilated by short range mi-gration to the edge dislocations in the walls or by mutualinteraction of vacancy and interstitial type defects.Since local plastic strain amplitude within the PSB is

high, two dislocation loops expanding from the neighborwalls can interact within a channel. Provided two oppo-

Quantitative Model of the Surface Relief Formation in Cyclic Straining . . . 677

Fig. 3. Schematics of dislocation arrangement and in-teractions in PSB.

site loops meet on the neighbor planes rows of vacanciesand rows of interstitials arise. Polák and Man [11] re-cently derived production rates of vacancies and intersti-tials in a PSB subjected to local plastic shear strain am-plitude γP (see Fig. 4). Dislocation loops in the PSB areemitted from the walls into the neighbor channels whosewidth is dch. Some loops can meet within the channel,interact and create the unit dipole corresponding to therow of vacancies or interstitials. The production rate ofpoint defect of type j (vacancy v or interstitial I) in acycle pj is

pj =

(dcjdN

)p

= 2sjb

dchγ2p . (1)

sj is a proportionality factor, smaller than 1 that is di�er-ent for unit vacancy dipoles and unit interstitial dipoles.It takes into account whether both dislocations from theopposite walls in a half-cycle are emitted simultaneously.It depends also how far the row of point defects can beextended by the movement of the non-conservative jogson two screw dislocations formed due to encounter of twoloops within a half-cycle period. Since formation energyof the interstitial is much larger than formation energyof a vacancy the length of the interstitial dipole formedduring half cycle will be smaller than that of vacancydipole. Therefore we expect that pI � pv and majorityof point defect will be vacancy type defects.Point defects produced during cyclic straining in the

channels of the PSB in the form of unit dipoles can be an-nihilated athermally by encountering an edge dislocation.The slip plane of the edge dislocation must be neighborplane to the slip plane containing the row of point de-

Fig. 4. Schematics of dislocation interactions and for-mation of point defects in PSB, (a) projection in (12̄1)plane, (b) projection in primary (111) slip plane.

fects. On encounter with a row of point defects edgedislocation climbs one atomic distance in appropriate di-rection. When cj is the concentration of point defect oftype j the annihilation rate of a defect of type j (vacancyor interstitial) in a cycle aj was evaluated to [11]:

aj =

(dcjdN

)a

= 2γpcj . (2)

The point defect concentration in a PSB with stabilizedwall dislocation structure vs. the number of cycles is ob-tained by equating production and annihilation rates andsolving for cj :

cj = sjb

dchγp [1− exp(−2γpN)] . (3)

Provided point defects are annihilated only athermally,their concentration increases with increasing number ofcycles until saturated value is reached

cj,eq = sjb

dchγp. (4)

In case point defects do not migrate to sinks the satu-rated point defect concentration is proportional to thelocal plastic shear strain amplitude.

4. Migration of point defects

and formation of the surface relief

Since vacancy production rates are higher, in the fol-lowing we shall consider only vacancies. They are steadilyproduced in the whole volume of the PSB with the pro-duction rate p (increase of the vacancy concentration in

678 J. Polák, J. Man

a cycle) and simultaneously are annihilated by sweepingdislocations. The width of the PSB is w and its lengthis l (see schema of PSB in Fig. 5). The annihilation co-e�cient A is the fraction of vacancies swept by mobiledislocations during one cycle. It was estimated in Sect. 3to A = 2γp. Due to vacancy migration the vacancy con-centration cv is a function of the distance x (see Fig. 5).Concentration pro�le can be obtained by solving appro-priate di�usion equations [12]. Migration of vacancies tothe walls is neglected (see Sect. 5).

Fig. 5. PSB of the width w and length l embedded inthe matrix (arti�cially inclined perpendicularly to thesurface) and the pro�le of vacancy concentration. Va-cancies migrate to the matrix and are annihilated thereat edge dislocations.

As soon as the vacancies arrive in the matrix they canbe annihilated at edge dislocations. Analytical solutionwas obtained for homogeneously distributed edge dislo-cations. Annihilation of vacancies at homogeneously dis-tributed sinks follows �rst order kinetics and the rate ofannihilation is proportional to edge dislocation densityρe and di�usion coe�cient Dv.By solving di�usion equation in the PSB and di�u-

sion equation and annihilation at the homogeneously dis-tributed edge dislocations having the density ρe in thematrix under appropriate boundary conditions the pro-�le of the steady vacancy concentration during cyclingwas evaluated [12]:

cv =p

A

(1− cosh(ax)

cosh(aw/2) + a√ρe

sinh(aw/2)

)for x ≤ w/2 (5)

and

cv =p

A

exp(−√ρe(x− w/2)

)1 +

√ρea coth(aw/2)

for x ≥ w/2, (6)

where a =√

AτDv

is the reciprocal characteristic di�usiondistance which takes into account vacancy annihilation.

Knowing the steady pro�le of vacancies in the PSB andin the adjoining matrix we can evaluate the growth rateof the extrusion (x < w/2) and intrusion (x > w/2). Inthe volume element ldx (l is the length of PSB in thedirection of the Burgers vector, see Fig. 5). The rate ofthe volume change due to migration and annihilation ofvacancies is equal to the di�erence of the vacancy �ux inthis volume and out of this volume. Vacancy �ux J is

J = −Dv∂cv∂x

. (7)

Since absorption of each vacancy corresponds to the ar-rival (in the PSB) or removal (in the matrix) of one atomfrom the lattice the internal compression stress (in thePSB) or the internal tensile stress (in the matrix) arise.Three-dimensional internal stresses are added to the ex-ternal cyclic stress σ applied both to PSB and to thematrix. It represents mean stress in the direction of theloading. The mean stress can be relaxed plastically byprocesses of cyclic creep only in the direction of the Burg-ers vector of primary dislocations. The relaxation willproceed di�erently in PSB and in the matrix.

Internal compression stress can be completely relaxedin the PSB by expansion of the material to the surfaceby process of cyclic creep. The change of the height h ofthe surface in the direction of the Burgers vector, duringone cycle in the distance x from the center of the PSB isthus

∆h

∆N= −τ lDv

d2cvdx2

. (8)

Inserting Eq. (5) into Eq. (8) and performing the deriva-tive we obtain extrusion growth rate. The extrusionheight hE is proportional to the number of loadingcycles N :

hE =plN cosh(ax)

cosh(aw/2) + a√ρe

sinh(aw/2)for x < w/2. (9)

The highest extrusion is produced at the PSB/matrixinterface (x = w/2).

Since the critical yield stress in the matrix is higherthan the critical yield stress in the PSB and the cycliccreep during cycling in the matrix starts only if the com-ponent of the mean stress in the direction of the actingcyclic stress reaches the critical value equal to the dif-ference between the yield stress of the matrix and PSB.The intrusion growth will thus start with some delay, i.e.when the critical elastic strain in the particular sheet ofthe matrix is built up.At the number of cyclesN the depth dI of the intrusion

(dI is positive) is [12]:

dI = rImN exp(−√ρe

(x− w

2

))− dc

if rImN exp(−√ρe

(x− w

2

))> dc for x > w/2, (10)

and

dI = 0 if rImN exp(−√ρe

(x− w

2

))≤ dc

for x ≤ w/2, (11)

Quantitative Model of the Surface Relief Formation in Cyclic Straining . . . 679

where dc is parameter which characterizes the delaybetween the start of the intrusion growth relative tothe onset of the extrusion growth. It is equal to themaximum depth of the hypothetical intrusion (at thePSB/matrix interface) which would arise provided theyield stress of the PSB and the matrix were the same.rIm is the maximum intrusion growth rate (at thePSB/matrix interface) equal to

rIm =pl

√ρea

coth(aw/2) + a√ρe

. (12)

No intrusion will arise until critical number of cycles Ncis reached

Nc =dcrIm

(13)

and maximum intrusion depth dIm at numberof cycles N is

dIm = rImN − dc. (14)It increases linearly with the number of loading cycles.Intrusion width ∆xI is

∆xI =1√ρe

ln

(NrImdc

). (15)

Figure 6 shows the pro�le of the PSM (only right part ofthe symmetrical pro�le) consisting of the extrusion andintrusion as calculated using Eqs. (9)�(11). The param-eters shown in caption to Fig. 6 correspond to cyclicallystrained copper at room temperature.

Fig. 6. PSM pro�le (right part of a symmetrical pro-�le) consisting of extrusion and intrusion calculated us-ing Eqs. (9)�(11). It corresponds to cyclic strainingof copper at room temperature for 1000 cycles. Theextrusion height in the center hEc = 3000 b, criticaldistance is dc. Edge dislocation density in the matrixρe = 1 × 10−13 m−2. All dimensions are in units of theBurgers vector modulus b.

The pro�le of the extrusion accompanied by two intru-sions has been calculated assuming total plastic relax-ation of the internal compression and tensile stresses in

the PSB and its neighborhood without considering theelastic stresses and strains. In reality, taking into ac-count the elastic strains the transition from extrusion tointrusion will be much smoother than shown in Fig. 6.

5. Discussion

Quantitative relations describing the shape of extru-sions and parallel intrusions were derived under severalsimplifying assumptions. The most limiting assumptionsrelate to the dislocation arrangement in the PSB andin the neighbor matrix. We have supposed the ladder-like dislocation arrangement of the PSB which allows themajority of produced point defects to escape to the ma-trix and participate on the production of PSMs. Evenin ladder-like structure only fraction of vacancies escapesto the matrix since another fraction is absorbed in dislo-cation rich walls in the PSB. If the dislocation structureis more irregular as e.g. in 316L steel [18] but still con-tains the dislocation rich and dislocation poor volumesthe fraction of vacancies systematically leaving PSB tothe neighbor matrix decreases further. Production rateof vacancies p is reduced and PSM builds up with smallerrate. In Eq. (9) and (10) the production rate p should bereplaced by e�ective production rate peff .Much higher e�ect on the shape of the PSM has the

assumption of the homogeneous distribution of edge dis-locations in the matrix serving as sinks for vacancies thatmigrate from the PSB. Some typical distribution of dis-locations in the neighborhood of PSBs is illustrated inFig. 1. In copper (Fig. 1a) the matrix consists of patchesand channels with approximately equal sizes. This dis-location structure of the matrix would lead to the widerintrusions since vacancies could migrate larger distancesfrom PSB/matrix interface before they are annihilatedat dislocations. Dislocation structure of the matrix inferritic stainless steel is close to the assumption of homo-geneous distribution. In case the edge dislocation distri-bution in the matrix close to the PSB/matrix interfaceis highly inhomogeneous both extrusion height and in-trusion depth will vary along the length of the PSM orno intrusion at all will be formed. Variable height ofextrusions and depth of intrusions has been often ob-served [1, 19�21] (see also Fig. 2b).Comparison of the quantitative predictions of the

model with experimental data [19�21] shows reason-able agreement concerning the height of extrusions andintrusions and also their shape. Great achievement ofthe model is the explanation of the experimental �ndingsthat extrusion starts early in the fatigue life and only af-ter signi�cant delay intrusions develop. The model alsoexplains the dependence of the intrusion depth on thelocal dislocation arrangement in the matrix. In case lowedge dislocation density is present in the matrix close toPSB/matrix boundary the formation of intrusion is de-layed signi�cantly or intrusions do not arise at all.Sharp intrusions grow and become crack-like defect

which creates its own cyclic plastic zone. Due to irre-

680 J. Polák, J. Man

versibility of the cyclic plastic strain at the tip of the in-trusion it becomes primary stage I fatigue crack. Whenthe growth rate of the stage I crack due to strain irre-versibility becomes higher than the growth rate of theintrusion the nucleation of the fatigue crack is accom-plished and the crack grows in stage I.

Acknowledgments

The present work was realized in CEITEC withresearch infrastructure supported by the project No.CZ.1.05/1.1.00/02.0068. The support by the projectsRVO: 68081723 and grants 13-23652S 13-32665S of theGrant Agency of the Czech Republic is gratefully ac-knowledged.

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