identifying creep mechanisms in plastic flow

Terence G. LangdonDepartments of Aerospace & Mechanical Engineering and Materials Science University of Southern California, Los Angeles,U.S.A.

Identifying creep mechanisms in plastic flowDedicated to Professor Wolfgang Blum on the occasion of his 65th birthday

The mechanisms occurring in plastic flow at elevated tem-peratures may be divided into three distinct classes depend-ing upon whether they are intragranular dislocation mech-anisms, grain boundary sliding including superplasticityor diffusion creep occurring through vacancy flow. Thecharacteristics of these various mechanisms are describedand procedures are outlined for distinguishing between thedifferent processes. Using this approach, it is shown thatthere is good experimental data supporting the occurrenceof both Harper – Dorn creep and Nabarro – Herring diffu-sion creep as distinct creep processes. Recent results re-ported from computer simulations, combined with experi-mental observations, suggest the possible occurrence ofgrain boundary sliding at low temperatures in nanostruc-tured and ultrafine-grained materials.

Keywords: Diffusion creep; Grain boundary sliding; Har-per –Dorn creep; superplasticity

1. Introduction

Creep deformation is defined as the nonrecoverable plasticstrain which occurs in a material when it is subjected to aconstant stress (or a constant load) over an extended periodof time. Interest in this subject first arose about one-hun-dred years ago with the first published report by Phillips[1] of the creep deformation occurring in a range of materi-als from India-rubber to glass and metal wires. Over the en-suing years, there have been many detailed descriptions ofthe creep behavior of a very wide range of metallic and non-metallic materials and these various reports have ranged

from theoretical models of creep to experimental observa-tions in the laboratory and detailed reports of the occur-rence of creep in real-life industrial applications. ProfessorWolfgang Blum has made numerous contributions to ourunderstanding of the creep processes over a period of manyyears and it is appropriate therefore, on the occasion of his65th birthday in 2005, that this paper should be dedicatedto Professor Blum in recognition of his seminal contribu-tions in the field.

When creep occurs in a crystalline structure, there is gen-erally a region of primary creep in which the creep rate de-creases as a function of time, a secondary or steady-state re-gion in which the rate of creep remains reasonably constantand then a tertiary region of accelerating creep leading ulti-mately to failure. It was recognized from the early 1950sthat the various possible creep mechanisms may be quanti-fied in a meaningful way by determining the precise func-tional relationship between the creep rate occurring in thesteady-state region and external parameters such as the test-ing stress and temperature. The application of this proce-dure led ultimately to a comprehensive review of creep fora very wide range of metals and metallic alloys [2] and thesame general approach will be followed in this paper. Ac-cordingly, the following section provides a description ofthe fundamental creep mechanisms and the next sectionsdescribe procedures that may be used to uniquely identifythese mechanisms and their significance over a very widerange of testing conditions.

2. Creep mechanisms in crystalline solids

It has been established that the rate of creep deformationdepends primarily upon three separate parameters. The first

T. G. Langdon: Identifying creep mechanisms in plastic flow

522 Carl Hanser Verlag, München Z. Metallkd. 96 (2005) 6

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two parameters are the applied stress, r, and the absolutetesting temperature, T, which relate to the testing conditionsimposed on the sample. The third parameter is associatedwith the microstructure of the material and in this respect,although the creep rate depends to some extent upon factorssuch as the stacking-fault energy in face-centered cubicstructures, the creep rate is quantified most easily in termsof the grain size of the material, d.

It is convenient to express the steady-state creep rate, _ee,through a relationship of the form

_ee ¼ ADGbkT

bd

� �pr

G

� �nð1Þ

where D is the diffusion coefficient (= Do exp ( –Q/RT)where Do is a frequency factor, Q is the appropriate activa-tion energy for the diffusive process and R is the gas con-stant), G is the value of the shear modulus at the testing tem-perature, b is the Burgers vector of the material, k isBoltzmann’s constant, p and n are constants defined as theinverse grain size exponent and the stress exponent, respec-tively, and A is a dimensionless constant.

Since the various theoretical mechanisms developed toexplain creep behavior predict creep rates of the formshown in equation (1), it follows that each mechanism maybe defined uniquely in terms of the values predicted for thefour parameters Q, p, n and A. In practice, however, the val-ue of A is related to structural features and it is more conve-nient in practice to express the various creep mechanismssolely in terms of the values predicted for Q, p and n.

Creep occurs in crystalline solids because of the presenceof defects in the atomic lattice, where these defects may belocalized at points (vacancies), along lines (dislocations)or at planes (grain boundaries). In general terms, disloca-tion mechanisms occur primarily within the grains andthese intragranular creep mechanisms represent the majorflow processes under most creep conditions. However, thepresence of grain boundaries is important because it maylead to grain boundary sliding where adjacent grains aredisplaced with respect to each other with the movement tak-ing place at, or very close to, their common interface. Va-cancy flow becomes important in diffusion creep wherethe vacancies flow, under the action of an applied stress, be-

tween different sets of grain boundaries in a polycrystallinematrix. For simplicity, these various theoretical mechan-isms are summarized in Table 1 where the mechanisms aredivided into the three separate groups of intragranular dislo-cation mechanisms, grain boundary sliding and diffusioncreep with vacancy flow and, in the first four columns ofTable 1, the various processes are delineated in terms ofthe predicted values for Q, p and n: the fourth and fifth col-umns in Table 1 relate to microstructural features of theflow processes which are discussed in more detail in the fol-lowing sections.

It is convenient to consider the basic characteristics ofthese various creep mechanisms.

3. Characteristics of the various creep mechanisms

3.1. Intragranular dislocation mechanisms

For coarse-grained polycrystals, most (if not all) of thecreep strain comes from the intragranular movement of dis-locations. An example of this process is shown in Fig. 1where dislocation loops are emitted and expand outwardsfrom a reasonably uniform distribution of dislocationsources (for example, Frank– Read sources) located on dif-ferent planes [10]. In practice, there is an interference be-tween the stress fields of the loops on different planes andthe leading screw dislocations, having opposite sign, read-


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Table 1. Characteristics of creep mechanisms.

Mechanism Q p n Grain elongation? Grain boundary offsets? Reference

Intragranular dislocation mechanisms:

Dislocation climb Ql 0 4.5 Yes No Weertman [3]Dislocation glide Qi 0 3 Yes No Weertman [4]

Harper – Dorn creep Ql 0 1 Yes No Harper and Dorn [5]

Grain boundary sliding (GBS):

GBS in creep (d > k) Ql 1 3 No Yes (Rachinger sliding) Langdon [6]GBS in superplasticity (d < k) Qgb 2 2 No Yes (Rachinger sliding) Langdon [6]

Diffusion creep and vacancy flow:

Nabarro – Herring creep Ql 2 1 Yes Yes (Lifshitz sliding) Nabarro [7], Herring [8]Coble creep Qgb 3 1 Yes Yes (Lifshitz sliding) Coble [9]

Fig. 1. The principle of intragranular glide and climb in high tempera-ture creep [10].

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ily cross-slip and annihilate. By contrast, the edge disloca-tions lying on different planes are blocked by these mutualinteractions and they pile up until the back stresses exertedon the sources are ultimately relieved by the leading edgedislocations climbing together to annihilate each other. Thisannihilation process permits an additional loop to beemitted from the source so that the mechanism becomes re-generative and capable of producing large creep strains. Byassuming that the dislocations can efficiently absorb andemit vacancies, it was shown in an early model of creep thata process where climb is rate-controlling leads to a steady-state creep rate in which n = 4.5 and Q = Ql, where Ql isthe activation energy for self-diffusion [3]. A later modelof climb, where the piled-up arrays of dislocations decom-pose into dipoles or multipoles, led to the same values forn and Q [11]. This type of behavior is generally termedpower-law creep. It is important to note also that these pro-cesses operate intragranularly so that there is no depend-ence on grain size and p = 0.

Dislocation climb is generally the rate-controlling pro-cess in pure metals because glide occurs rapidly and climbis the controlling step. In metallic alloys, however, soluteatoms may form preferentially as impurity atmospheresaround the dislocations, thereby producing a situationwhere glide is slower than climb and viscous drag becomesthe rate-controlling process. Under creep conditions con-trolled by dislocation glide, it has been shown that n = 3,Q = Qi and again p = 0, where Qi is the activation energyfor interdiffusion of the solute [4].

In practice, the rate-controlling process in solid solutionmetallic alloys is dependent both upon the alloy and uponthe precise testing conditions. Thus, there may be a transi-tion with increasing stress from dislocation climb to viscousglide [12] and at even higher stresses the dislocations maybreak away from their solute atom atmospheres so thatn > 3 [13].

As indicated in Table 1, there is an additional dislocationmechanism which may operate intragranularly and this isHarper – Dorn (H – D) creep based originally on the experi-mental results obtained in the creep testing of pure alumi-num by Harper and Dorn [5]. The precise flow mechanismin H–D creep has been the subject of much speculation[14 – 19] and there are subsequent analyses and results bothconfirming [20, 21] and refuting [22, 23] the validity of theH-D creep process. A procedure for confirming or invali-dating the occurrence of H–D creep is described in a latersection (§ 5) but it is important to note that H–D creep is aNewtonian viscous process with n = 1 and with values ofQ = Ql and p = 0.

3.2. Grain boundary sliding (GBS)

Grain boundary sliding (GBS) occurs when two grains, inresponse to an external stress, slide over each other withthe displacement occurring at or very close to their mutualinterface. The principle of GBS is depicted schematicallyin Fig. 2 where grains 1 and 2 are displaced at the boundaryby a sliding vector S. This sliding vector is made up of threemutually orthogonal displacements: there is a displacementu parallel to the loading axis and displacements v and w per-pendicular to the stress axis and either perpendicular to thesurface or in the plane of the surface, respectively. Two an-gles are also needed to define the orientation of the grainboundary with respect to the stress axis: these are h and wrepresenting the angle between the boundary and the stressaxis in the surface plane and the angle between the bound-ary and the specimen surface on a section cut perpendicularto the surface, respectively.

The v displacements are revealed most readily by usinginterferometry and an example is shown in Fig. 3 for acoarse-grained magnesium alloy tested under creep condi-tions where there are clear offsets in the interference fringesat the points where they impinge upon the grain boundaries[24]. By contrast, marker lines are generally used to recordthe values of offsets occurring in the plane of the surfaceand this is illustrated schematically in Fig. 4 where themarker line AB, scribed parallel to the tensile axis inFig. 4a, becomes displaced to reveal the magnitude of wafter creep testing in Fig. 4b. The Appendix provides a briefsummary of the procedures for using measurements of u, vor w to determine the magnitude of the strain due to GBS,egbs, and hence the overall contribution of GBS to the totalstrain, defined as n = egbs/et where et is the total strain.


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Fig. 2. The principle of grain boundary sliding: S is the sliding vectorbetween grains 1 and 2 and u, v and w are the sliding offsets.

Fig. 3. Evidence for grain boundary sliding in a magnesium alloyusing interferometry: the v offsets can be measured from the displace-ments in the interference fringes at the grain boundaries [24].

Fig. 4. Schematic illustration of (a) a longitudinal marker and (b) theoffset w after grain boundary sliding.

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A simple consideration of GBS leads to the conclusionthat sliding cannot occur as an isolated process. In realpolycrystalline materials, the grains have unique shapes, asis evident in Fig. 4, and GBS can take place only if it isaccommodated by the intragranular movement of disloca-tions. These dislocations will be nucleated at points ofstress concentrations, such as at triple points or grainboundary ledges, and they will move across the adjacentgrain until they encounter an obstacle.

An early investigation of the two-phase Zn – 22 % Al eu-tectoid alloy suggested that the precise nature of the accom-modating process is dependent upon the grain size of thematerial. Thus, it is well known that coarse-grained materi-als exhibit relatively limited ductility but when the grainsize is very small, typically < 10 lm, it is possible toachieve superplasticity in the form of very high tensile elon-gations. Figure 5 shows a simple deformation mechanismmap, constructed in the form of the normalized grain sized/b versus the normalized shear stress s/G where s is theshear stress, based on experimental data recorded for theZn – 22 % Al alloy when testing at a temperature of 503 K[25]. The experimental region III in Fig. 5 denotes conven-tional creep behavior where an intragranular dislocationmechanism is rate-controlling, region II denotes the super-plastic region associated with very high tensile ductilitiesand region I is a region of limited ductility now generallyassociated with the presence of impurities in the grainboundaries [26, 27]: the regions labeled Nabarro– Herringand Coble creep in Fig. 5 are not derived from the experi-ments but are estimates based on the theoretical relation-ships for these two processes of diffusion creep as discussedin the following section. When creep occurs at high tem-peratures through the process of dislocation climb, sub-grains are formed within the grains and it is well-estab-lished, for a very wide range of metals [2] and ceramics[28], that the average size of these subgrains, k, is relatedto the applied stress through a relationship of the form

k

b¼ f

s

G

� ��1ð2Þ

where f is a constant having a value of ~ 10. Superimposedin Fig. 5 is a line delineating equation (2) for the situationwhere the specimen grain size, d, is placed equal to the sub-grain size, k. This line is in excellent agreement with the ex-

perimental line marking the transition from conventionalcreep in region III to superplastic behavior in region II,thereby demonstrating that superplastic ductilities requiregrain sizes which are sufficiently small that they are equalto or smaller than the equilibrium subgrain size.

The transition between these two types of behavior is un-derstood most easily by reference to Fig. 6. In Fig. 6a whered > k, the occurrence of GBS, which is due to the move-ment of dislocations along the boundary, leads to a stressconcentration at the triple point A and accommodation bythe movement of dislocations into the next grain. For thiscondition, these dislocations pile up at a subgrain boundaryas at B and the rate of GBS is ultimately controlled by therate of climb of these dislocations into the subgrain bound-ary. It has been shown that this mechanism leads to equation(1) with n = 3, Q = Ql and p = 1 [6]. Conversely, the situa-tion for the superplastic condition is shown in Fig. 6b whered < k and the stress concentration at C leads to a pile up ofdislocations at the opposite grain boundary at D so thatthese dislocations climb into the grain boundary. For theseconditions, corresponding to superplastic deformation, ithas been shown that n = 2, Q = Qgb and p = 2, where Qgb isthe activation energy for grain boundary diffusion [6]: thesevalues of n, Q and p are in excellent agreement with typicalexperimental behavior in superplasticity [29, 30].

The mechanism of superplasticity depicted in Fig. 6b isa process in which superplastic flow occurs exclusivelythrough the occurrence of GBS and there is also some intra-granular flow serving only to accommodate the GBS andmaking no contribution to the total strain. This approach isconsistent with a detailed analysis of measurements of egbs

in superplasticity [31] and it is supported also by two sepa-rate sets of experimental data. First, there is evidence thatdislocations move intragranularly during superplastic flowbecause it was shown that some of these dislocations be-come trapped in the coherent twin boundaries of a super-plastic copper-based alloy [32]. Second, experiments showthe strains recorded in the individual grains of a Pb– 62 %Sn alloy during superplastic flow are oscillatory in natureand they make no net contribution to the total strain [33].


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Fig. 5. Deformation mechanism map for the Zn –22 % Al eutectoid al-loy at 503 K: the broken line shows equation (2) when the grain size dis placed equal to the subgrain size k [25].

Fig. 6. Schematic illustration of Rachinger grain boundary sliding: (a)sliding at coarse grain sizes when d > k and (b) sliding at fine grainsizes in the superplastic condition when d < k.

Fig. 7. Accommodation within a grain at three different increments ofstrain for a superplastic material with d < k [33].

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The latter result is consistent with the model depictedschematically in Fig. 7 where, at three different but similarincrements of strain, intragranular dislocations pass throughthe grain to accommodate GBS on the adjacent boundaries.

Finally, it is necessary to note an important characteristicof this type of GBS, whether occurring in the conventionalcreep regime with d > k or in the superplastic regime withd < k. For both conditions, the grains retain essentially theiroriginal shape and they become displaced with respect toeach other so that there is a net increase in the number ofgrains measured along the tensile axis. This type of GBSis designated Rachinger sliding [34] and it differs in an im-portant way from Lifshitz sliding [35] which is discussed inthe following section.

3.3. Diffusion creep and vacancy flow

The diffusion creep process was the first mechanism ofcreep developed formally in the form of a specific equationfor the strain rate and, despite some subsequent questionsregarding its viability, it remains the best developed andmost fully documented of all creep processes. The principleof diffusion creep is illustrated schematically in Fig. 8.

Under the action of an external stress of the form shownin Fig. 8a, an excess of vacancies is created along thosegrain boundaries lying perpendicular to the tensile axis andthere is a corresponding depletion of vacancies along thosegrain boundaries experiencing a compressive stress. Thesedifferences in the vacancy concentrations lead to a stress-di-rected flow of vacancies, as illustrated by the broken linesin Fig. 8a. If the vacancies flow through the crystalline ma-trix, the process is known as Nabarro – Herring (N – H) dif-fusion creep [7, 8] and the rate of flow is given by equation(1) with n = 1, Q = Ql and p = 2, whereas if the vacanciesflow along the grain boundaries the process is known as Co-ble diffusion creep [9] and the rate of flow is again given byequation (1) with n = 1, Q = Qgb and p = 3. The flow of va-cancies is equivalent to a net flow of atoms in the oppositedirection and it leads, therefore, to an increase in the lengthof the grains measured along the tensile axis and a corre-sponding reduction in the width of the grains measured per-pendicular to the tensile axis.

Figure 8b illustrates an additional feature of diffusioncreep. If the material undergoing diffusion creep containsan array of uniformly-distributed particles, the lengtheningof the grains through vacancy diffusion will lead to the de-velopment of zones denuded of particles along those grainboundaries acting as vacancy sources and there will be acorresponding build up of particles along those grainboundaries acting as vacancy sinks. In principle, therefore,

the presence of denuded zones on the transverse boundariesand particle build up on the longitudinal boundaries pro-vides additional support for the occurrence of the diffusioncreep process.

Since the grains change their shapes in diffusion creep bybecoming elongated along the tensile axis, it follows thatthere is some lateral displacement of the grains with respectto each other and, in order to maintain specimen coherencywithout opening voids, surface marker lines will show off-sets at the points where they cross the grain boundaries.These offsets are identical in appearance to those arisingfrom Rachinger GBS but their origin is physically and me-chanistically different. In diffusion creep, sliding accom-modates the changes in grain shape and there is no net in-crease in the number of grains measured along thetensile axis. This process is known as Lifshitz sliding [35].

4. An overview of the creep mechanisms

It is apparent from the preceding section that there are sev-eral fundamental creep mechanisms which relate to physi-cally different processes and which give different valuesfor the parameters Q, p and n in equation (1). In generalterms, it can be seen that the intragranular dislocationmechanisms and the diffusion creep mechanisms both leadto an elongation of the grains along the tensile axis whereasthere is no elongation of the individual grains in RachingerGBS where the grains slide over each other and ultimately,at high elongations in superplastic flow, they become dis-placed with respect to each other as new grains move be-tween them. This latter effect is readily visible in super-plastic alloys such as the Pb– 62 % Sn eutectic alloy byrecording the same area at successively higher strains usingscanning electron microscopy [33]. On the other hand, thedislocation mechanisms operate intragranularly and theydo not lead to any grain boundary offsets in marker lineswhereas Rachinger GBS and Lifshitz GBS both lead toboundary offsets and these offsets are similar in appearanceand thus not easy to differentiate. This information is sum-marized in the fifth and sixth columns of Table 1.

When coarse-grained polycrystalline metals are de-formed in high temperature creep at reasonably high stresslevels, the stress exponent, n, is generally � 3 and it is rela-tively easy to identify the rate-controlling mechanism. Butat very low stresses, and especially when the grain size issmall, other creep mechanisms become important and itthen becomes difficult to distinguish unambiguously be-tween the various creep processes such as H– D creep, Ra-chinger GBS and diffusion creep. These problems have ledto the publication of many conflicting views in the scientific


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Fig. 8. Principle of diffusion creep: (a) the va-cancy flow lines within a square grain and (b) thedevelopment of denuded zones along transverseboundaries and the build up of particles alonglongitudinal boundaries.

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literature. For example, Ruano et al. [20] argued against theoccurrence of N– H diffusion creep and their paper includesthe following conclusion:

“It is shown that all creep data in metals ascribed to theNabarro –Herring (N – H) diffusional creep mechanismcannot in fact be effectively described by the N– H creeptheory. Rather, the creep mechanism is associated witheither Harper – Dorn (H – D) dislocation creep or grainboundary sliding (GBS).”

In a similar vein, a comprehensive study by Blum andMaier [22] of the creep of pure aluminum, at a temperatureof 923 K very close to the melting point, led to the follow-ing conclusion:

“The necessary condition for Harper – Dorn creep is notfulfilled in Al. This indicates that Harper – Dorn creep as aunique dislocation mechanism with Newtonian viscositymay not exist.”

These and other similar reports have led many to ques-tion the viability of H– D creep and N–H diffusion creepas separate and distinct creep mechanisms. It is important,therefore, to develop unique experimental tests that may beundertaken in order to confirm or negate the occurrence ofthese two creep processes. These tests are outlined in thefollowing section.

5. Examining the viability of Harper –Dorn creepand Nabarro–Herring diffusion creep

Harper – Dorn creep and Nabarro – Herring diffusion creepare both Newtonian viscous processes with a stress expo-nent of n = 1 and both mechanisms are dependent upon thelattice diffusivity with an activation energy of Ql. However,the mechanisms differ because H–D creep occurs intragran-ularly with p = 0 and N– H creep is dependent upon thegrain size with p = 2. Both mechanisms lead to an elonga-tion of the grains but this is accompanied by Lifshitz slidingin N– H creep whereas no grain boundary offsets are pro-duced in H– D creep. These differences can be used to de-velop a method for unambiguously distinguishing betweenH– D and N– H creep [36].

The difference between these two creep mechanisms isillustrated schematically in Fig. 9 [37]. Figure 9a showsthree adjacent grains in a polycrystalline matrix where thetensile axis is vertical and marker lines are scribed parallelto the stress axis at AA0 and BB0, respectively. If thespecimen deforms by H– D creep, the grains become elon-

gated along the tensile axis but the marker lines retain theiroriginal positions so that they remain continuous, withoutany offsets, across the transverse boundaries. By contrast,the same grain elongation in N– H creep is attained by re-moving atoms from the longitudinal boundaries and platingatoms on the transverse boundaries, thereby leading to a sit-uation where the separation between the two lines remainsunchanged in the lower grain but the markers become closertogether in the upper two grains and offsets are produced atthe transverse boundaries as illustrated in Fig. 9c. A similarsituation arises when the marker lines are scribed perpendic-ular to the tensile axis as shown in Fig. 10a except that themarker lines in the two grains on the right now move apartin N– H creep because of the plating of atoms on the trans-verse boundaries. Thus, there are no offsets in H– D creepas in Fig. 10b but there are well-defined offsets in N– Hcreep as in Fig. 10c. These experimental differences in themarker line configurations, taken together with the differ-ence in the values of p for the two mechanisms, providethe potential for determining the viability of reports ofH– D and N– H creep.

5.1. On the viability of Harper – Dorn creep

There have been numerous reports of the occurrence ofH– D creep in a wide range of materials and much of thisinformation is summarized in a review of H– D creep [38].Questions have been raised concerning several of the ex-perimental features associated with H– D creep: for exam-ple, there are both early [39, 40] and more recent [23] ques-tions concerning size effects in H– D creep, in particular therelatively small cross-sections of the samples with respectto the large grain sizes, and there are also questions regard-ing the role of the initial dislocation density [41] and purityeffects [42, 43]. However, there are at least a limited num-ber of reports which appear to provide unambigous evi-dence for the occurrence of the H– D mechanism and it isnevertheless important to check carefully on the viabilityof H– D creep as a separate and distinct creep process. Thisanalysis is undertaken most easily by examining the origi-nal creep data, published by Harper and Dorn in 1957 [5],which served to first establish the H– D mechanism as aseparate but unknown deformation process.

The data of Harper and Dorn [5] are shown in Fig. 11 in alogarithmic plot of the steady-state creep rate against theapplied stress. These results were obtained on pure alumi-num tested at 920 K and datum points are shown both forpolycrystalline samples having a grain size of 3 mm and


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Fig. 9. The use oflongitudinal markersto distinguish be-tween Harper –Dorncreep and diffusioncreep: (a) markerlines prior to creep,(b) marker linesafter Harper –Dorncreep and (c) markerlines after diffusioncreep [37].

Fig. 10. The use oftransverse markersto distinguish be-tween Harper –Dorncreep and diffusioncreep: (a) markerlines prior to creep,(b) marker linesafter Harper –Dorncreep and (c) markerlines after diffusioncreep [37].

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an aluminum single crystal. Three conclusions may bereached from this plot. First, the datum point for the singlecrystal is coincident with the points obtained from the poly-crystalline samples thereby demonstrating that p = 0. Sec-ond, the points lie along a line having a slope whichchanges from n � 4.5 at the higher stresses to n � 1 at thelower stresses. The value of n � 4.5 is consistent with theexpectations for control by dislocation climb in aluminum[2] and the transition to n � 1 reveals the occurrence ofa Newtonian viscous flow process at the lowest stresses.The transition to n � 1 occurs at a stress level of r �10 – 1 MPa. Third, and as shown by the lower broken line,the experimental region having n � 1 lies at strain rateswhich are more than two orders of magnitude faster thanthe theoretical prediction for N–H creep when testing witha grain size of d = 3 mm. All of these findings are thereforeconsistent with H– D creep as a distinct deformation mech-anism.

Harper and Dorn [5] also tested a single specimen withtransverse marker lines similar to those shown in Fig. 10.This specimen was creep tested at a stress of 9.3�10 – 2 MPa which is within the transition region betweenn � 4.5 and n � 1 and they measured the offsets at 200boundaries after creep to obtain a contribution from grainboundary sliding of n � 12 %. This low value of n is notconsistent with the expectations of N– H creep where, asdocumented in Figs. 9 and 10, grain boundary offsets mustoccur to accommodate the grain elongation. Furthermore,a series of experiments was conducted later by Harper et al.[44] where transverse marker lines were scribed on a num-ber of specimens of pure Al prior to creep testing at differ-ent stresses at the same temperature of 920 K. These speci-mens had a grain size of 3.25 mm and they were used toobtain values of n as a function of the applied stress. The re-sults from these experiments are shown in Fig. 12. Thus, thevalues of n initially increase with decreasing stress to amaximum value of n � 48 % at r = 2.3�10 – 1 MPa butthereafter the values of n decrease with decreasing stress

and they lie within the range of n � 5– 15 % within thelow stress region where n � 1. These low values of n pro-vide a clear and unambiguous demonstration that the creepdata reported by Harper and Dorn [5] and Harper et al.[44] at low stresses where n � 1 are not due to the adventof Nabarro – Herring diffusion creep since N–H creep re-quires high values for the sliding contribution n. Thus, theymust represent a new and different creep mechanism.

5.2. On the viability of Nabarro-Herring creep

There are numerous reports of creep experiments in whichthe data are in reasonable agreement with the behavior an-ticipated for N– H creep. Nevertheless, questions have beenraised concerning the viability of this mechanism [20, 45].

It was recognized many years ago that Lifshitz GBS mustaccommodate the elongation of grains in N–H creep andthis led to a measurement of the sliding contribution, n, ina Magnox ZR55 (Mg– 0.55 % Zr) alloy after testing underconditions attributed to diffusion creep [46]. The creep testwas conducted at 673 K to a strain of 13.3 % under a con-stant stress of 2.0 MPa using a specimen with a grain sizeof ~ 80 lm. A complete analysis of the creep behavior ofthis specimen was reported by Harris [47] and it was shownthat the initial creep rate was consistent with the rate pre-dicted theoretically by the N– H creep model to within afactor of ~ 4 and, based on other creep data for the same al-loy [48, 49], the specimen was tested within a region wheren � 1.

The Magnox ZR55 alloy is convenient for measuring nbecause the alloy contains hydride stringers within thegrains which provide convenient markers for measurementsof the sliding offsets. An example of the appearance of thespecimen after creep testing is shown in Fig. 13 where thetensile axis is vertical [36]. From measurements of the off-sets in the hydride stringers, equivalent to taking measure-ments of the w offsets as indicated in Fig. 4, it was esti-mated that the sliding contribution in this sample wasn � 60 % [46]. Subsequently, there were additional meas-urements of n under conditions of diffusion creep with re-ported values of n � 50 % in a Mg– 0.62 % Mn alloy [50]and n � 51 % in a Mg – 0.55 % Zr alloy [51]. Thus, all ofthese measurements are mutually consistent and they givea sliding contribution of n � 50 –60 % under conditions ofN– H diffusion creep.


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Fig. 11. Creep data of Harper and Dorn [5] plotted as the steady-statecreep rate, _ee, versus the applied stress, r: the lower broken line showsthe prediction for Nabarro –Herring creep with a grain size ofd = 3 mm.

Fig. 12. The contribution of grain boundary sliding, n, reported byHarper et al. [44] as a function of the applied stress, r: the vertical bro-ken line at r = 10 – 1 MPa denotes the approximate transition betweenn � 1 and n � 4.5 shown in Fig. 11.

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In practice, the mechanism of vacancy flow and diffusioncreep is accommodated by Lifshitz GBS and the two creepprocesses are complementary in the sense that they do notmake separate contributions to the total strain incurred inthe sample. Indeed, it was recognized many years ago thatthe predicted value of n in diffusion creep may range from0 % to 100 % depending only upon the precise proceduresused to separately define the apparent strain due to GBSand the apparent strain due to vacancy flow [52]. This con-clusion differs, therefore, from a more recent analysis thatpurports to predict a sliding contribution in diffusion creepof n � 60 % [53] but it is consistent with other recent inter-pretations [54, 55].

If the values of n � 50 –60 % cannot be easily quantifiedon theoretical grounds, it is necessary instead to examinethe significance of the procedure used in measuring egbs.The hydride stringers in Fig. 13 lie essentially parallel tothe tensile axis and the value of egbs was determined in theconventional way from measurements of the offset w ateach separate boundary. It has been shown in a detailedanalysis that this experimental procedure leads to a valueof n � 50 – 60 % even when all of the strain is due to GBS[31]. It is reasonable to conclude, therefore, that vacancyflow and Lifshitz GBS contribute equally to the overallstrain in diffusion creep, they act together as complemen-tary processes and, depending only upon the precise defini-tions used to quantify these strains, they may both be re-garded as contributing all of the strain in diffusion creep.

It is apparent from Fig. 13 that there are well-defined de-nuded zones lying along the transverse grain boundaries.This is consistent with the anticipated behavior for diffusioncreep as documented in Fig. 8b but the presence of thesezones cannot be taken as unambiguous evidence for the oc-currence of N– H creep. There are arguments and experi-mental data both supporting [56 – 60] and opposing [45,61– 63] a link between denuded zone formation and the oc-currence of diffusion creep and in practice it should benoted there are reports of the presence of denuded zonesafter annealing treatments in the absence of any creep test-ing [49, 51, 64] thereby demonstrating conclusively thatdiffusion creep is not an essential requirement for the for-mation of denuded zones. It is reasonable to conclude, asin an earlier proposal [65], that the denuded zones visiblein Fig. 13 are consistent with, but not conclusive evidencefor, the expectations of diffusion creep.

An important characteristic of diffusion creep is, as indi-cated in Table 1, that the grains become elongated along thetensile axis. Accordingly, measurements of grain shape canbe used to distinguish between the Lifshitz sliding in diffu-sion creep and the Rachinger sliding in GBS where there isno grain elongation. Two conclusions may be drawn froman analysis of the microstructure shown in Fig. 13. First,measurements of the widths of the denuded zones suggest atotal strain of ~ 9 % which is reasonably consistent with theimposed strain of 13.3 % [47]. Second, the grains are visiblyelongated along the tensile axis in Fig. 13 and measurementsof the grain aspect ratio in this and a similar photomicro-graph gave an average value of ~ 1.25 which is consistentwith the grain aspect ratio of ~ 1.28 which is anticipated fora specimen elongation of 13.3 % [37]. Thus, these observa-tions are mutually consistent and they support the occurrenceof Nabarro –Herring diffusion creep in the Mg– 0.55 % Zr al-loy shown in Fig. 13.

6. Discussion

The preceding discussion confirms the occurrence of Na-barro – Herring diffusion creep and Harper – Dorn creep asdistinct creep mechanisms, at least under some experimen-tal conditions. It is reasonable to conclude, therefore, thatTable 1 provides a reasonable summary of the major creepmechanisms in high-temperature plastic flow. Distinguish-ing between these various mechanisms is not always easyand, in general, several different tests are needed to providean unambiguous identification. For example, Harper – Dorncreep requires not only a stress exponent of n = 1, a lack ofany dependence on grain size so that p = 0 and experimentalcreep rates which are measurably faster than those pre-dicted by Nabarro– Herring creep but also it requires an ab-sence of any grain boundary offsets so that the measuredsliding contribution, n, is negligibly small.

It should be noted that the occurrence of Lifshitz slidingas an accommodation mechanism in Nabarro – Herringcreep leads to grain boundary offsets which are identical inappearance to the offsets arising from Rachinger sliding inGBS. Thus, GBS and Nabarro – Herring creep cannot bedistinguished through offset measurements and it is neces-sary instead to determine the average grain aspect ratiosince the grains are not elongated in GBS.*

Some predictions can be made concerning the effect ofreducing the grain size to the submicrometer or nanometerlevel. Since p = 2 in superplastic deformation, it followsthat a reduction in grain size will increase the strain rate as-sociated with optimum superplasticity. This trend was pre-dicted in an early report [66] where it was proposed thatprocessing by severe plastic deformation, as in equal-chan-nel angular pressing, may be used to refine the microstruc-ture to the submicrometer grain size and consequently to


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Fig. 13. Appearance ofa Mg–0.55 % Zr alloyafter testing to a strainof 13.3 % at 673 K underconditions associatedwith diffusion creep: thetensile axis is vertical[36].

* There is a misunderstanding in the recent report by Wadsworth et al.[63] where they claim the use of displacements in scribed markers isnot “a unique microstructural method with which to identify diffu-sional creep”. Their statement is correct because Lifshitz slidingand Rachinger sliding give identical grain boundary offsets but theyhave misunderstood the original proposal [36] which referred speci-fically to using displacements in scribed markers to distinguish be-tween the occurrence of Harper –Dorn creep where there are no off-sets and N–H creep where offsets are present.

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achieve high strain rate superplasticity. This effect has beendemonstrated in several subsequent reports covering arange of alloys [67 – 70].

All of the mechanisms in Table 1 are diffusion-controlledand therefore, in principle at least, the associated strainrates will become insignificant at low temperatures. This issupported by early experimental evidence showing, for ex-ample, that grain boundary sliding is not important in lowtemperature deformation [71]. However, very recent evi-dence has suggested the possibility that an alternative typeof GBS may occur at relatively low temperatures in nano-structured materials where the grain sizes are extremelysmall. The possibility of achieving GBS at low tempera-tures was first predicted from the use of three-dimensionalmolecular dynamics computer simulations of nanocrystal-line solids [72 – 74] and subsequently it was observed ex-perimentally in electrodeposited Cu with a grain size of~ 28 nm [75]. More recently, the occurrence of GBS hasbeen invoked to explain the development of increasing duc-tility with increasing imposed strain in Cu specimens testedat room temperature and processed using equal-channel an-gular pressing [76] and this proposal is consistent with thereport of GBS in ultrafine-grained Cu deformed at roomtemperature [77].

All of these results suggest, therefore, that GBS may playan important role in nanostructured and ultrafine-grainedmaterials having grain sizes of < 1 lm. It has been proposedthat this unexpected result may be a consequence of an ex-ceptionally high effective grain boundary diffusion coeffi-cient in materials having unique grain boundary structures[78] and/or it may arise because of the presence of a highdensity of extrinsic dislocations in the non-equilibriumgrain boundaries that are an inherent feature of processingusing severe plastic deformation [79]. An example of therole of these extrinsic dislocations is shown in Fig. 14where there is a high density of dislocations adjacent to thegrain boundary after processing in Fig. 14a but these dislo-cations re-arrange after annealing at a low temperature togive a non-equilibrium grain boundary having a high den-sity of dislocations as shown in Fig. 14b. There is experi-mental evidence, from observations using high-resolutionelectron microscopy, for a high dislocation density in a zoneadjacent to the boundaries of an Al– Mg alloy, where thiszone was reported to have an estimated width of ~ 5 nm[80]. In this model, it is anticipated that GBS occurs moreeasily in the non-equilibrium boundaries because of the pre-sence of a high density of extrinsic dislocations. However,more work is now needed to obtain a complete understand-ing of the role of the boundaries in these materials wherethe grain size is very small.

7. Summary and conclusions

1. Plastic flow at elevated temperatures takes placethrough creep mechanisms that divide into three distinctclasses: (i) intragranular dislocation mechanisms, (ii)grain boundary sliding and (iii) diffusion creep. Thecharacteristics of these various mechanisms are tabu-lated and described.

2. Procedures are outlined for distinguishing betweenthese various mechanisms, especially between pro-cesses having several similar characteristics. Using thisapproach, it is shown that there is very good evidence

supporting both Harper – Dorn creep and Nabarro – Her-ring diffusion creep as unique and distinct deformationprocesses.

3. Very recent data, including both computer simulationsand experimental observations, suggest the possibilityof the occurrence of grain boundary sliding at low tem-peratures in materials where the grain size is exception-ally small. These observations are attributed to theoccurrence of sliding at non-equilibrium boundariescontaining an excess of extrinsic dislocations.

This work was supported by the National Science Foundation of theUnited States under Grant No. DMR-0243331.

Appendix

Figure 2 illustrates three orthogonal displacements, u, v andw, arising from grain boundary sliding between two adja-cent grains. The sliding contribution, n, may be estimatedby determining the value of egbs from individual measure-ments of any of these three displacements.

It is apparent from Fig. 4 that the individual sliding dis-placements are related through the expression

u ¼ vtan w

þ wtan h

ð3Þ

If individual measurements of u are taken along a longitu-dinal marker line at every point where the marker line in-tersects a grain boundary, the sliding strain is given by[81]

egbs ¼ nl �uul ð4Þ

where n is the number of grains per unit length, �uu is theaverage value of u and the subscript l denotes taking meas-urements along a longitudinal line.

If a longitudinal line is used to record measurements of w,the sliding strain is given by [82]

egbs ¼ k0nl �wwl ð5Þ

where �ww is the average value and k0 is a constant having avalue estimated as 1.5.


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Fig. 14. Schematic illustration of a grain boundary (a) after severeplastic deformation and (b) after subsequent annealing at a low tem-perature.

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If measurements are taken of the offsets v perpendicularto the specimen surface, the easiest procedure is to take themeasurements at randomly selected boundaries and thesliding strain is given by [82]

egbs ¼ k00nr�vvr ð6Þ

where k00 is a constant having a value of 1.1 for a polishedsurface and 1.5 for an annealed surface, �vv is the average val-ue and the subscript r denotes randomly selected bound-aries.

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(Received December 17, 2004; accepted March 8, 2005)

Correspondence address

T. G. LangdonDepartment of Aerospace & Mechanical Engineering and MaterialsScience, University of Southern CaliforniaCA 90089-1453 Los Angeles, USATel.: +1 213 740 0491Fax: +1 213 740 8071E-mail: [email protected]


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identifying creep mechanisms in plastic flow

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