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Materials Science and Engineering A 528 (2011) 1431–1435

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

eformation mechanism maps for micro-grained, ultrafine-grained,nd nano-grained materials

arghalli A. Mohamed ∗

epartment of Chemical Engineering and Materials Science, University of California, Irvine, CA 92697-2575, USA

r t i c l e i n f o

rticle history:eceived 16 August 2010eceived in revised form 13 October 2010ccepted 15 October 2010

a b s t r a c t

There are several deformation mechanisms that depend on grain size and are controlled by grain boundarydiffusion. These mechanisms include: Coble creep, superplastic flow (micrograin superplastic flow andhigh-strain rate superplastic flow), and nanograin deformation. By combining the rate-controlling equa-tions of these mechanisms and by making assumptions regarding triple-junction creep, a deformationmap based on grain size was constructed. It is demonstrated that this map can account for the loca-

eywords:oble creepeformation mapigh-strain rate superplasticityicrograin superplasticity

tions of experimental data representing three types of deformation behavior: micrograin superplasticity,high-strain rate superplasticity, and nanograin deformation.

© 2010 Elsevier B.V. All rights reserved.

anocrystalline materialsriple-junction creep

. Introduction

Materials are classified into four groups in terms of the valuef grain size: large-grained materials, micrograined materials,ltrafine-grained (UFG) materials, and nanocrystalline materials.arge-grained materials are characterized by grain sizes largerhan 20 �m, micrograined materials by grain sizes in the range–10 �m, ultrafine-grained materials by grain sizes in the range00–900 nm, and nanocrystalline (nanograined) materials by grainizes less than 200 nm. The grain size ranges associated with thebove classifications are approximate and are based on considera-ion of observations reported in literature.

Over the past several decades, the creep behavior of materialsas been the subject of many studies. These studies have led to three

mportant results. First, it is well established that the creep behaviorf a polycrystalline material may, in general, be represented by anxpression of the form [1].

˙ = A(

D Gb

kT

)(b

d

)s( �

G

)n

(1a)

ith

= Do exp(

− Q

RT

)(1b)

∗ Tel.: +1 949 824 5807; fax: +1 949 824 2541.E-mail address: famohame@uci.edu

921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2010.10.048

where �̇ is the shear creep rate, A is a dimensionless constant, D isthe diffusion coefficient that characterizes the creep process, G isthe shear modulus, b is the Burgers vector, k is Boltzmann’s con-stant, T is the absolute temperature, d is the grain size, s is thegrain size sensitivity, � is the applied shear stress, n is the stressexponent, Q is the activation energy for the diffusion process thatcontrols the creep behavior, and Do is the frequency factor fordiffusion. Eq. (1) is referred to as power-law creep. Second, vari-ous materials parameters and experimental conditions that favorthe operation of several particular deformation mechanisms wereindentified. For example, it has been shown that the mechanism ofviscous glide (stress exponent = 3) [2] controls high-temperaturedeformation in a large-grained solid-solution alloy under the fol-lowing conditions [3,4]: (a) a large atom misfit, (b) a high value forthe stacking fault energy, (c) a high concentration of solute atoms,(d) a small elastic modulus, and (e) high normalized stresses. Ifthese conditions are not met, dislocation climb [1,4,5] will controlthe deformation behavior of the alloy (stress exponent = 5). Third, ithas been demonstrated that deformation mechanism maps [6–8]represent an effective approach that can be utilized not only toprovide an overall presentation of the domains of rate-controllingmechanisms but also to identify conditions describing transitions indeformation behavior. A deformation map is constructed by using

the relevant rate equations for known mechanisms. Of the availabledeformation maps that were proposed [6,8], the deformation mapthat plots normalized grain size, d/b against normalized stress, �/G,has attracted attention partly because it can be easily constructed[7].

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432 F.A. Mohamed / Materials Science a

Very recently, considerable efforts have been devoted tossessing deformation processes in nanocrystalline (nanograined)aterials. As a result, several deformations mechanisms have been

roposed (for review, see Refs. [9,10]). These mechanisms areifferent in terms of concept and details. In addition, extensivexperimental data on the mechanical behavior of nanocrystallineaterials such as nanocrystalline Ni and nanocrystalline Cu have

ecome available [10,11]. The availability of such information onhese materials serves as motivation to construct deformation

echanism maps that includes a deformation field for nanocrys-alline (nanograined) materials.

Accordingly, the objective of the present paper is two-fold: (a) toescribe briefly deformation mechanisms that most likely controlreep in micrograined, ultrafine-grained, and nano-grained mate-ials, and (b) to construct a deformation map that is based on grainize and that includes a field representing nano-grained materials.

. Analysis and discussion

A deformation map based on grain size [7] is associated with tworimary advantages. First, the map is divided into fields of grainize/stress space wherein a single mechanism is rate controlling.or simple systems, the individual fields are separated by straightines. As a result, the construction process is straightforward andimple. Second, in engineering applications, stress and grain sizere the permissible variables since the operating temperature isn general specified. The map can be constructed for independent

echanisms [7] as well as sequential mechanisms [12]. For inde-endent mechanisms (parallel), the total deformation rate is giveny the sum of the individual rates [13]. Therefore, the fastest mech-nism controls the deformation behavior. On the other hand, forequential process, the reciprocal of the total creep rate is given byhe sum of the reciprocal of the individual rates [13]. Therefore, thelowest mechanism dominates the deformation behavior.

.1. Deformation mechanisms

The deformation map considered in the present work will beimited to presenting independent deformation mechanisms thatre sensitive to grain size and are characterized by an activationnergy equal to that for grain boundary diffusion. A review ofxisting information and recent developments suggests the aboveharacteristics are met by the following.

.1.1. Coble creepEarly theoretical considerations have indicated that under the

onditions of low homologous temperatures, very small normal-zed grain sizes, and small normalized stresses, Coble creep [14]

ay control the creep behavior of fine-grained materials. Coblereep, in which creep strain is produced by the diffusion of vacan-ies along the grain boundaries, is represented by the followingquation [7,14]:

•�kT

DgbGb

)= ACo

[(b

d

)3 ( �

G

)](2)

here ACo ∼ 200, Dgb is grain boundary diffusion coefficient. Aomparison between Eqs. (2) and (1) shows that s = 3, and n = 1Newtonian flow).

.1.2. Micrograin superplasticity

It is well documented that when some micrograined alloys are

eformed in tension tests at temperatures greater than about 0.4he melting point, Tm, they exhibit superplastic behavior, which is

anifested by large elongations, greater than 400 pct and some-imes in excess of 2000 pct. This phenomenon is referred to as

gineering A 528 (2011) 1431–1435

micrograin superplasticity whose occurrence requires: (a) a fineand equiaxed grain size (less than 10 �m) that does not undergosignificant growth during high-temperature deformation, and (b)mobile, high-angled grain boundaries that resist tensile separation.These requirements along with the strong sensitivity of steady-state creep rates measured during superplastic flow to changesin grain size have indicated that boundaries play an importantrole, which is related to their ability to contribute to deformationthrough the process of boundary sliding. Over the past four decades,considerable efforts have been made to characterize the nature andsignificance of such a role in terms of deformation mechanisms.As a result of these efforts, a number of deformation mechanismswere developed or speculated [15–20]. The most successful mech-anisms are based on grain boundary sliding (GBS) accommodatedby dislocation motion. These models are different in assumptionsand details. For example, in the model of Mukherjee [17], largeledges or protrusions on the grain boundary surface provide mostthe obstruction to boundary sliding. As a result, dislocations aregenerated at the obstructing ledge. Then, the generated disloca-tions move into the grain and pileup against the opposite boundarywhere they climb and are annihilated. On the other hand, the modelof Gifkins [18] involves sliding by dislocation movement in themantle (a narrow region adjacent to boundaries) and accommoda-tion occurs by the glide and climb of dissociated dislocations alongboundaries; there is no dislocation activity in the core. Despitevarious differences in assumptions and details, as illustrated bythe above examples, all models based on GBS accommodated bydislocations can be represented by the following rate-controllingequation that predicts the deformation characteristics reported forthe superplastic region, in which ductility exhibits a maximumvalue:

�̇ = CDoGb

kT

(b

d

)2( �

G

)2exp

(−Qgb

RT

)(3a)

where C is a constant that can be estimated for each model and allother terms have been defined previously.

Recent experiments were conducted on the superplasticZn–22%Al eutectoid that contained nanometer-scale dispersionparticles [21]. These particles were introduced in the matrix of thealloy via powder metallurgy followed by cryomilling. Transmissionelectron microscopy observations made on specimens crept at astrain rate near the center of the superplastic region have revealedthat the initial microstructure is dislocation free; that after defor-mation, only some grains contain dislocations, which interact withdispersion particle; and that the configurations of the lattice dis-locations in the interiors of these grains are suggestive of viscousglide and single slip. The above characteristics are consistent withthe model of Ball and Hutchison [16] that during a burst of boundarysliding, a group of grains slide as a unit until blocked by an unfavor-ably oriented grain; a triple point is present. This process producesstress concentration at the triple point. The local stress concentra-tion may be relieved by the generation and movement of latticedislocations in the blocking grain. When the interiors of the block-ing grain is free of obstacles, dislocations move and then pile up atthe opposite grain boundary until their back stress prevents fur-ther generation of dislocations. The dislocations at the head of thepile-up climb into and along the grain boundary [22], and the con-tinual replacement of the dislocations would permit further grainboundary sliding. Based on the analysis of Ball and Hutchison [16],the rate-controlling equation for micrograin superplastic flow can

be represented by an equation of the following form:( •

�kT

DgbGb

)= 600

[(b

d

)2( �

G

)2]

(3b)

nd Engineering A 528 (2011) 1431–1435 1433

2

t(

wtuathitattgE(

2

trsnmicwcide[ps

atdsosr

�

waibp

2

die

Fig. 1. (a) A deformation mechanism map at 473 K showing the fields of Coble creep(Eq. (2)), superplasticity (Eq. (3b)), triple-Junction creep (Eq. (4b)), and nanograindeformation (Eq. (5)). The solid line in red represents Eq. (7b). (b) A deformation

F.A. Mohamed / Materials Science a

.1.3. Triple junction diffusional creepWang et al. [23] derived a rate equation for triple junction creep

hat can be represented by the following equation:

•�kT

DtjGb

)= Atj

[(b

d

)4 ( �

G

)](4a)

here Atj is a dimensionless constant and Dtj is the triple junc-ion diffusion coefficient. There are two difficulties associated withsing Eq. (4a) in calculations or prediction: (a) Atj is not known,nd (b) Dtj is not available. In absence of any definitive informa-ion about these two parameters, two crude assumptions are madeere to allow the inclusion of deformation by triple-junction creep

nto the deformation map based on grain size. First, it is assumedhat Dtj can be replaced with Dgb as an upper limit. Second, it isssumed that a transition from Coble creep to triple-junction creepakes place at d = 10 nm. This assumption is based on the view [24]hat the triple-junction process is likely to be important when therain size is less than 10 nm. Applying these assumption and usingqs. (4) and (8) leads to Atj = 200. Eq. (4a) becomes:

•�kT

DtjGb

)= 200

[(b

d

)4 ( �

G

)](4b)

.1.4. Nanograin deformationA recent analysis of experimental data reported for nanocrys-

alline materials [10] has revealed that there are severalequirements that a successful deformation mechanism needs toatisfy in order to account for the deformation characteristics ofc-materials. These requirements include the following: (a) defor-ation is rate dependent, (b) an activation volume whose value is

n the range 10b3–40b3, (c) an apparent activation energy that islose to the activation energy for boundary diffusion but decreasesith increasing stress, (d) the magnitudes of deformation rates that

over wide ranges of temperatures, stresses, and grain sizes, and (e)nverse Hall–Petch behavior (strength or hardness decreases withecreasing grain size). Satisfying one or two requirements are nec-ssary but not sufficient. Consideration of proposed mechanisms9,23,25–29] in light of these requirements has shown [10] that theredictions of the model of dislocation-accommodated boundaryliding [9] are consistent with the above requirements.

The basic concept used in developing the model of dislocation-ccommodated boundary sliding [9] is that plasticity in nanocrys-alline materials is the result of grain boundary sliding accommo-ated by the generation and motion of dislocations under localtresses, which are higher than applied stresses due to the devel-pment of stress concentrations. By considering the details ofliding and dislocation climb along the boundary, the followingate-controlling equation was derived [9]:

•= 9(

b

d

)3(

Dgb0

b2

)exp

(−Qgb

RT

)[exp

(2��b3

kT

)− 1

](5)

here � is the activation volume. On the basis of thebove discussion, nanograin deformation refers to deformationn nanocrystalline (nanograined) materials that is controlledy dislocation-accommodated boundary sliding (no dislocationileup).

.2. Construction of the map

The map used in the present study plots normalized grain size,/b, against normalized shear stress, �/G, so that the map is dividednto fields of normalized grain size/normalized stress space [7]. Inach field, a single deformation mechanism is rate controlling. At

mechanism map showing the effect of changing temperature from 423 to 523 K onthe boundary between (a) superplasticity and nanograin deformation, and (b) Coblecreep and nanograin deformation. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of the article.)

the boundary between two fields, the strain rates are equal. In addi-tion, when more than two mechanisms act independently [13],efforts are directed to selecting only those boundaries that char-acterize the fastest mechanisms at any point. Applying the aboveapproach, which is well documented elsewhere [7], to the mecha-nisms that are represented by Eqs. (2), (3b), (4b) and (5) leads to theconstruction of the normalized grain size/normalized shear stressmap shown in Fig. 1(a). The following comments and observationsprovide description of the details of the map:

(a) The map in Fig. 1(a) was constructed for 473 K. However,all deformation mechanisms considered in constructing thedeformation map in Fig. 1(a) are characterized by a true activa-tion energy that is equal to that of grain boundary diffusion.As a result, the boundaries in the map are expected to beindependent of temperature. Present calculations validate this

expectation except for boundaries ab, ac and cd that are iden-tified in Fig. 1(b). For example, as shown in Fig. 1(b), boundaryab shifts to smaller grain sizes with increasing temperature inthe experimental range 423–523 K; this range approximatelyrepresents those associated with testing Zn–22%Al, Pb–62%Sn,

1 nd Engineering A 528 (2011) 1431–1435

(

(

d

a

t

mssbtnsdtab

glocsrctepfi

Fig. 2. (a) A deformation mechanism map showing the fields of Coble creep (Eq. (2)),superplasticity (Eq. (3b)), triple-Junction creep (Eq. (4b)), and nanograin deforma-tion (Eq. (5)). The serrated solid line represents an approximate boundary betweenmicrograin superplasticity and HSR superplasticity. (b) A Deformation mechanismmap showing the superimposition on the map of (a) experimental data represent-ing three types of deformation behavior: micrograin superplasticity, high-strain

434 F.A. Mohamed / Materials Science a

and some Al alloys such as 5083Al. However, this difference inthe position of boundary ab, as shown in Fig. 1(b) and others (acand cd) is so small that these boundaries can be virtually con-sidered independent of temperature. This consideration is notunreasonable in view of: (a) errors encountered in measuringgrain size, and (b) approximations and assumptions involvedin developing rate-controlling equations.

b) The domain that represents deformation in nano-grainedmaterials via dislocation—accommodated boundary sliding(no dislocation pileup) is labeled as “nanograin deformation.”On the other hand, the domain that represents deformationin micro-grained materials via dislocation—accommodatedboundary sliding (dislocation pileup) is labeled as “micrograinsuperplasticity.”

(c) The boundary between Coble creep [14] and triple-junctioncreep [23] and the boundary between nanograin deformationand triple-junction creep are represented by dashed lines sinceno definitive information on parameters such the value of thedimensionless constant, Atj, is available. However, any newinformation may change the extent of field of triple-junctioncreep but not its relative position with respect to that of Coblecreep; triple-junction creep will always predominate at grainsizes smaller than those characterizing Coble creep.

d) Included in the map is a straight line (red) that represents theboundary between superplasticity and climb-controlled behav-ior. The position of this line is determined from the suggestion[30] that the transition from micrograin superplasticity to creepcontrolled by dislocation climb occurs under the following con-dition:

(grain size) ∼= ı (subgrain size) (6)

The equation that gives the dependence of subgrain size onpplied stress is given by [1]

ı

b= 10

�/G(7a)

Combining Eq. (6) with Eq. (7a) gives the following expressionhat describes the red line in Fig. 1(a):

d

b= 10

�/G(7b)

The above suggestion is consistent with two pieces of infor-ation. First, in Ball and Hutchison’s model for micrograin

uperplasticity [16], the dislocations that accommodate boundaryliding and are emitted from the triple point need to traverse thelocking grain and climb into (or along) the opposing boundary. Inhe presence of subgrains in the blocking grain, dislocations willot reach the opposing boundary, resulting in limited boundaryliding and the loss of superplasticity. Second, earlier results oneformation maps have shown that for superplastic flow [30,31],he line represented by Eq. (7b) falls very close to the bound-ry between the field of superplastic flow and climb-controlledehavior.

Micrograin superplasticity is characterized by very large elon-ations that are equal to or greater than 500%. However, thesearge elongations are usually attained at strain rates in the rangef 10−5–10−3 s−1. Such a strain-rate range is low for commer-ial forming of structural materials. However, recent advances inuperplasticity have led to a new area referred to as high-strainate (HSR) superplasticity, which is very attractive for commer-

ial applications. In particular, several studies have demonstratedhat a variety of metallic materials, including aluminum alloys, canxhibit superplasticity at relatively high strain rates (≥10−2 s−1). Arimary requirement for achieving HSR superplasticity is an ultra-ne grain size that can be produced via refinement. Equal-Channel

rate (HSR) superplasticity, and nanograin deformation. There is excellent agreementbetween the experiment data and the predictions of the map.

Angular pressing (ECAP) [32] and cryomilling followed by consol-idation [33] are two primary processes that can be used for thispurpose. Consideration of data reported for superplasticity in Alalloys indicates that HSR superplasticity is noted when the grainsize of the material is the range 300–1000 nm. In Fig. 2(a), theupper bound of about 1000 nm for observing HSR superplasticity ismarked by a “serrated” horizontal line.

Having finalized the construction of the deformation map inFig. 2(a), attention was placed on examining whether the mapgiven in Fig. 2(a) can account for the locations of experimental datathat represent three types of deformation behavior: micrograinsuperplasticity, HSR superplasticity, and nanograin deformation.In performing this examination, the following information wasused:

(a) Data on deformation of nc-Ni and nc-Cu with regard to thevalues of grain size and applied stress were taken from [11].

(b) Data on micrograin superplasticity in Zn–22%Al (a) andPb–62%Sn were taken from several sources [34–37].

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[[[[[[

F.A. Mohamed / Materials Science a

(c) Data on HSR superplasticity in Zn–22%Al (b), and Zn–22%Al (c)were taken from [38,39], respectively.

d) Data on HSR superplasticity in 5083Al was taken from [33].(e) The values of the shear modulus for nc-Ni, nc-Cu, Zn–22%Al,

and Pb–62%Sn, were documented in [9,40,35,36], respectively.(f) The values of the Burgers vectors for nc-Ni, nc-Cu, Pb–62%Sn,

and Zn22%Al, were taken from [9,40,35,37], respectively.

By using the aforementioned information, the experimentalata representing three types of deformation behavior, micrograinuperplasticity, HSR superplasticity, and nanograin deformation,ere superimposed in terms of normalized stress and normalized

rain size in Fig. 2(b). As clearly shown by the figure, each set ofata fall into the correct field. In addition, the locations of the fieldsre consistent with those expected experimentally. For example, atigh-normalized stresses in the range 2 × 10−3–4 × 10−2, the tran-ition from nanograin deformation to HSR superplasticity occursith increasing grain size.

ummary

. By combining several independent deformation mechanismsthat depend on grain size and are controlled by grain boundarydiffusion, a deformation mechanism map based on grain size wasconstructed.

. Experimental data reported for nanocrystalline Cu andnanocrystalline Ni along with those available for HSR superplas-ticity and micrograin superplasticity fall into the correct fieldsof the map.

cknowledgments

This work was supported by US National Science FoundationGrant No.: DMR-0702978). Thanks are also due to my former Grad-ate student, Dr. Heather Yang, for her assistance.

[[[[[[

gineering A 528 (2011) 1431–1435 1435

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