Mechanics of Materials 71 (2014) 62–73

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Mechanics of Materials

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Dislocation reactions, plastic anisotropy and foreststrengthening in MgO at high temperature

http://dx.doi.org/10.1016/j.mechmat.2014.01.0010167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Present address: INSA Lyon, LaboratoireMATEIS, UMR CNRS 5510, 25 avenue Jean Capelle, 69621 VilleurbanneCedex, France. Tel.: +33 472 438235; fax: +33 472 438539.

E-mail address: jonathan.amodeo@insa-lyon.fr (J. Amodeo).

J. Amodeo a,⇑, B. Devincre b, Ph. Carrez a, P. Cordier a

a Unité Matériaux et Transformations, UMR CNRS 8207, Université de Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq, Franceb Laboratoire d’Etude des Microstructures, UMR 104, CNRS-ONERA, 29 avenue de la Division Leclerc, 92322 Chatillon, France

a r t i c l e i n f o

Article history:Received 6 November 2013Received in revised form 21 December 2013Available online 18 January 2014

Keywords:MgOPlasticityDislocation Dynamics simulationForest strengthening

a b s t r a c t

The collective properties of dislocations in MgO are investigated in the high temperatureregime and at constant strain rate with 3D Dislocation Dynamics simulations. Intersectionsbetween slip systems 1/2h110i{110} and 1/2h110i{100} allow essentially two types ofjunction reactions. These junctions are energetically stable and are expected to promotestrong forest strengthening at high temperature. Large-scale DD simulations show thatMgO strain hardening at high temperature may be dominated by forest reactions. Impor-tant parameters for dislocation density based modeling of MgO plasticity are finally calcu-lated and verified to be consistent with experimental observations.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction Dislocations in the 1/2h110i{110} slip system are found

MgO is a model ceramic with NaCl-structure intensivelystudied in the past decades for its potential uses as arefractory material (Tf > 3000 K at ambient pressure). Al-loyed to a small amount of iron (�10%), MgO is also thesecond most abundant phase of the Earth’s lower mantleafter the magnesium-silicate perovskite phase MgSiO3.Therefore, the study of MgO plastic properties is of greatinterest in both Geophysics and Materials Sciences.

At ambient pressure, plastic strain in MgO single crystalis expected to be governed either by lattice friction, impu-rities strengthening or dislocation–dislocation elasticinteractions depending on the investigated temperaturerange (Barthel, 1984; Haasen et al., 1985; Sato and Sumino,1980).

At low temperature, a large lattice friction is measuredon the two experimentally observed slip systems1/2h110i{110} and 1/2h110i{100}, respectively.

to glide at the lowest stress level (Barthel, 1984). In both slipsystems, the kink-pairs nucleation process is the glide con-trolling rate mechanism and plastic deformation is gov-erned by the slow motion of long screw dislocations(Appel and Wielke, 1985). At intermediate temperatures,lattice friction is less important and dislocations interac-tions with solute elements start to affect the MgO flow stress(Appel et al., 1984; Gorum et al., 1960; Srinivasan and Stoeb-e, 1974). The MgO mechanical properties in this intermedi-ate temperature regime was recently overviewed byMesserschmidt (2010). Lastly, at temperatures higher thanthe athermal transition temperature Ta (respectively about600 K and 1500 K for the 1/2h110i{110} and the1/2h110i{100} slip systems), new features are expected toinfluence MgO plastic properties. Deformation microstruc-tures are now made of curved dislocation segments andsmall debris (Clauer and Wilcox, 1976; Haasen et al.,1986). Whatever the active slip system and the impurityconcentration, MgO flow stress is low about 10 MPa in thistemperature regime (Amodeo et al., 2011; Copley and Pask,1965; Hulse et al., 1963).

Very few quantitative analysis exist on the elementarymechanisms controlling MgO plasticity in the high

J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73 63

temperature regime (Copley and Pask, 1965; Day andStokes, 1964). This matter of fact is problematic as MgOhas gained in recent years renewed interest to investigatesmall scale plasticity (Dong et al., 2010; Gaillard et al.,2006; Howie et al., 2012; Korte and Clegg, 2011; Korteet al., 2011) and in geophysics (Amodeo et al., 2012;Cordier et al., 2012; Girard et al., 2012; Merkel et al.,2002). For instance, Cordier and collaborators (2012) haverecently shown that the mechanical properties of MgO inthe athermal regime (i.e. for T > Ta) are of primary impor-tance to understand the rheology of the Earth’s lower man-tle. For these reasons, it is expected that DislocationDynamics (DD) simulations dedicated to MgO plasticitycan provide new inputs and hopefully can improve ourunderstanding of the Earth’s mantle flow mechanisms.

Three-dimensional DD simulations are today acknowl-edged as a unique tool to investigate collective propertiesof dislocations and to access crystal plasticity at the micro-structure level (i.e. at the mesoscopic scale). DD simula-tions have shown interest in lots of different studies, butmainly in the case of metal plasticity (Bulatov et al.,2006; Devincre et al., 2008; Dimiduk et al., 2006; Madecet al., 2003). In a recent study, it was shown that MgO yieldstress can be precisely evaluated with DD simulations forMgO single crystals and large grains polycrystals (Amodeoet al., 2011). Here, we present new calculations performedin the athermal regime to examine the contribution of for-est reactions to MgO plastic flow at ambient pressure andin laboratory conditions of strain rate.

The effect of forest interactions (i.e. dislocation–disloca-tion contact reactions) is explored in two steps. First, wesystematically investigate the reactions occurring betweentwo intersecting dislocations depending on their slip sys-tems and their relative orientations. Then, large-scale DDsimulations are performed to quantify the contribution offorest interactions to plastic strain depending on the activeslip systems. Finally, we calculate physical parameters,which are essential ingredients of crystal plasticity modelsat large scale.

2. Simulations methodology

2.1. Dislocation Dynamics simulations

Dislocation Dynamics simulations have been performedwith the free and open source simulation code microMegas(mM). Full description of this computer model has beenpresented in numerous studies (see for example, Devincreet al., 2001; Durinck et al., 2007; Monnet et al., 2004), andwas recently described with some details in Devincre et al.(2011). More specific information on the local rules appliedin the simulation to model junction zipping-unzipping canbe found in Carrez et al. (2005), Devincre (1996), andDevincre et al. (2006). In the following, we recall somebasics and calculation specificities of the present work.

The mM simulation relies on a discretization of bothspace and time. For each slip systems, dislocation linesare discretized into a set of straight segments with discretecharacter including screw, edge and mixed directions.Those segments move on a 3D cubic lattice with glide

directions here reproducing the symmetry of MgO crystallattice. Forces on dislocation segments are calculated usingstandard formula taken from the dislocation elastic theory(Devincre and Condat, 1992; Hirth and Lothe, 1982). Theeffective resolved shear stress calculated at the center ofdislocation segments takes into account the internal stressfield associated with the whole set of dislocation linespresent in the simulated volume, the applied stressaccounting for the loading conditions and a line tensioncorrection term accounting for the elastic energy lost whenreplacing locally curved dislocation sections by straightsegments. Local rules are prescribed in the simulation toaccount for contact interactions, such as annihilations orthe locking of sessile dislocation junctions in specific crys-tallographic directions. For the present study, the length ofthe Burgers vector 1/2h110i for dislocations in MgO is setto 2.99 Å and isotropic elasticity is used with a shear mod-ulus l(T) = 140 � 0.0255 * T (Isaak et al., 1989, 1990; Karkiet al., 2000) and a constant Poisson ratio m = 0.18.

2.2. Dislocation intersection mapping

The first simulations reported in Section 3 are dedicatedto the mapping of dislocation–dislocation intersections.This type of simulations depicts possible reactions betweentwo intersecting dislocations and provides preliminaryinformation on the strength of forest interactions betweenslip systems. Such type of calculations have already beenmade in several materials including FCC and BCC structures(Kubin et al., 2003; Madec et al., 2002a; Madec and Kubin,2004; Wickham et al., 1999), HCP structures (Capolungo,2011; Monnet et al., 2004) and other materials like olivine(Durinck et al., 2007) and ice (Devincre, 2012). Carrez andcollaborators also published a first numerical investigationof the dislocation interactions between 1/2h110i{110} slipsystems in MgO (Carrez et al., 2005).

The methodology to calculate a map of dislocation–dis-location intersections with DD simulations is now standardand proceeds as follows. Two straight dislocations seg-ments, crossing at their centers, are introduced in a largesimulation cell and relaxed by considering only dislocationglide. Depending on the initial configuration, different kindof reactions can be observed like junction, annihilation,crossed or repulsive states. These simulations are com-pletely driven by elastic energy minimization and no ap-plied stress is imposed. The tested configurations aredefined by the two angles (u1,u2) existing in betweenthe initial segments directions and the intersection be-tween their glide planes (Fig. 1).

2.3. Massive DD simulations

The second set of simulations reported in Section 3 aimsat reproducing the collective properties of dislocations inthe athermal regime during compression tests at constantstrain rate (referred as massive DD simulation in the fol-lowing). As we want to simulate plastic deformation forT > Ta, use is made of a free flight mobility law:

vðsÞ ¼ bsB

ð1Þ

Fig. 1. Two parent dislocation lines, with Burgers vectors b*

1 and b*

2,interact each other leading to the formation of a dislocation junction withBurgers vector b

*

j ¼ b*

1 þ b*

2. The nature of the two parent dislocations isdefined by the angles (u1,u2).

Table 1The rotations matrix applied to the MgO crystal in the simulation frame tocontrol periodic boundaries artifacts.

1 4 84 7 48 4 1

24

35

Table 2Indexation of the 12 MgO slip systems considered in the present work.

1/2h110i{110} 1/2h110i{100}

i Plane ~b i Plane ~b

1 ð110Þ 12 ½11 0� 7 (001) 1

2 ½11 0�2 (110) 1

2 ½11 0� 8 (001) 12 ½11 0�

3 ð011Þ 12 ½011� 9 (100) 1

2 ½011�4 (011) 1

2 ½011� 10 (100) 12 ½101�

5 ð101Þ 12 ½101� 11 (010) 1

2 ½101�6 (101) 1

2 ½101� 12 (010) 12 ½101�

64 J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73

This mobility law is supposed to be dislocation charac-ter and slip plane independent. The viscous drag coefficientB accounts for dissipating processes and refers to the atom-istic mechanisms that control dislocation mobility at hightemperature in ionic crystals (i.e. phonon drag and interac-tions between charged dislocations and their compensat-ing charged clouds). Without any reliable value of B inMgO (Kardashev et al., 1985) a common value measuredin ionic crystals of B = 10�5 Pa s is considered (Robinson,1972; Singh et al., 2008). Results of DD simulations arenot expected to depend on the details of this free flightmobility law in materials with forest strengthening.

In massive DD simulations, a random distribution ofedge and screw Frank–Read (F–R) sources with a densityq is taken as initial dislocation microstructure. The lengthof sources, l = 10 q�1/2, is set large enough to avoid the sim-ulated flow stress to be controlled by critical stress for dis-location multiplication at sources. As we will focus here onforest strengthening, the dislocation cross-slip and climbmechanisms were not taken into account in the simula-tions. Dislocation self-annihilation is a possible artifact ofDD simulations when using periodic boundary conditions(Bulatov et al., 2000; Madec et al., 2004; Queyreau,2008). Among the different existing solutions for this prob-lem, a rotation of the crystal directions with respect to theh100i surfaces of periodicity and the use of a non-cubicshape for the simulation cell are used in the present work.Hence, whatever the investigated slip systems, the simu-lated crystal is rotated three times around differenth111i directions in the simulation cell frame. This opera-tion corresponds to the rotation matrix reproduced inTable 1.

Because DD simulation does not allow for large strainsand therefore for large increase of the dislocation density,simulations starting with different initial dislocation den-sities must be performed to test forest strengthening at dif-ferent virtual level of strain (Madec et al., 2002b). For thisreason, we used a scaling method that provides differentdislocation densities from a reference simulation by scal-ing all the characteristic lengths of the investigated prob-lem with a unique factor. Here, considering a referencesimulation volume Vref, the dislocation density is defined

by qref = Nref * lref/Vref. Nref and lref refer respectively to thenumber of F–R sources and their length. Then, an alterna-tive simulation will be parameterized by a scaling factor fthat impose an initial dislocation density:

q ¼ qref fref =f� �2 ð2Þ

Outcomes of this scaling method and other key param-eters used in the massive DD simulations are summarizedin Table 3.

3. Results

3.1. Mapping of dislocation–dislocation intersections

3.1.1. Preliminary analysisIn MgO and usually in materials with the NaCl crystal-

line structure, plastic deformation is known to take placein the two 1/2h110i{110} and 1/2h110i{100} slip systemfamilies. Each family is made of 6 independent slip systemsnoted hereafter with index i as listed in Table 2.

Geometrical examination and Frank’s rule for disloca-tion reactions (Friedel, 1967) give preliminary elementsof analysis on the possible dislocation reactions that mayexist between slip systems. Considering all slip systemsnoted in Table 2, 66 combinations of dislocation interac-tion exist (i.e. 15 per slip system family plus 36 reportedas ‘‘crossed’’ configurations between 1/2h110i{110} and1/2h110i{100} slip systems). Accounting for the MgOcrystal symmetries, those 66 combinations can be groupedin 7 different classes of intersections (i.e. 2 per slip systemfamily and 3 for the ‘‘crossed’’ configurations). Within suchclasses, 1 type per slip system family corresponds to co-planar or orthogonal intersections. Such reactions havevery low probability and are not energetically favorable.Therefore, they are not expected to strongly contribute toforest strengthening. For this reason, they will not be de-tailed further in the present study. Readers interested insuch intersections may refer to Carrez et al. (2005) formore details. Then, one remains only with:

J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73 65

� One type of intersections between 1/2h110i{110} slipsystems:

This type of intersections (intersection between slipsystems i = 2 and 4 for instance) corresponds to contactinteractions between dislocations in slip systems with Bur-gers vectors oriented at 60� or 120� each other (Copley andPask, 1965; Kear et al., 1959). In this case, the expectedjunction is of edge character, directed along the h111idirections, with a 1/2h110iBurgers vector. Its potentialglide plane is {112}, which is not an identified glide planein MgO. For this reason, such junction was considered assessile in the simulations. This reaction is called the60�/120�{110} junction in the following.

� One type of intersections between 1/2h110i{100} slipsystems:

This orthogonal intersecting slip systems configuration(systems i = 8 and 9 for instance) can form1/2h110i{110} edge junctions along the h100i directions.It is later mentioned as the 90�{100} junction.

� Three types of ‘‘crossed’’ intersections between1/2h110i{110} and 1/2h110i{100} slip systems:(i) Intersections involving slip systems with the same

Burgers vectors (systems i = 1 and 7 for instance)are specials. They correspond to the particular caseof collinear annihilations. Collinear annihilation isthe reaction promoting the strongest of all possibleforest strengthening. In the last years, the excep-tional properties of this reaction was the matter ofseveral investigations (Devincre et al., 2005; Madecet al., 2003).

(ii) Intersections between slip systems with 90� Burgersvectors (slip systems i = 1 and 8 for instance) mayform junctions with mixed character, potentiallyglissile in the {100} planes. Nevertheless, theh100i Burgers vector of such junctions is too largeto be energetically favorable and this intersectiononly leads to repulsive and crossed-states.

(iii) Intersections between slip systems with 60�/120�Burgers vectors (slip systems i = 3 and 12 forinstance) can form mixed junctions with Burgersvector 1/2h110i. Those junctions are glissile in the{100} planes.

3.1.2. Dislocation intersection maps in MgODD simulation is a powerful numerical tool to quantify

the domains of reactions occurrence in the above listedintersections. The intersection maps reported in Fig. 2 arebased on interaction simulations between dislocation seg-ments with length l = 1 lm intersecting at the center of acubic simulation cell with 3 lm linear dimension. To calcu-late these maps, 676 (u1,u2) initial configurations are sys-tematically tested to explore a wide range of possibleintersection configurations (we test 26 different segmentcharacters for each slip systems). Following the work ofMadec et al. (2002b), the simulation results are systemat-ically compared to the predictions of a simple elastic mod-el (Hirth and Lothe, 1982) neglecting the contribution of

dislocation curvature to the relaxation problem. It mustbe noted that for all the configurations we tested in thecase of MgO, a good match is found between the DD simu-lation results and the predictions of this simple elasticmodel.

Fig. 2a and b shows respectively the results of60�/120�{110} and the 90�{100} intersection maps. Inthese maps, the junction domains are well delimited withround shapes. In such lobes, the junction length (and con-sequently the junction strength) is decreasing rapidlywhen approaching the domain frontiers. The junction do-mains represent in both maps approximately 40% of theintersections and the remaining relaxed configurationsare repulsive interactions or crossed-states, both in similarproportions.

Fig. 2c and d refer to ‘‘crossed’’ intersections i.e. involv-ing slip systems from the two MgO slip system families.For sake of brevity, only ‘‘crossed’’ configurations leadingto energetically favorable reactions are represented. Theintersection map inferred from ‘‘crossed’’ slip systems withBurgers vectors oriented at 60�/120� each other (Fig. 2c),e.g. 1/2[011](011) and 1/2[101](010), is characterizedby a junction domain larger than those observed in the60�/120�{110} and 90�{100} intersections. Like in othermaterials, the collinear interaction map (Fig. 2d) showsthat the annihilation domains are much larger than theattractive regions calculated with conventional elastic pre-dictions (68% of favorable geometrical configurations).Hence, collinear annihilation is potentially a very efficientforest strengthening mechanism if slip system activities al-low for its occurrence.

These intersection map results are interesting and showthat, in agreement with the forest model (Saada, 1960),strong dislocation line ‘‘pinning’’ induced by junction reac-tions and collinear annihilations take place in MgO plasticdeformation if the temperature is large enough to facilitatedislocation mobility. While intersection map calculationsallow identifying dislocation–dislocation reaction types,they strictly do not quantify forest strengthening. This iswhy, more complex DD simulations, accounting for thecollective properties of dislocations are now presented.

3.2. Massive DD simulations

Numerical straining experiments at constant strain ratehave been performed to study plastic deformation and for-est strengthening in MgO. As the literature always relatesmechanical tests where dislocations glide either in the1/2h110i{110} slip system family or in the 1/2h110i{100}slip system family (Appel et al., 1984; Copley and Pask,1965; Day and Stokes, 1964), we restricted simulations tosymmetric multi-slip conditions with initial dislocationdensities including only one set of slip system family. Inthese simulations, the initial dislocation density qinit isequally distributed between each of the six possible slipsystems per family.

First, compression tests with the 1/2h110i{110} slipsystems were made using a h100i loading axis. This orien-tation allows to activate dislocation glide in four of the sixpossible slip planes with a Schmid factor S = 0.5. In thosesimulations, the MgO shear modulus was set to

Fig. 2. Dislocation intersection maps (a) the 60�/120�{110} reaction, (b) the 90�{100} reaction, (c) the ‘‘crossed’’ 60�/120� reaction and (d) the ‘‘crossed’’collinear reaction. Solid lines delimit repulsive (�) and attractive (+) domains and dashed lines refer to junction lobes calculated with a simple elasticmodel. (a)–(c) Each figure pixel represents a (u1,u2) initial configuration leading to a junction in the simulation. Colors refer to the junction length lj (fromlight gray for shortest junctions up to yellow for largest junctions). In the middle of the junction lobe, (0,0), the dislocation segments are parallel to thejunction direction, hence lj = l. (d) Black solid circles refer to geometrical configurations favorable to annihilations. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.).

66 J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73

114.5 GPa to reflect a temperature equal to 1000 K. It mustbe noted that such temperature is significantly above the1/2h110i{110} slip system athermal temperature whichis approximately equal to 600 K. The reference dislocationdensity and simulated volume are respectively qref = 1012

m�2 and Vref = 10.70 � 11.70 � 11.80 lm3.A second set of calculations was carried out with the

1/2h110i{100} slip system family. To compare with exper-imental observations such simulations have been madeusing a shear modulus of 89 GPa consistent with a temper-ature of 2000 K. Again, it must be noted that suchtemperature is above the tested slip systems athermaltemperature which is approximately equal to 1500 K. Theloading axis is along the h110i directions. Four of six1/2h110i{100} slip systems are thus activated with aSchmid factor

ffiffiffi2p

=4 (�0.35). The reference dislocationdensity is still qref = 1012 m�2, but the dimensions of thesimulation reference volume was modified to accountfor different dislocation self-annihilation directions,Vref = 8.00 � 9.09 � 10.32 lm3.

3.2.1. Strain rate calibrationIn DD simulations, considering a dislocation viscous

mobility law like Eq. (1), one may define two deformationregimes respectively at low and high strain rate (Kubinet al., 1998). At low strain rate, the simulated flow stressis mostly insensitive to strain rate variations if the disloca-tion dynamics is governed by forest interactions. This lowstrain rate case is called ‘‘quasi-static’’ regime and suitsto the mechanical behavior of FCC metals in laboratorytests, typically in the range of 10�4 s�1 up to 1 s�1. Risingthe strain rate, DD simulation switches to the ‘‘dynamic’’regime. In this high strain rate regime, a large density ofimmobile dislocations pinned at forest obstacles do notanymore rule plastic strain, but rather the latter is con-trolled by dislocations velocity during their free flight. Con-trary to the ‘‘static’’ regime, the ‘‘dynamic’’ regime is thenstrongly sensitive to strain rate.

In order to verify that our simulations refer to conven-tional ‘‘quasi-static’’ conditions (flow stress independentof strain rate), three increasing strain rates were

J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73 67

investigated: 1 s�1, 10 s�1 and 100 s�1. These simulationsare made considering the 1/2h110i{110} reference setupdescribed above.

The shear stress s versus plastic shear cp curves we getwith these simulations are presented in Fig. 3a. Similar re-sults are found at 1 s�1 and 10 s�1 strain rate and the dis-location dynamics observed are found very closed in thesimulated volume. Discrepancies start to appear only at100 s�1 with a slight increase of the flow stress. It mustbe noted that similar behavior was observed with the sim-ulations considering the 1/2h110i{100} slip systems. Theslopes of the simulated plastic deformation versus timeplotted with a linear regression are respectively 0.98 s�1,9.96 s�1 and 99.40 s�1 (Fig. 3b). These values are in goodagreement with the imposed strain rate meaning that thestrain rate regime is achieved whatever the simulationconditions.

From the above remarks and considering a dislocationdensity qref = 1012 m�2, DD simulations show that MgOsingle crystal deforms within the ‘‘static’’ regime below10 s�1 imposed strain rate. Between 10 s�1 and 100 s�1,simulations enter the ‘‘dynamic’’ regime. In this regime,the calculated flow stress is questionable since it is con-trolled by dislocation mobility equation which is poorlyknown in the MgO at high temperature. Nevertheless, itis interesting to note that the transition range we repro-duced between the two regimes is close to the one calcu-lated in previous simulations made in FCC and BCCcrystals (Devincre and Kubin, 1997; Kubin et al., 1998;Madec, 2001; Queyreau, 2008). This result suggests thatthe strength of forest interactions in MgO at high temper-atures is closed to those found in FCC and BCC metals.

3.2.2. Dislocation avalanches and plastic anisotropySimulations made at the reference density (qref = 1012

m�2) show that the flow stress is about 10 MPa for the both1/2h110i{110} and 1/2h110i{100} slip systems families.During these simulations, the dislocation microstructureevolves through the formation and destruction of lots ofjunctions, which impose a jerky motion to mobile disloca-tions. As for FCC metals (Devincre et al., 2008), the ather-mal deformation regime of MgO is thus characterized bythe discontinuous motion of dislocations, through consec-utive long waiting periods and few glide events sometimecalled ‘‘dislocation avalanches’’. This non-linear dynamics

Fig. 3. Shear stress s versus plastic shear cp (a) and plastic deformation

justify the use of DD simulations to do up-scaling and toidentify parameters for continuum models.

To illustrate dislocation avalanches in MgO, we super-imposed eight successive views taken from a thin foil of1 lm, oriented along the [110] direction (Fig. 4). From thissetup, one can follows the jerky glide of a 1/2[110](110)dislocation passing through the whole thin foil (from posi-tions marked 1–8). After entering the thin foil (bottom ofFig. 4, position 1), the dislocation alternates gliding andpinned configurations due to the formation and thedestruction of junctions. This mechanism is especially welldepicted at positions 3 and 4 where junctions are clearlyidentified as pinning obstacles to dislocation glide. The dis-location is finally stopped in position 8, where two strongjunctions and a dipole finish to impede its progression. Itmust be noted in Fig. 4 that only the progression of one dis-location is reproduced, but during large plastic burst, thesimulated dislocation avalanches involve several mobiledislocations in parallel glide planes.

In accordance with Section 3, numerous 60�/120�{110}and 90�{100} junctions, oriented along the h111i and theh100i directions can be observed in the simulated disloca-tion microstructures (Fig. 5). Whatever the activated slipsystem, both dislocation and junction densities (respec-tively q and qj) increase during deformation, whereas theratio qj/q quickly saturates in between 30% and 40% as al-ready observed in FCC similar investigations (Devincreet al., 2008).

Furthermore, we note that during simulations, the den-sity of junctions per slip system qj,i does not increasehomogeneously with strain. Such tendency is observedeven for slip systems stressed with the same Schmid factor.Furthermore, this phenomenon is found in all simulationswhatever the tested slip system families or the initial dis-location density. For the sake of brevity, only the case ofthe 1/2h110i{110} reference simulation is analyzed inthe following.

Fig. 6b illustrates the different evolution of the qj,i dis-tribution calculated for each activated (i = 1 to 4) and inac-tive slip systems (i = 5 and i = 6) in the early stages of the1/2h110i{110} with the reference density. From deforma-tion as small as ep � 0,04%, it is clear that in the large peri-odic volume element used in massive DD simulation, astrong deviation of the number of junctions is detectedon mechanically equivalent slip systems. This unbalanced

ep versus time t (b) for strain rates of 1 s�1, 10 s�1 and 100 s�1.

Fig. 4. 8 successive views of a thin foil of 1 lm thick, oriented along the[110] direction, are superimposed. Static dislocation lines are repre-sented in gray and gliding dislocation segments in red. Numbers 1–8 referto successive positions of the dislocation involved within a plastic burst(avalanche). Black arrows point out small displacements (bow out) ofneighboring dislocations. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.).

68 J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73

evolution is not load controlled, but is rather ruled by het-erogeneities in the microstructure formed during thedeformation. Using different initial dislocation source dis-tribution allows checking this point. Indeed, it is found thata tiny heterogeneity in the initial dislocation distributionquickly promotes unbalanced formation of junctions be-tween active slip systems. Such differences help or restrictthe mobility of dislocations and affect the multiplicationrate of dislocation in the slip systems (Fig. 6a). The largeris the dislocation density on a given slip system (for in-stance slip system i = 1 in Fig 6), the more such system actsas forest obstacles to the other slip systems and decreases

Fig. 5. Two example of dislocation microstructure for simulations with the 1Dislocations are represented in gray and junctions are in black.

the density of mobile dislocations. This is why, as for qj,i,the plastic shear distribution between slip system cp,i isalso anisotropic (Fig. 6a) and the total plastic strain is gov-erned at the end of simulations by the activity of one orfew dominating slip systems (e.g. i = 1 and 2 in Fig. 6).

3.2.3. Forest strengthening in MgOThe simulations of reference reported in the previous

sections with both 1/2h110i{110} and 1/2h110i{100} slipsystem families highlight the important role of disloca-tion–dislocation contact interactions in the athermaldeformation regime. In metals, the forest model (Saada,1960) is the most pertinent model accounting for the influ-ence of dislocation microstructure on the plastic flow. Thismodel justifies the well-known Taylor form:

sf ¼ albffiffiffiffiqp ð3Þ

where a is a temperature independent coefficient whichaverages the effect of all possible dislocation–dislocationinteractions, qf is the forest dislocation density (i.e. thedensity of dislocations cutting to the glide plane of mobiledislocations) and sf the athermal stress required to over-come the forest dislocations network.

However, Eq. (3)refers only to a simplified expression ofthe dislocation line tension which neglects small variationsassociated to a logarithmic term in the definition of thedislocation elastic energy (Hirth and Lothe, 1982). Bothexperiments and simulations have proved this logarithmiccontribution to be significant (i.e. systematic deviationsfrom Eq. (3)are observed), especially for large values ofqf. This is why, alternative forms of the Taylor equationwhich account for the exact energetics of dislocation linetension exist. For instance, Devincre et al. (2006) proposeda modified forest expression with the form:

sf ¼ lbln 1=b

ffiffiffiffiffiffiffiffibqf

q� �

ln 1=bffiffiffiffiffiffiffiffiffiffiffibqref

q� �ffiffiffiffiffiffiffiffibqf

qð4Þ

where b is a forest strengthening coefficient equivalent toa , but which does not depend on the dislocation density.

/2h110i{110} slip system (a) and the 1/2h110i{100} slip system (b).

Fig. 6. Plastic shear per slip system cp,i (a) and evolution of the junction density per slip system qj,i (b) during simulations with the 1/2h110i{110} slipsystem family and the reference density. In the legend, numbers refer to the slip system index presented Table 1. Curves with identical colors refer toorthogonal slip systems. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73 69

In order to see if the forest model also applies for MgOsingle crystal deformation in the athermal regime, simula-tions using the scaling method described in Section 2 wereperformed. This scaling methodology allows crossing abroad range of dislocation densities without increasingCPU costs. Details on the key parameters used for these cal-culations are summarized in Table 3.

Here, we define the effective forest density qf as theglide planes intersecting dislocation density, what leadsto qf = 5/6q and qf = 2/3q respectively for simulations in{110} and {100} slip planes. In the case of {100}, the dis-location density attributed to coplanar slip systems is sub-tracted. As expected, the stress–strain curves presented inFig. 7 show that the larger the dislocation density, thehigher the flow stress. As an effect of the plastic instabilitydiscussed above, plastic deformation is progressively local-ized in one or two ‘‘easy’’ slip system. This explains whydislocation dynamics is modified from multi-slip to easyslip conditions and why a smooth softening effect is ob-served at the end of the stress–strain curves (Fig. 7). Toquantify the influence of each slip system on foreststrengthening, this plastic anisotropy phenomenon mustbe handled carefully. The forest coefficients a and b weretherefore calculated using the following protocol: (i) stressand dislocation density measurements are performed afteryield, but after relatively low simulated plastic deforma-tions and before the occurrence of plastic anisotropy (ii)the imposed strain rate regime must be achieved.

Table 3Key parameters of the massive DD simulations made at different scales. Thesame parameters are applied for simulations with slip systems1/2h110i{11 0} and 1/2h110i{100}. f, qinit, V, dt, _e and U are respectivelythe simulation scaling factor, the initial dislocation density, the simulationcell volume, the time step, the imposed strain rate and the dislocationdiscretization length. The parameters used for the reference simulations arenoted in bold.

f qinit (m�2) V (lm3) dt (�10�10 s) _e (s�1) U (lm)

31.6 1 � 1011 �31600 2 1 0.6310 1 � 1012 �1000 0.66 10 0.204.47 5 � 1012 �89 0.30 50 0.093.16 1 � 1013 �32 0.066 100 0.06

Evolution of the resolved shear stress on active slip sys-tems regarding the square root of the forest density is plot-ted in Fig. 8. These results are compared with predictionsof Eqs. (3) and (4) in order to fit, the a and b coefficientsvalues. From this plot we see that the standard Taylorequation (Eq. (3)) fits well DD simulation results with avalues of 0.24 when considering the 1/2h110i{110} slipsand 0.28 with the 1/2h110i{100} slips. Comparison withEq. (4)gives even a better agreement and leads to b valuesof 0.07 and 0.10 (using a reference dislocation density of1012 m�2) for the 1/2h110i{110} and 1/2h110i{100} slipsystem families respectively (see Fig. 8). As expected, themain benefit of Eq. (4) appears at the extreme values ofthe forest density.

The close correspondence between the a (or b) forestcoefficients found with the two tested slip system familiesconfirms statistically the first predictions made withthe dislocations intersection maps. The dislocationreactions taking place during the 1/2h110i{110} and1/2h110i{100} slip systems simulations, i.e. mainly the60�/120�{110} and the 90�{100} junctions, have compara-ble stability and therefore impose close forest strengthen-ing. Finally, the low level of plastic deformation simulatedwith DD simulations and the softening instability observedin the MgO simulations make direct comparison to exper-iment a difficult exercise. One can only say that forestinteraction, which is the only strengthening mechanismconsidered in the present study, reproduce realistic flowstress for MgO plasticity in the athermal regime.

4. Discussion

4.1. Comparison to experiments and other DD simulationsstudies

Intersections between slip planes in high temperaturecompression tests with MgO single crystal have been men-tioned very early, especially for the {110} slip planes (Cop-ley and Pask, 1965; Day and Stokes, 1964; Groves andKelly, 1963). In compression tests, it was shown thatMgO deforms with 1/2h110i{110} slips thanks to the acti-vation of only one or few glide directions whereas other

Fig. 7. Stress–strain curves for simulations with the 1/2h110i{110} slip systems (a) and the 1/2h110i{100} slip systems (b).

Fig. 8. Simulated flow stress evolution as function of forest density forthe two MgO slip system families. Symbols represent DD simulationresults. Straight and dashed lines correspond to the least-squaresresolution of Eqs. (3) and (4), respectively.

70 J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73

observed slip systems seem to be blocked. Qualitativeinterpretation of such plastic anisotropy is today basedon the assumption that strong interactions exist betweenslip systems.

In this work, two different approaches based on the dis-location elastic theory and making use of DD simulations(dislocation intersection mapping and large-scale massivesimulation) were used.

First, the intersection maps presented in Section 3 leadto the definition of several different interaction geometrieswhich support the formation of strong junctions when dis-locations intersect between two different slip systems. The60�/120�{110} and the 90�{100} intersection maps exhibitcomparable statistics of junction occurrence with 40% offavorable configurations (Fig. 2a and b). Both intersectionmaps are highly comparable to the well-known Lomer-Cottrell reaction observed in FCC crystals (Madec et al.,2002a). This similitude comes partly from the direction1/2h110i and the amplitude of the existing dislocationBurgers vector in MgO that is closed to the FCC case. Inaddition, it was shown that the dimensions of the junctiondomains obtained in all MgO intersection maps are largerthan the Hirth junction solution existing in FCC crystals(Kubin et al., 2003; Wickham et al., 1999) and also is largeror equivalent to most junction solutions found in BCC crys-tals (Kubin et al., 2003; Wickham et al., 1999). Hence, for-est strengthening in MgO is expected to be large in the

absence of strong lattice friction and comparable to FCCand BCC metals results. In addition, we found that numer-ous collinear annihilations can occur in MgO plastic defor-mation if the two 1/2h110i{110} and 1/2h110i{100} slipsystem families are simultaneously activated. This possi-bility could induce very large forest strengthening.

Secondly, massive DD simulations have been performedto quantitatively determine forest strengthening in MgO.Experimental observations (Washburn et al., 1960) of anincrease of the dislocation density in the four h111i direc-tions when only the 1/2h110i{110} slip systems are acti-vated is well reproduced with the DD simulations. Thisobservation is simply explained by the formation ofh111i sessile 60�/120�{110} junctions that block disloca-tion glide in these directions.

The formation of MgO dislocation microstructures inmulti-slip conditions involves the accumulation of manyjunctions and is found to systematically promote plasticanisotropy. This ‘‘autocatalytic’’ phenomena, justified inSection 3, may explain frequent macroscopic observationsmade in MgO deformation experiments (Copley and Pask,1965; Day and Stokes, 1964; Hulse and Pask, 1960).

The increase of flow stress with the increase of disloca-tion density found in massive simulations follows Taylor’stypes relations (Eqs. (3)and (4)) and validates the existenceof a strong forest strengthening in MgO at high tempera-ture. The calculated Taylor coefficient a is 0.24 and 0.28with the 1/2h110i{110} and 1/2h110i{100} slip systemfamilies respectively. Such values agree with experimentalmeasurements of about 0.2 (Davidge and Pratt, 1964).However, too few measurements of a have been performedin MgO to make a strong statement. In addition, calculationof the b coefficients is made. The latter coefficient is impor-tant as it provide information for ceramic micromechanicalmodels at much lower and higher dislocation density thanconsidered in DD simulations.

Taylor coefficients (a or b) compare well for the two slipsystem families observed in MgO. This result is easilyunderstood by the similarity of dislocation microstruc-tures, in terms of junction’s strength and probability ofoccurrence. While the 60�/120�{110} junction in MgOand the FCC Lomer–Cottrell junctions lead to similar do-main of existence in intersection maps, the value of a forthe 1/2h110i{110} slip systems of MgO is lower than the

J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73 71

0.30–0.35 values measured (or calculated) in aluminum,copper or silver (Basinski and Basinski, 1979; Hansen andHuang, 1998). Such difference can be explained by (i) a lackof collinear annihilations, in MgO conventional strain con-ditions, which is considered to be an important ingredientof forest strengthening in FCC metals (Madec et al., 2003),(ii) a higher number of slip systems per family in FCC (12instead of 6 in MgO) that increase the number of slip sys-tems reaction causing the formation of junctions.

4.2. Toward a description of the strain hardening rate

Several deformation experiments in MgO report mea-surements about strain hardening rate coefficient h = ds/dcin temperature ranges comparable to the one tested in thepresent simulations (Copley and Pask, 1965; Hulse and Pask,1960; Routbort, 1979). Unfortunately, as for the flow stressquestion, there is today no strong argumentation on thephysical process controlling hardening in MgO.

Following the seminal work of Kocks and Mecking(Kocks and Mecking, 2003; Mecking and Kocks, 1981),one can define a simple relation which links a to h basedon the derivative of Eq. (3) with respect to plastic strain :

h ¼ ðalbÞ2

2sdqf

dcð5Þ

Here, no attempt has been made to formally calculatethe dislocation storage rate dqf/dc existing in the secondpart of Eq. (5). This quantity results from a complex com-petition between different storage and recovery mecha-nisms, which are poorly known in MgO at hightemperature. Rather, we made only a rough evaluation ofthis quantity directly from the DD simulations assumingthat plastic anisotropy is not problematic when dislocationstorage is averaged on several active slip systems.Hence, we estimated dqf/dc about 2.09 1014 m�2 and4.15 1014 m�2 from reference simulations in the1/2h110i{110} and the 1/2h110i{100} slip system family,respectively. With these evaluations and the previouslycalculated values of a, work hardening rate evaluationsare reported in Fig. 9.

These calculated values of h are about l/250 at 1000 Kfor 1/2h110i{110} and of l/130 at 2000 K for1/2h110i{100}. Such strain hardening coefficients are

Fig. 9. Calculations (red symbols) of the strain hardening rate coefficienth are compared to experimental data (black symbols) for both slipsystems families 1/2h110i{110} and 1/2h110i{100}. (For interpretationof the references to color in this figure legend, the reader is referred to theweb version of this article.).

larger than reported experimentally (Copley and Pask,1965; Hulse and Pask, 1960; Routbort, 1979). Possibleexplanation for such over-evaluation may be that the sim-ulated dislocation storage rates are necessarily overesti-mated in the absence of dislocation recovery mechanism.Indeed, h is experimentally measured at large deformation,where recovery mechanisms are efficient (Routbort, 1979).Nevertheless, it is interesting to note that DD simulationsgive the right order of magnitude for h without any fittingprocedure. This matter of fact reinforces the idea of aprominent role of forest interactions on MgO plastic defor-mation in the athermal regime.

5. Conclusion

The main goal of this work was to identify possibleexisting reactions between dislocations in MgO and toquantify their influence on plastic flow at high tempera-ture. Intersection maps calculations emphasize the impor-tance of two energetically favorable junction reactions, i.e.the 60�/120�{110} and the 90�{100} junctions. These junc-tions act as strong pinning point in dislocation microstruc-tures and justify strong forest strengthening and hardeningin large-scale massive DD simulations of MgO at hightemperature.

While this study gives a better understanding of the ele-mentary mechanisms responsible for MgO plastic proper-ties at high temperature, we also believe that the presentresults qualitatively apply for other materials with samecrystallographic geometry like alkali halides (NaCl, KCl,LiF, etc.) where similar plastic behavior was observedexperimentally (Takeuchi et al., 2009).

The present results are important for Materials Sciencewhere MgO plays a significant role as a model ceramic. Inaddition, MgO plasticity investigations are of primaryimportance for Earth Sciences considering the abundanceof this phase in the lower mantle.

Here we show that 1/2h110i{110} and 1/2h110i{100}slip systems strengthen in similar proportions (a of 0.24and 0.28, respectively) at ambient pressure and in theathermal regime of deformation. This result differsstrongly from MgO mechanical behavior below Ta, whereMgO deforms at stresses one order of magnitude lower in1/2h110i{110} than in 1/2h110i{100}. Based on a multi-scale modeling approach, Cordier and collaborators(2012) have recently shown that in the Earth’s conditionsof pressure, temperature and strain rate, MgO deformsmainly in the athermal regime (Cordier et al., 2012). Nev-ertheless, their description of MgO plasticity was limitedto the early stage of deformation. In calculating forest-strengthening coefficients for MgO at high temperature,the present study provides new important inputs regard-ing MgO strain hardening properties. To explore largerdeformation properties and afford more advanced infor-mation on the mantle rheology, the development of phys-ically justified crystal plasticity models is of primaryimportance. In this context, the question of MgO strainhardening properties under the Earth’s conditions of pres-sure, temperature and strain rate is an upcomingchallenge.

72 J. Amodeo et al. / Mechanics of Materials 71 (2014) 62–73

Acknowledgements

The authors are grateful to L.P. Kubin and R. Madec forhelpful discussions. B. Devincre acknowledges the supportof the M-ERA.net project FASS. Ph. Carrez acknowledgesthe support of the French ANR program (ANR ProjectDIUP). P. Cordier is supported by funding from the Euro-pean Research Council under the Seventh Framework Pro-gramme (FP7), ERC Grant N 290424 – RheoMan.

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