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Acta Materialia 59 (2011) 4323–4334

Mechanisms of grain boundary softening and strain-rate sensitivityin deformation of ultrafine-grained metals at high temperatures

Naveed Ahmed, Alexander Hartmaier ⇑

Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-Universitat Bochum, Stiepeler Str. 129, 44801 Bochum, Germany

Received 17 September 2010; received in revised form 19 March 2011; accepted 19 March 2011Available online 22 April 2011

Abstract

Two-dimensional dislocation dynamics and diffusion kinetics simulations are employed to study the different mechanisms of plasticdeformation of ultrafine-grained (UFG) metals at different temperatures. Besides conventional plastic deformation by dislocation glidewithin the grains, we also consider grain boundary (GB)-mediated deformation and recovery mechanisms based on the absorption ofdislocations into GBs. The material is modeled as an elastic continuum that contains a defect microstructure consisting of a pre-existingdislocation population, dislocation sources and GBs. The mechanical response of the material to an external load is calculated with thismodel over a wide range of temperatures. We find that at low homologous temperatures, the model material behaves in agreement withthe classical Hall–Petch law. At high homologous temperatures, however, a pronounced GB softening and, moreover, a high strain-ratesensitivity of the model material is found. Qualitatively, these numerical results agree well with experimental results known from the lit-erature. Thus, we conclude that dynamic recovery processes at GBs and GB diffusion are the rate-limiting processes during plastic defor-mation of UFG metals.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Dislocation dynamics; Polycrystals; Plasticity; Diffusion; Modeling

1. Introduction

Grain boundary (GB) strengthening is a well-knownphenomenon in polycrystalline materials at low homolo-gous temperatures. The Hall–Petch relation describes anincrease of flow stress with decreasing grain size. However,an increase in temperature renders polycrystals softer andthe Hall–Petch coefficient in aluminum has been observedto decrease with increasing temperature [12]. For ultra-fine-grained (UFG) metals at high homologous tempera-tures a transition from GB strengthening to GB softeningis observed experimentally for copper [4] and austeniticsteel [24]. A very general feature observed for face-centeredcubic as well as for body-centered cubic metals is anincrease in the strain-rate sensitivity of the material withdecreasing grain size [15,14]. In the case of body-centred

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.03.056

⇑ Corresponding author. Tel.: +49 234 32 29314; fax: +49 234 32 14984.E-mail address: Alexander.Hartmaier@rub.de (A. Hartmaier).

cubic metals, e.g. a-iron [15], this trend is only observedat elevated temperatures. In the present work, we want toshed some light on the mechanisms behind such generaltrends in material behavior.

In general, plastic deformation results from the motionof dislocations in the lattice (conventional plasticity) orfrom the sliding or rotation of GBs (GB-mediated defor-mation). Further deformation processes such as deforma-tion twinning or diffusional creep (Nabarro–Herringcreep) will not be considered in this paper. Many mechan-ical properties of materials are dominated by long-rangeelastic interactions between dislocations and other defects.Two-dimensional dislocation dynamics (2D-DD) modelscapture these interactions of dislocations with themselvesand with other defects, and provide fundamental insightsinto a variety of mechanical properties, including crack-tip plasticity [7,30], void growth [16,26], the role ofinterfaces [9], grain size effects [3,2], and GB and obstacleeffects [18,8].

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Table 1Material constants of copper along with the model parameters used in allsimulations.

Parameter Value

Burgers vector (b) 0.25 nmShear modulus (G) 44 GPaPoison ratio (m) 0.3Source strength (snuc) 0.2 GPaGlide mobility (Mgl) 500 Pa sDiffusion constant (D0) 5 � 10�15 m3/sActivation energy (Qdif) 57 kJ/mole

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In the present model, conventional plastic deformationprocesses as well as GB-mediated deformation are takeninto account. The latter process is modeled by the absorp-tion of lattice dislocations into GBs. As this occurs, theirBurgers vector is split up into a component parallel and acomponent orthogonal to the GB. The resulting GB dislo-cation with the Burgers vector parallel to the GB is movingconservatively and thus causes GB sliding; the second GBdislocation climbs under production or annihilation ofvacancies. The motion of this climb dislocation thuschanges the local concentration of vacancies within theGB network. This requires solving the diffusion equationand coupling it to dislocation motion via source and sinkterms. The local vacancy concentration in turn exerts anosmotic force on the climb dislocations [23]. In the presentpaper we employ such a coupled 2D-DD and diffusionkinetics model to investigate the deformation behavior ofmetals and the significance of the considered deformationmechanisms for different grain sizes and at different tem-peratures. This model yields material parameters that canbe—in a qualitative sense—directly compared to experi-mental data in the literature. It is noted immediately thatwith such a simple 2D-DD model we cannot obtain a quan-titative comparison with experimental findings, neither inthe resulting quantities, nor in the loading conditions.Rather, the model is employed to investigate which trendsin the strength of UFG metals must be expected under theassumption that certain deformation mechanisms areactive. Hence, while the inherently qualitative nature of2D-DD models must be regarded as a weakness, theirstrength is that the effect of specific deformation mecha-nisms under well-controlled loading conditions can bestudied; whereas disturbing effects, such as secondarydeformation mechanisms or changes in the microstructure,are deliberately disregarded. As a consequence, the modelresults may only be compared to experimental data underconditions where only those deformation mechanisms areactive in the real material. This holds true irrespective ofthe absolute values of the thermal and mechanical loadingconditions of the model whose range is frequently limitedby numerical efficiency. In the same sense, the model resultscannot be interpreted to be material specific, but rather theobserved trends in the deformation behavior are genericallycompared to a large class of UFG metals. By such qualita-tive comparison between experiment and model we want toaddress the questions under which conditions metals doexhibit GB strengthening and why GB softening andenhanced strain-rate sensitivity does occur at elevatedtemperatures.

This paper is organized as follows. In the following sec-tion the model of plasticity applied here is introduced andthe method of coupling dislocation dynamics and diffusionkinetics is described. After that we present the results fromnumerical mechanical testing along with microstructuresand internal stresses obtained for the deformation of poly-crystals at different temperatures. These results are thencritically discussed with respect to experimental findings

of general trends in the deformation behavior of UFG met-als in the literature.

2. Model

Classical statistical models of crystal plasticity [20,5]consist of kinematic equations of state. The deformationbehavior of the material is described by structure evolutionlaws. In contrast, in 2D-DD models structure evolution iscontrolled by the motion of discrete dislocations thatcauses plastic strain. In addition to such conventionaldeformation mechanisms, in our model we include GB-mediated deformation processes and GB diffusion. In thefollowing, the basic ingredients of our model concerningplasticity, diffusion kinetics and dislocation GB interac-tions are described.

2.1. Conventional plasticity

Conventional plastic deformation is taken into accountby generation and glide of lattice dislocations within grainsin the typical way of 2D-DD models (see e.g. [27,13]). Inthis approach, discrete dislocations are considered as singu-larities in an elastic continuum that interact by their elasticfields. With the help of analytical or—as in our case—numerical methods the stress tensor at each point in thedomain is calculated. From this stress field the drivingforce on each individual dislocation, i.e. the Peach–Koehlerforce, is calculated and the dislocations are moved accord-ing to a mobility law. This dislocation motion, finally, pro-duces plastic strain.

In this work, the velocities of dislocations are consideredto be directly proportional to the driving force, giving riseto a viscous law of dislocation motion [17]. After the veloc-ities of all dislocations are calculated, the equations ofmotion are integrated by an Euler method with an explicittime step. The formulation of plasticity by discrete disloca-tion dynamics gives an inherently viscoplastic materialbehavior. All data that are used for the numerical simula-tions in this work are summarized in Table 1; furtherdetails of the method can be found in Ref. [13].

The non-conservative motion of climb dislocations fur-thermore produces or absorbs point defects and thuschanges the local concentration of vacancies. The chemicalpotential l of vacancies is the parameter that describes the

N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334 4325

change in free energy G of the system by changes in themole fraction n of vacancies at constant temperature T

and pressure p, as:

lðcÞ ¼ @G

@n

� �p;T

¼ kBT lncðrÞco

� �; ð1Þ

where c(r) is the local concentration of vacancies at posi-tion r and co is the equilibrium concentration of vacancies.The symbol kB represents the Boltzmann constant. As theresult of non-conservative motion of dislocations, the localconcentration of vacancies changes, which leads to an os-motic force:

Fosm ¼ �lbXðb� lÞjb� lj ¼ �

kBTbXðb� lÞjb� lj ln

cðrÞco

� �ð2Þ

on the dislocations [23]. Here, b is the Burgers vector withnorm b, l is the line direction of the dislocation, and X is theatomic volume, which is set equal to b2 in this work. Thus,dislocations will climb under the Peach–Koehler force andproduce or absorb vacancies until the mechanical drivingforce is in equilibrium with the osmotic force. Hence, thetotal climb force on a dislocation is given by the sum ofthe climb component of the Peach–Koehler force and theosmotic force.

Dislocations are produced by Frank–Read-like disloca-tion sources in the following way. A test–dislocation–dipole, representing the two-dimensional analog of aFrank–Read source, is created at the source position. Ifthis dipole collapses under the local shear stress, self-consistent dislocation nucleation from this source underthe given stress is not possible. If, however, the local shearstress is large enough at least to hold the dipole in equilib-rium, a new dislocation dipole is created, and these disloca-tions participate in the simulation in the usual manner.This method consequently gives an upper limit for the pro-duction rate of dislocations, because every dislocationdipole that is at least stable will be produced. The criticalshear stress snuc for this dislocation production processdepends only on the dipole separation and is related tothe yield strength of the material. Additionally to suchFrank–Read-like sources, we consider a finite initial dislo-cation density in our simulations. These dislocations aredistributed randomly within the domain and then relaxedto their force equilibrium positions at zero applied stress.

If two dislocations with opposite Burgers vectorapproach each other closer than a critical distance of 6b

[11], they annihilate each other and are removed from thesimulation. If two dislocations with non-parallel Burgersvectors approach closer than that distance, they areassumed to form a lock and are subsequently immobilized.

2.2. Diffusion kinetics

To describe GB diffusion, we use Fick’s law with addi-tional source and sink terms taking into account thevacancy production or absorption by climbing disloca-

tions. The diffusion equation is solved numerically on agrid that describes the GB network with the help of smallelements inside which a constant vacancy concentration isassumed. Each GB segment is subdivided into 100 ele-ments. The rate of change of the local vacancy concentra-tion caused by dislocation climb follows the relation:

@csðrðiÞ; tÞ@t

¼ bðiÞGBCDvðiÞ

b2Leldgb

; ð3Þ

where Lel and dgb are the GB element length and thickness,respectively. v(i) is the velocity of the ith GB climb disloca-tion at position r(i) and time t with Burgers vector bðiÞGBCD.b is the norm of the Burgers vector of lattice dislocations.For every position along the GB, the diffusion equationand dislocation dynamics source and sink terms are cou-pled by:

@cðr; tÞ@t

¼ DDcþ @csðr; tÞ@t

; ð4Þ

where D is the Laplace operator and D is the temperature-dependent diffusion coefficient. The latter is given byD ¼ D0 expð�Qdif=kB T Þ, where D0 is the diffusion constantand Qdif is the activation energy of GB diffusion. Withinthis work, the temperature is always kept constant duringindividual simulation runs.

The iterative coupling of dislocation kinetics and the dif-fusion process is shown schematically in the flowchart inFig. 1. We note that dislocation dynamics and diffusionkinetics define two different timescales that have to betaken into account during the simulations, i.e. the time stephas to be small enough to capture fast processes during dis-location motion and the simulation time has to be longenough such that slow diffusive processes have enough timeto reach a dynamic equilibrium. This requires the applica-tion of rather high temperatures in the numerical simula-tions presented below, in order to speed up diffusion andto minimize the ratio of the two different timescales. Wenote that the microstructure of real materials would notbe stable at the temperatures studied here, nor would thematerials sustain the stress levels applied in the numericalsimulations. However, since we are using high tempera-tures and high stresses only to accelerate the numerical per-formance of the simulations, and since the model allows usto control the deformation mechanisms to those occurringin experiments at lower temperatures, the numerical resultswill be compared later on to experimental results obtainedat considerably lower temperatures, where the microstruc-ture is stable and the strength of the material is rather high.

2.3. Dislocation–grain boundary interaction

When a moving lattice dislocation intersects with a GB,there are various possibilities of interactions. The GB caneither block the motion of dislocations, or it can act asreflecting, refracting or absorbing medium for dislocations.In all interaction mechanisms (transmission, reflection,

Fig. 1. Simplified flowchart of the coupled dislocation dynamics and diffusion kinetics model in which dislocation motion changes the concentration fieldof vacancies and diffusion kinetics quantifies the osmotic force on dislocations. This iterative interaction couples both systems.

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blocking and absorption) it is required that the sum of allBurgers vectors is constant, i.e. bin = bout + bgb, wherebin , bout and bgb are the Burgers vectors of the incoming,the emitted or the dislocation stored in the GB, respec-tively. The conservation of the total Burgers during the dis-location–GB interaction ensures the consistent treatmentof elastic interactions between lattice dislocations and GBdislocations.

Polycrystalline material is simulated by subdividing thedomain into hexagonal grains of equal size. Two very dif-ferent types of GBs for these hexagonal grains are consid-ered in this work. The first type mimics blocking GBs andacts as an impenetrable barrier to incoming lattice disloca-tions. The second one behaves as a random GB andabsorbs incoming dislocations. In this case, the lattice dis-location is split into two components, one with the Burgersvector parallel and one with the Burgers vector orthogonalto the GB. The first GB dislocation with the parallelBurgers vector is termed “GB glide dislocation”, thesecond is “GB climb dislocation”. In both types of disloca-tion–GB interaction (blocking and absorbing), local stressconcentrations close to the GBs are produced. Such stressconcentrations can activate dislocation sources and hencecause dislocation transmission, refraction or reflection.Theoretical models and experimental evidence corroboratethis kind of dislocation–GB interaction [6,22,10].

During the simulations, GB glide dislocations have thesame kinetics as lattice dislocations. This conservativeGB dislocation motion causes GB sliding. In contrast,GB climb dislocations produce or absorb vacancies during

their motion. As described above this change in thevacancy concentration field is taken into account as addi-tional source and sink terms in the diffusion equation. Fur-thermore, GB climb dislocations are not only subject to thePeach–Koehler force, but also to an osmotic force. Thisnon-conservative climb hence causes grain rotation andgrain shape changes which, however, we do not considerexplicitly since we use a small strain formulation. Conse-quently, in the case of absorbing GBs, lattice dislocationsnot only produce conventional plastic deformation withinthe grains, but also GB-mediated deformation. However,it is also important to note that within our model neitherGB sliding nor GB diffusion occurs unless conventionalplastic deformation takes place inside the grain and latticedislocations are absorbed by GBs.

Due to their elastic interaction, lattice dislocations orig-inating from the same dislocation source and intersectingthe GB at the same position repel each other. Hence, dislo-cation pile-ups are forming on slip planes that intersectGBs. This holds true for blocking GBs as well as forabsorbing GBs. Since the Burgers vector of the absorbeddislocation is conserved, subsequent dislocations are inany case repelled by internal stresses. Only after both com-ponents of the GB dislocations, i.e. the glide and the climbpart, have moved away from the intersection point can sub-sequent dislocations again be absorbed. GB climb disloca-tions exhibit a rather slow kinetics, since their motion iscoupled to the slow diffusion process. Thus, GB climb dis-locations, and indirectly diffusion kinetics, limit the rate atwhich dislocation pile-ups at GBs are removed (see Fig. 2).

Fig. 2. Schematic diagram of the dislocation–GB interaction as it isincorporated into our model. As the result of this interaction a latticedislocation splits into a GB climb dislocation and a GB glide dislocationwhen it impinges on a GB. The total Burgers vector is conserved duringthis operation, which also ensures that the elastic interaction of GBdislocations with lattice dislocations is treated consistently.

Table 2Eight different slip systems are used in the presented simulations. For eachindividual grain we consider two slip systems, inclined by 60�. The anglesof the slip systems together with their Schmid factors, under uniaxialloading are given.

Grain number Slip system Angle (deg) Schmid factor

1 1 60 0.431 2 �60 0.432 3 13 0.222 4 73 0.283 5 81 0.153 6 �39 0.494 7 �35 0.474 8 25 0.38

N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334 4327

Since dislocation pile-ups at GBs exhibit a back-stress ondislocation sources and limit the dislocation productionrate within the grain, it is seen already here that GB–dislo-cation interactions will have a strong influence on conven-tional plastic deformation.

3. Results

In this section we present the results of the numericaldeformation simulations of polycrystals at low and highhomologous temperatures. The polycrystalline material ismodeled by dividing the domain into four equally sizedhexagonal grains. For each grain two slip systems are con-sidered, which are inclined to each other at an angle of 60�.In each grain the slip systems have different orientations asgiven in Table 2. In all simulations, we employ materialconstants mimicking the properties of copper (see Table1). Note that we reduced the activation energy of GB diffu-sion by 50% to speed up the simulations. Nonetheless, weprefer to give temperatures as absolute values and not inreduced units. Furthermore, the terms low and highhomologous temperatures are used to discriminate betweenthe two distinctly different temperature regimes in which wefind GB strengthening and GB softening, respectively.

3.1. Low temperatures

The deformation behavior of metals in the low-temper-ature regime is investigated by calculating the Hall–Petchconstant at different temperatures and for the two differenttypes of GB behavior: blocking and absorbing GBs. Inthese numerical experiments, five different grain sizes

(d = 100, 150, 200, 250 and 300 nm) are used. The initialdislocation density amounts to 4 � 1014 m�2 and the dislo-cation source density is set to 2 � 1014 m�2. During thesimulations quasi-static stress–strain curves are calculatedby subjecting the relaxed initial dislocation configurationsto an external stress that moves dislocations, creates dislo-cation sources and thus produces plastic strain. In this pro-cess the strain rate is constantly monitored. When itpermanently drops below 10 s�1 the configuration is con-sidered as being relaxed and the applied stress is consideredto be the flow stress for the obtained plastic strain. The var-iation of plastic strain with respect to time can be observedin Fig. 3 (top). This procedure is continued at differentstress levels to obtain stress–strain curves.

The stress–strain curves show that the plastic strain at aspecific stress increases monotonously with temperatureand grain size. Thus, the flow stress at a constant plasticstrain decreases with increasing temperature and grain size.It is also seen that the polycrystals with blocking GBs arealways stronger than those with absorbing GBs.

The Hall–Petch constants under different conditions arecalculated by considering the flow stresses at 0.2% plasticstrain, obtained by linear interpolation between the closestdata points. In Fig. 3 (bottom) the flow stresses are plottedas a function of the square root of the inverse grain size,such that the slope of the resulting regression lines corre-sponds to the Hall–Petch constant. These values are pro-vided in Table 3, where it is seen that the polycrystalssoften continuously with increasing temperature.

3.2. High temperatures

To study the behavior of the model material at high tem-peratures, simulations at a constant applied stress havebeen conducted in the temperature range from 800 to1100 K. The stress level has been set to 800 MPa. Due tothe limitations in the timescale discussed above, our modelis restricted to rather high stresses and strain rates. How-ever, since in the model the deformation mechanisms arewell controlled to those occurring in experiments at lowertemperatures, the results of the simulations are comparedto experimental data obtained at considerably lower tem-peratures, where also the microstructure is stable, as it is

Fig. 3. Top: Quasi-static stress–strain curves of polycrystals with differentgrain size d at different temperatures. Bottom: The Hall–Petch constant ofthe model material at 0.2% plastic strain decreases with increasingtemperature. The Hall–Petch constants at different temperatures aresummarized in Table 3.

Table 3The Hall–Petch constants of UFG material at 0.2% plastic strain atdifferent temperatures.

Deformation conditions Temperature Hall–Petch constant (KHP)

Without GB plasticity 0.304 MPa m1/2

With GB plasticity 300 K 0.154 MPa m1/2

With GB plasticity 500 K 0.08 MPa m1/2

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in our model. During these simulations a steady-statedeformation develops (see Fig. 4 (top row)) and the strainrate in this steady-state regime is monitored as a functionof applied stress and temperature. Model materials withgrain sizes of 100, 200 and 300 are studied. At these hightemperatures only GBs of the absorbing type areconsidered. The dislocation source density is set to2 � 1014 m�2, and no initial dislocations are considered.The time evolution of the plastic strain at different temper-atures in the 100 and 300 nm grain sized material is shownin Fig. 4 (top row). Initially, the strain evolution at differenttemperatures is identical for materials with the same grainsize. However, at larger strains temperature-dependentGB-mediated deformation processes gradually change the

strain evolution. At the lowest temperatures investigatedin this regime, the plastic strain rate becomes rather small,as we also observed during the relaxation steps in thequasi-static deformations at low temperatures. From theplastic strain evolution at high temperatures two generalobservations can be made: (i) the comparison of plasticstrains in the small-strain regime shows that the contribu-tion of conventional plastic deformation increases withthe grain size; and (ii) temperature-dependent softening ismore pronounced in fine-grained material.

To understand this deformation behavior of the modelmaterial at high temperatures, different contributions toplastic strain are evaluated in the following. Besides con-ventional plastic deformation inside the grains, our modelalso captures two types of GB-mediated deformation pro-cesses, namely the contribution of diffusion-assisted GBdislocation climb and GB sliding caused by conservativemotion of GB glide dislocations. The time evolution ofthe plastic strain caused by GB diffusion and GB disloca-tion climb is shown in Fig. 4 (middle row) for different tem-peratures and grain sizes. These results illustrate howstrongly such diffusive processes depend on temperatureand also grain size (note the different timescales for the dif-ferent grain sizes). The plastic slip contribution of GB slid-ing is shown in Fig. 4 (bottom row). Here, it is seen that forthe smallest grain size investigated (100 nm) the plasticstrain rate increases with temperature, whereas for thecoarser-grained material (300 nm) the steady-state plasticstrain rate due to GB sliding is temperature independent.

Another important observation that can be drawn fromFig. 4 is that each of the GB-mediated deformation pro-cesses contributes at most 10% to the total plastic deforma-tion of the material. Hence, even for the finest grains at thehighest temperatures more than 80% of the plastic slip iscaused by conventional dislocation glide inside the grains.Since, however, the plastic strain rate strongly dependson temperature and grain size, it is concluded already herethat GB-mediated deformation must play an importantrole in assisting conventional plastic deformation andmaintaining the glide of dislocations inside the grains.The processes by which GB-mediated deformation assistsconventional deformation and the consequences on thedeformation behavior of UFG metals will be discussed inSection 4.

The steady-state plastic strain rates that are taken fromthe linear parts of the plastic strain vs. time curves are plot-ted in Fig. 5, where it is seen that the temperature depen-dence of the strain rate for the fine-grained modelmaterial is much higher than for its coarse-grained counter-part, which must be attributed to the shorter diffusionpaths. These results show that at high temperatures finegrains soften the material and that there is a transitionfrom GB hardening to GB softening. This is also seen fromthe data points obtained for the lowest temperature inves-tigated here, where the plastic strain rates for both grainsizes are very similar. Furthermore, an extrapolation ofthe data seems to suggest that there is a transition, i.e. that

Fig. 4. The time evolution of total plastic strain (top row), strain caused by diffusion (middle row) and strain produced by GB sliding (bottom row) attemperatures of 800 and 1100 K in 100 nm (left column) and 300 nm (right column) grain sized material.

N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334 4329

at lower temperatures the plastic strain rate in the coarsegrained material will be higher, consistent with the GBhardening found at low temperatures.

Finally, the strain-rate sensitivity of the model materialis studied. To accomplish this, the steady-state strain rateshave been calculated for two different applied stresses of700 and 800 MPa. All other parameters, i.e. temperaturesand grain sizes, are the same as used in the previous numer-ical experiments. The results of strain-rate sensitivity forthe grain sizes of 100, 200 and 300 nm in the temperaturerange from 800 to 1100 K are summarized in Fig. 6. Firstly,it is seen from these results that for the fine-grained mate-rial, the strain-rate sensitivity increases quite strongly withtemperature. In contrast to that, for the coarse-grainedmaterial the strain-rate sensitivity remains constant. Sec-

ondly, the strain-rate sensitivity of the fine-grained materialremains considerably higher than that of the coarse-grained material at all temperatures.

3.3. Microstructures

The evolution of dislocation microstructures and inter-nal stresses with plastic strain for a model material withgrain size of 100 nm is shown in Fig. 7. It is seen that themicrostructures and stress states at the same plastic strainlevels are comparable even for different temperatures.However, the microstructure evolution occurs much fasterat the higher temperature. Since the diffusion coefficient isthe only temperature-dependent parameter in our model,this must consequently be attributed to the faster GB diffu-

Fig. 5. The plastic strain rate _epl of a polycrystal with 100 nm grain sizereveals a stronger temperature T dependence than that of a polycrystalwith 300 nm grain size. Lines result from regressions of the data pointsand are extrapolated towards smaller temperatures (dotted lines) toindicate the expected cross-over.

Fig. 6. The strain-rate sensitivity m of a polycrystal with grain size 100 nmexhibits a stronger temperature dependence than that of a polycrystal withgrain size 300 nm.

4330 N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334

sion, even though by itself this diffusion contributes only asmall amount to the plastic deformation. A striking featureis that at the triple points extremely high stresses occur,which are caused by GB sliding and diffusion. Even thoughthe model allows for the transmission of GB dislocations toneighboring GBs at triple points, it shows that there is noelastic driving force to move GB dislocations from oneGB segment to another. Therefore, all GB dislocations pileup at the triple points and cause high stresses there. Thesehigh internal stresses also interfere with dislocation motioninside the grains.

Furthermore, it can be observed in Fig. 7 that inside thegrains with a size of 100 nm there is only a small dislocationdensity, which means that almost all lattice dislocationsreach the GB and are absorbed there. Later it will be seenthat for larger grains the dislocation density within the grainis somewhat larger and the dislocation structure more com-plex (cf. Fig. 9). It is also noted that only few dislocationpile-ups at the GBs are observed in the microstructure.

The stress fields shown in the different micrographsreveal that the internal stresses in the microstructure

become increasingly polarized, i.e. increasingly large areasexhibit stresses at the upper or lower end of the entire spec-trum. Hence also the stress gradients within the materialbecome more pronounced. The reason of this polarizationof the internal stress is the increase in dislocation densitywith plastic strain and the high concentrations of disloca-tions at GBs.

Although, the microstructures obtained for a constantplastic strain at different temperatures in Fig. 7 appear tobe quite similar, a quantitative analysis reveals some inter-esting features. The change in the ratio of the number oflattice dislocations to the number of total dislocationsNlat/Ntot with time is shown in Fig. 8. It is seen that atthe start of the simulations, the ratio is almost the samefor both temperatures, but that it varies as time progressesand the steady-state deformation is reached. During thissteady-state deformation, the ratio of lattice dislocationsamounts to roughly 15% at 800 K, whereas it drops to val-ues between 8% and 10% at 1100 K. This indicates that athigher temperatures the dynamic equilibrium between dis-location production within the grain and dislocationabsorption at GBs is shifted towards the absorption pro-cess. Again, this must be attributed to the faster diffusionat larger temperatures that allows the climb part of theGB dislocation to detach faster from the location wherethe dislocation has been absorbed. Hence, the back-stresson the following dislocations is smaller and they approachthe GB faster, to be absorbed there.

The micrographs of the coarser-grained material with agrain size of 300 nm are shown in Fig. 9. The micrographsat different simulation temperatures again exhibit quitesimilar microstructures and also similar internal stressstates. As opposed to the observations made for the fine-grained material, a large number of lattice dislocations istrapped by locks within the grain interior.

4. Discussion

Having presented the numerical results, their generalimplications on deformation of UFG metals will now becritically discussed. The numerical results presented aboveshow that the coupled 2D-DD/GB diffusion model predictsa transition from GB hardening at low temperatures to GBsoftening at high temperatures for grain sizes of 100-300 nm, i.e. for UFG metals. At low temperatures theHall–Petch coefficients result from the response of themodel material to quasi-static loading. A wider overviewon the possibilities to study the fundamental processes ofplastic deformation with 2D-DD models at low tempera-tures has been presented in a previous work [1]. At elevatedtemperatures the plastic strain rate and the strain-ratesensitivity has been calculated from constant stress simula-tions. Within our model, we consider pre-existing disloca-tions, dislocation nucleation from Frank–Read-likedislocation sources inside the grains, and dislocation anni-hilation inside the grains as well as dislocation absorptionat GBs. The latter process transforms lattice dislocations

Fig. 7. Microstructure evolution along stress states of 100 nm grain sized copper at 800 and 1100 K at different plastic strains (epl = 0.5%, 0.7% and 0.9%).The color field indicates the local tensile stress in loading direction in GPa, as given in the color bars. The simulation times t, at which the strain levels havebeen reached are also given. Dislocation positions are marked by crosses; positions of dislocation sources are indicated by open circles.

N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334 4331

into GB dislocations with a glide part that causes GB slid-ing, and a climb part whose motion is controlled by GB dif-fusion. Our model does not consider nucleation ofdislocations at GBs or direct production of GB disloca-tions. Furthermore, due to the large difference in the veloc-ities of climb and glide dislocations, applications of themodel are limited to relatively high temperatures, wherethe velocity ratio is moderate and does not cause too severenumerical problems. Hence, this 2D model is not expected

to yield quantitative material-specific results, but it can beused to analyze the processes that cause the observedbehavior of the model material. Furthermore, due to thesesimplifications the model is capable of predicting materialproperties that can be directly assessed experimentally,meaning that the consistency of the model with experimen-tal data can be judged immediately.

The comparison of quasi-static stress–strain curves fordifferent model assumptions reveals that UFG material

Fig. 8. The variation of the total number of dislocations vs. latticedislocations, Nlat/Ntot, at 800 K and 1100 K in 100 nm grain sizedmaterial.

4332 N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334

without GB plasticity and the related dynamic recoverymechanisms (dislocation absorption into GBs) exhibitsmore GB strengthening than UFG material with GB plas-ticity (cf. Fig. 3). Moreover, an increase in temperaturefrom 300 to 500 K significantly reduces GB strengthening.The softening of UFG material from 300 to 500 K is due tothe higher GB diffusion rates at higher temperatures. Sincethe motion of GB climb dislocations is controlled by diffu-sion kinetics, faster diffusion means that these dislocationscan faster retreat from the position in the GB where theyhave been absorbed. This releases the back-stress on thefollowing dislocations of the same slip system and henceallows them to approach the GB faster. Note that even ifthe first dislocation on a slip plane is absorbed into theGB, the following dislocations will be repelled by their elas-tic interaction with this dislocation. This means that signif-icant dynamic recovery only takes place if the residualBurgers vector of an absorbed dislocation is spread effi-ciently along the GB by diffusive processes. Consequently,faster diffusion kinetics cause more dynamic recovery bydislocation absorption in GBs, which is also seen in thedecrease of lattice dislocation density with temperature(Fig. 8). Due to the enhanced rate of dislocation absorp-

Fig. 9. Microstructure of 300 nm grain sized material at 800 and 1100 K at 100color bar. Dislocation positions are marked by crosses; positions of dislocatio

tion, the GBs loose their function as strong obstacles forlattice dislocations, which leads to softening. This softeningcauses a reduction in the Hall–Petch coefficient withincreasing temperature (see Table 3). In the same way,the flow stress at a specific plastic strain decreases withincreasing temperature and enhanced GB plasticity. Thesevariations in flow stress and the Hall–Petch constants withtemperature are consistent with experimental observations[12]. It shall be stressed here that GB plasticity must beconsidered a helping mechanism, because it enables plasticdeformation inside the grains but by itself contributes atmost 20% to the plastic strain.

The deformation behavior of materials with differentgrain sizes at elevated temperatures (800-1100 K) showsthat the softening increases more rapidly with temperaturefor smaller grain sizes. These numerical results show anagreement with the plastic deformation behavior that wasobserved under quasi-static loading at low temperatures:an increase in temperature enhances the role of GB defor-mation mechanisms and recovery processes which leads toGB softening at high temperatures. Hence, the resultsobtained for high temperatures support the conclusionsdrawn above. Similar observations of GB strengtheningat low temperature and GB softening at high temperaturefor UFG metals have also been reported from experiments[4,24].

Furthermore, the variation in strain-rate sensitivity withtemperature and grain size obtained from our model qual-itatively agrees well with experimental observationsobtained for copper, aluminum, nickel and a-iron[4,15,19,14,28]. These results corroborate the statementthat GB-mediated deformation along with dynamic recov-ery processes are the origin of the extraordinary mechani-cal properties of UFG materials. Although the numericalresults qualitatively reproduce the experimental trends,the calculated values of strain-rate sensitivity are far toohigh in comparison. The reason of this difference is thatin these simulations, uniformly sized and shaped grainsalong with flat GBs are considered, whereas in real materi-als the GBs have facets which offer more resistance to GB

ns. The color field indicates the local tensile stress in GPa, as given in then sources are indicated by open circles.

N. Ahmed, A. Hartmaier / Acta Materialia 59 (2011) 4323–4334 4333

sliding. Additionally, the grain structure in our simulationsis stationary, which is clearly a simplification. For example,molecular dynamics simulations show that the motion oftriple points has a significant influence on the deformationbehavior of UFG materials [29]. Finally, the two-dimen-sional projection of a three-dimensional process naturallyhas severe limitations.

The dislocation structures found in our models are char-acterized by low dislocation densities in the grain interiorand high dislocation densities at GBs. However, the dislo-cation density in the grain interior increases with the grainsize. Additionally, the GB dislocation density also changeswith temperature, due to the increased dislocation absorp-tion rate at higher temperatures. The evaluation of localstresses exhibits a significant stress concentration at tripleGB junctions. These observations are consistent with theo-retical expectations that the deformed microstructures con-sist of hard GBs and a soft region inside the grains [21,25].

Finally, let us discuss the limitations of the appliedmodel. As stated above, the comparison of numerical andexperimental data is only possible in a qualitative sense.This refers not only to the observed material quantities,but also to the temperatures and loads employed in themodel. It must be made clear that at the high temperaturesstudied numerically, the microstructure of real materialswould not be stable. Furthermore, the material wouldnot be able to sustain the high loads employed in the sim-ulations. However, these boundary conditions are neces-sary for numerical reasons, but since we can control thedeformation mechanisms and the microstructural stabilityof our model material, the numerical results can be qualita-tively compared to experiments under conditions in whichthe same deformation mechanisms as those considered inthe model are active.

5. Conclusions

We employed coupled dislocation dynamics and diffu-sion kinetics simulations to investigate the mechanicalbehavior of UFG metals over a wide temperature range.Two types of dislocation–GB interactions are considered:(i) GBs acting as barriers to dislocation motion, and (ii)GBs absorbing lattice dislocations and transforming theminto GB glide and climb dislocations under conservationof the total Burgers vector. Thus, GB-mediated plasticityand recovery are the essential features that are introducedinto our model. GB diffusion processes that are necessaryto achieve a thermodynamically consistent description ofdislocation climb are considered by solving the diffusionequation with additional source and sink terms that takeinto account vacancy production and annihilation byclimbing dislocations.

Our results show that at low homologous temperaturesthe flow stress of the model material exhibits an inverseproportionality with the square root of grain size, in agree-ment with the Hall–Petch relation. The Hall–Petch con-stant of material decreases with increasing temperature.

At high temperatures, finally, the Hall–Petch constantbecomes negative and the material exhibits GB softeningrather than GB strengthening. This variation in the Hall–Petch constant with temperature is due to the accelerationof GB diffusion, which increases the rate of GB-mediateddeformation and in particular the dynamic recovery by dis-location absorption at GBs. We note here that GB-medi-ated processes themselves contribute only marginally tothe total plastic deformation, but that the dynamic recov-ery at the GBs helps to maintain the conventional plasticdeformation in the grain interior. The variation in theHall–Petch constant with increasing temperature and thetransition to GB softening at high homologous tempera-tures is consistent with experimental observations [12].

The influence of dynamic recovery at GBs is alsonoticed in the variation of the strain-rate sensitivity withgrain size and temperature. At constant temperature, thecalculated value of strain-rate sensitivity of a fine-grainedmaterial is considerably higher than that of a coarse-grained material. Additionally, the strain-rate sensitivityof a fine-grained material increases with temperature, whileit remains almost constant for coarse-grained materials.These modeling results on the strain-rate sensitivity arealso consistent with experimental data for a wide rangeof face-centered cubic as well as body-centered cubic met-als [15,14,28]. Hence, by comparing our model results withexperimental data in the literature we find a good qualita-tive agreement in the material behavior. From this we con-clude both that our model captures the essentialmechanisms of the deformation of UFG metals correctly,and that dynamic recovery processes at GBs and GB diffu-sion are the rate-limiting mechanisms for the deformationof UFG metals.

Acknowledgments

N.A. acknowledges support from the Higher EducationCommission of Pakistan (HEC). Both authors acknowl-edge funding from Thyssen Krupp AG, Bayer MaterialScience AG, Salzgitter Mannesmann Forschung GmbH,Robert Bosch GmbH, Benteler Stahl/Rohr GmbH, BayerTechnology Services GmbH, and the state of North-RhineWestphalia as well as the European Commission in theframework of the European Regional Development Fund(ERDF).

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