Computational Materials Science 141 (2018) 19–29

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Molecular dynamics investigation of c-axis deformation of single crystalTi under uniaxial stress conditions: Evolution of compression twinningand dislocations

http://dx.doi.org/10.1016/j.commatsci.2017.09.0150927-0256/� 2017 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: alig.sunil@gmail.com (S. Rawat), nilanjan@civil.iitkgp.ernet.in

(N. Mitra).

Sunil Rawat ⇑, Nilanjan MitraIndian Institute of Technology Kharagpur, Kharagpur 721302, India

a r t i c l e i n f o

Article history:Received 23 May 2017Received in revised form 5 September 2017Accepted 5 September 2017

Keywords:Compression twinningPhase transformationHCP metalsTitaniumMolecular dynamics

a b s t r a c t

Twinning plays an important role on the microstructure evolution of HCP metals. In this work, we per-form c-axis compression of single crystal Ti under uniaxial stress conditions using molecular dynamicssimulations. The objective is to demonstrate activation and evolution of compression twins along withdislocations when Ti single crystals are loaded in uniaxial stress conditions using two commonly utilizedinteratomic potentials. We find that activation of only f10 �11g compression twins with the Kim potentialis inconsistent with the Schmid criterion while activation of both f10 �11g and f11 �22g compressiontwins with the Hennig potential is consistent with the Schmid criterion. Twin variants activated donot contribute equally to the total twinned volume even with equal Schmid factor for both potentialsand thereby demonstrating inability of the Schmid criterion to predict dominance and evolution of twinvariants. The Kim potential shows large amount of dislocation density in comparison to the Hennigpotential. The strain rate sensitivity on activation of compression twin systems is observed with theHennig potential. The study can be utilized to further investigate the evolution of compression twinsrequired to develop/improve twin volume fraction evolution laws for crystal plasticity finite elementsimulations.

� 2017 Elsevier B.V. All rights reserved.

1. Introduction

Hexagonal-close-pack metal at ambient temperature and pres-sure conditions, Titanium (typically referred to as a-Ti) is used indifferent industries primarily because of its light weight, highstrength and stiffness apart from being bio-compatible and beingresistant to wear and corrosion. Due to its HCP crystal structure,it has a high propensity for deformation twinning compared tothe dislocation slip when subjected to c-axis loading. Deformationtwinning is an important plastic deformation mode in HCP metals[1,2] and plays an important role in texture evolution [3], strainhardening [4,5], ductility and fracture strength [6]. Based on thenature of the c-axis loading, the deformation twins are categorizedas tension twins (for c-axis extension) and compression twins (forc-axis contraction). The most commonly reported tension twinsare f10 �12g and f11�21g and the compression twins are {11 �22},{11 �24} and {10 �11} [6–8]. The {11 �22} compression twins aremost frequently observed at ambient temperature and pressure

conditions [9–11] whereas {11 �24} are reported to activate atlow temperatures [12]. The {10 �11} twins activate at elevated tem-peratures [9] and at room temperatures (have been observed inmicro-pillar compression experiments on single crystal titaniumalloy [11] and dynamic compression experiments on commerciallypure titanium [13]).

The twinning-induced plastic deformation in HCP metals suchas Mg and Ti are commonly investigated using Crystal plasticityfinite element (CPFE) simulations [5,14]. In these simulations, thetwin volume fraction is computed and tracked on each twin systemat each Gauss point. When the overall twin volume fraction at anyGauss point reaches a critical value, the volume associated withthat Gauss point is reoriented based on the dominant twin variantof a twin system. In this approach, it is assumed that the activationand evolution of twin systems and their variants follow the Schmidcriterion. In fact, the activation of twin systems in most of the casesis found to follow the Schmid criterion. However, ability of the Sch-mid factor to predict evolution and dominance of twin variants isstill not clear. This is primarily because the in situ measurementsof twin variant evolution are major challenges in experiments. Inthis regard, atomistic simulations can serve as a useful resolutionto investigate the evolution and dominance of twin variants under

Fig. 1. (a) C-axis compression under uniaxial strain conditions [16]. (b) C-axiscompression under uniaxial stress conditions.

20 S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29

the applied loading conditions [15]. Note that such information canbe useful to improve the twinning evolution laws for predictiveCPFE simulations.

Recently, Rawat et al. [16] compared the deformation behaviourof Ti single crystal under uniaxial strain condition using four com-monly utilized interatomic potentials: Finnis-Sinclair [17], embed-ded atom method (EAM) [18], 2NN modified embedded atommethod (MEAM) [19] and spline-based modified embedded atommethod [20] potentials and concluded that the Kim potential[19] should be considered for c-axis compressive uniaxial strainconditions. The superiority of the MEAM potentials over theembedded atom method and Finnis-Sinclair potentials is alsoestablished in literature since these potentials account for angulardependence of electron density to account for bond bending, havestacking fault energies (SFEs) (SFEKim = 213 mJ m�1 [19], SFEHen-nig = 170 mJ m�1 [20]) near the experimentally observed value(300 mJ m�1 [1]) and are also able to reproduce c/a ratio as wellas other properties simultaneously. Note that the SFE plays animportant role on the activation of preferred slip/twin systems aswell as dislocation mobility [21]. The EAM [18] and Finnis-Sinclair [17] potentials with very low stacking fault energies(SFEMishin = 56 mJ m�1 [18] and SFEAckland = 64 mJ m�1 [17]) maynot be appropriate to predict the deformation response of Ti asthey may not be able to reproduce the realistic dislocation corestructure [22–24]. Thereby only MEAM potentials are being con-sidered in this work to study the response of Ti single crystal underuniaxial stress type of loading situation. In the MEAM formalism,the pair interactions are computed as a function of nearest-neighbour distance from the known values of total energy andthe embedding function. The original 1NN (NN refers to nearest-neighbour) MEAM which considers only first nearest-neighbourdistance to compute the electron densities was found to have somestructural stability issues [25]. In the 2NN MEAM potential [19],the second nearest-neighbour interactions are considered to com-pute the electron densities to resolve the structural stability issues.The spline-based MEAM [20] considers the cubic spline functionsto compute the total energy of the system and the electron densitydescribed in this potential removes the constraint of fixed angularcharacter.

In this work, we simulate c-axis compression of perfect singlecrystal Ti at room temperature under uniaxial stress conditionusing two commonly utilized interatomic potentials [19,20] withan objective to investigate the deformation behaviour of singlecrystal Ti. It should be noted that the uniaxial stress conditionsare different from the uniaxial strain conditions [26] (highlightedin our previous publication [16]). The uniaxial stress conditionsare representative of loading conditions which occur in static com-pression experiments using Universal Testing Machine as well asdynamic compression experiments using Kolsky bar [27]; whereasuniaxial strain conditions are representative of shock loading situ-ations [26] in high velocity impact experiments [28] or by laserablation [29]. Fig. 1 shows a representative diagram for the twocases for ease of understanding of the readers.

For uniaxial stress conditions, the pressure along X- and Y-directions is kept at 0 bar and thereby the material can changeshape and size along those directions; whereas for uniaxial strainconditions the material is constrained to change shape and sizealong X- and Y-directions and thereby stresses can originate inthese directions. Note that since there are no experimental investi-gations at ultra high strain rates, as carried out in this manuscript,it is difficult to comment on suitability of one individual MEAMpotential over the other. In this manuscript, we present theresponse of single crystal Ti with both available MEAM potentialsunder uniaxial stress conditions and expect that the results wouldbe compared and validated in future by experimentalinvestigations.

2. Computational method

We use LAMMPS [30] to simulate the c-axis compression of per-fect single crystal titanium under uniaxial stress condition (rzz – 0,rxx = ryy = 0). The simulation domain contains 4.6 � 105 atoms andeach side length of the domain is 20 nm. The X-,Y- and Z-directionsof the simulation domain correspond to h2 �1 �10i, h01 �10i andh0001i, respectively. The periodic boundary conditions are usedin each direction and velocity-Verlet algorithm is used to integratethe equation of motion with a timestep of 2 femto-second. We usethe spline-based modified embedded atom method (MEAM)potential [20] (henceforth being referred to as the Hennig poten-tial) and the 2NN-MEAM potential [19] (henceforth being referredto as the Kim potential) to study the comparative deformationbehaviour of perfect single crystal titanium. To control the temper-ature and pressure, Nose-Hoover thermostat and barostat are used.The system is first equilibrated at 300 K and 0 bar pressure usingan NPT ensemble up to 40 ps and then a strain rate of 109 s�1 isapplied along the Z-axis (c-axis) to compress the system at a con-stant temperature of 300 K. The pressure along the X- and Y-directions is kept at 0 bar using an NPT ensemble thereby allowingthe material to change shape and size along these directions. Thesimulations at 109 s�1 strain rate are performed to save computa-tional time. To study the role of applied strain rate on the results,we also perform simulations at 108 s�1 strain rate. The atomic datais then post-processed using basal plane vector analysis [31] andcrystal analysis tool (CAT) [32]. The results are visualized in Ovito[33].

3. Results and discussion

3.1. Stress-time profile

Fig. 2 shows the axial stress as a function of time for single crys-tal titanium deformed at 109 s�1 strain rate with the Kim [19] andthe Hennig [20] potentials. Typically a sharp fall in stress indicatesstress relaxation due to defect nucleation in the material. Therebyit is observed that since the peak stress for the Kim potential [19] isvery high (�2 times) in comparison to that obtained with the Hen-nig potential, higher stress is required for the nucleation of defectswith the Kim potential compared to that with the Hennig potential.In fact, this type of behaviour is also expected since the Kim poten-tial has high stacking fault energy in comparison to the Hennigpotential.

Interestingly, the Kim potential shows only a single peak in thestress-time profile, whereas the Hennig potential shows doublepeaks in the stress-time profile (Fig. 2). The presence of doublepeaks with the Hennig potential indicates that the microstructuralevolution with the Hennig potential is quite different from that

Fig. 2. Axial stress as a function of time for single crystal titanium deformed at109 s�1 strain rate at 300 K temperature with the Kim and the Hennig potentials.

S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29 21

obtained with the Kim potential. To analyse the double peaks inthe stress-time profile with the Hennig potential, we perform (a)analysis for structural changes which gives information about theHCP structure and defects (stacking faults and unknown struc-tures) (b) structure factor analysis which also gives informationabout the nucleation of defects and structural changes (c) c-axismisorientation distribution which gives information about therotation of the c-axis. For comparison, we perform the same calcu-lations for the Kim potential as well.

3.1.1. Analysis for structural changesTo identify the structural changes due to deformation, we per-

form CAT analysis [32] which gives information about the HCPstructure (atoms belonging to parent HCP structure), stackingfaults (atoms belonging to stacking faults) and unknown structure(atoms with unknown coordination number) at any instant of time.Fig. 3 shows the snapshots for structural changes corresponding tothe time points in Fig. 2 for single crystal titanium deformed at109 s�1 strain rate with the Kim and the Hennig potentials.

As observed from Fig. 3 that the peak in stress-time profile(point C in Fig. 2) for the Kim potential occurs due to the creationof an unknown structure (snapshot C in Fig. 3). While for the Hen-nig potential, the drop in stress-time profile also occurs due to thecreation of an unknown structure (snapshot F in Fig. 3). However,the unknown structure leads to double peaks in stress-time profile.Note that the snapshot corresponding to point H in Fig. 3 shows thearrangement of some HCP atoms in a specific manner and the ori-entation of these atoms is quite different from the parent HCPatoms. This could be a result of the compression twins which canactivate due to the contraction of the c-axis under the applied load-ing conditions.

To understand the amount of unknown structure, stacking faultand HCP structure present at any instant of time, we count thenumber of atoms belonging to each structure type. Fig. 4 showsthe time evolution of number fraction corresponding to HCP struc-ture, stacking faults and unknown structure.

As observed from Fig. 4 that for both potentials, the numberfraction corresponding to the HCP structure decreases withincrease in deformation. This decrease in the HCP structure corre-sponds to the creation of an unknown structure and stacking faults.For the Kim potential, the creation of an unknown structure andstacking faults (which nucleate almost simultaneously) corre-sponds to the peak drop in stress-time profile (c.f. Fig. 2). For theHennig potential, the creation of an unknown structure corre-sponds to the first peak drop in the stress-time profile, while cre-

ation of stacking faults corresponds to the second peak drop inthe stress-time profile (c.f. Fig. 2). This means that the first peakin the stress-time profile could be due to lattice rotations andnot due to creation of defects.

3.1.2. Structure factorThe CAT analysis presented in previous section (Section 3.1.1)

reveals that the drop in stress occurs due to the creation of stackingfaults and unknown structure. However, it does not explain whyfirst peak in stress-time profile occurs for the Hennig potential?To confirm the structural changes obtained by CAT analysis andto explain the occurrence of first peak in stress-time profile withthe Hennig potential, we perform structure factor [S(k)=P

n exp ik:rn; k: reciprocal lattice vector, rn: atomic positions] cal-culations [34] for the Kim and the Hennig potentials. The structurefactor gives information about the structural changes and latticerotations.

Fig. 5 shows the structure factor as a function of time for theKim and the Hennig potentials.

For the Kim potential, the structure factor start decreasingrapidly at 164 ps and becomes zero after 167 ps indicating nucle-ation of defects in the system. While for the Hennig potential,the structure factor start decreasing at 50 ps but does not becomezero indicating no nucleation of defects and the system is still per-fect. The decrease in structure factor for (002) reflections corre-sponds to an increase in the structure factor for (201) reflectionsindicating elastic lattice rotations in the system. Note that theonset of decrease in structure factor corresponds to the first peakdrop in the stress-time profile (c.f. Fig. 2). This means that the firstpeak drop in the stress-time profile (Fig. 2) is due to the stressrelaxation resulting from the lattice rotations not due to the nucle-ation of defects. The rapid drop in structure factor starts at 85 psindicating the onset of plasticity in the system and becomes zeroafter 93 ps. This corresponds to the second peak in the stress-time profile (c.f. Fig. 2).

3.1.3. C-axis misorientationTo confirm further the occurrence of first peak in stress-time

profile (Fig. 2) with the Hennig potential due to lattice rotationsas well as to get the signature of possible activation of compressiontwins, we perform c-axis misorientation analysis. This analysisgives misorientation information between the initial and rotatedc-axes. Fig. 6 shows the distribution of c-axis misorientations forthe Kim and the Hennig potentials at different time points corre-sponding to elastic and plastic regimes of the stress-time profile(c.f. Fig. 2).

For the Kim potential, before the peak in the stress-time profile(i.e. in the elastic regime) (c.f. Fig. 2), the c-axis misorientations areless than 10�. While for the Hennig potential, the c-axis misorien-tations are less than 10� till 50 ps but after 50 ps, the misorienta-tions are increased (0 < h < 20�) indicating elastic lattice rotationsas suggested by the structure factor calculations (Section 3.1.2).For the Kim potential in the plastic regime, peaks occur in therange of 50–65� while for the Hennig potential in plastic regime,the peaks occur in the range of 55–75�. We know that the theoret-ical misorientations for {10 �11} h�1012 i and {11 �22} h�1 �123i twinsystems [8] are 57.2� and 64.4�, respectively. In addition to this,the misorientations induced by different variants of {11 �22}h�1 �123i twin system are 50.8�, 60� and 77.1� [35]. Therefore, thesepeaks are possibly due to the activation of compression twins. Notethat the peaks in the misorientation angle distribution for the Hen-nig potential correspond to the second peak in the stress-time pro-file (c.f. Fig. 2). This implies that the first peak in the stress-timeprofile does not correspond to the activation of compression twins.

Fig. 3. Snapshots for structural changes corresponding to the time points in Fig. 2 for single crystal Ti deformed at 109 s�1 strain rate with the Kim and the Hennig potentials.

22 S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29

From structural changes, structure factor and c-axis misorienta-tion analyses, it is clear that the first peak in the stress-time profileobtained with the Hennig potential is due to the stress relaxationresulting from the elastic lattice rotations while second peak inthe stress-time profile is due to the creation of defects.

3.2. Twin analysis

The snapshots in Fig. 3 (Section 3.1.1) show that some of theHCP atoms have different orientations from that of the parentHCP atoms indicating the possible activation of compression twins.In addition to this, the c-axis misorientation analysis presented in

previous section (Section 3.1.3) also gives the signature for theactivation of compression twins. Therefore, to confirm the activa-tion of compression twins, we perform the c-vector analysis. Forc-axis compression, all variants of {11 �22} h�1 �123i twin systemhave equal and highest (0.45) Schmid factor while all variants of{10 �11} h�1012 i twin system have equal but comparatively lowSchmid factor (0.42) (c.f. Appendix A). This suggests that if the acti-vation of compression twin systems follows the Schmid factor cri-terion, then the {11 �22} h�1 �123i twin system should activate firstin comparison to the {10 �11} h�1012 i twin system. In addition tothis, all six variants of these twin systems which have equal Sch-mid factor are expected to be activated.

Fig. 4. Time evolution of number fraction belonging to HCP structure, stackingfaults and unknown structure for the Kim and the Hennig potentials.

S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29 23

Fig. 7 shows the snapshots for the activated twin variants of{10 �11} h�1012 i twin system with the Kim potential and {10 �11}h�1012 i and {11 �22} h�1 �123i twin systems with the Hennig poten-tial. The color of a twin in Fig. 7 indicates the variant type of a twinsystem (Appendix A).

For the Kim potential, all six variants of only {10 �11} h�1012 itwin system activate while for the Hennig potential, all variantsof {10 �11} h�1012 i and {11 �22} h�1 �123i twin systems activate. Notethat {10 �11} h�1012 i twin system has low Schmid factor comparedto {11 �22} h�1 �123i twin system (c.f. Appendix A). This suggests thatactivation of twin systems for the Hennig potential follows theSchmid factor criterion, while it does not follow for the Kim poten-tial. However, activation of twin variants follows the Schmid factorcriterion for both the Kim and the Hennig potentials. Note also thatfor the Hennig potential, {11 �22} h�1 �123i twin system dominatesthe twinned volume. On an associated note, in the micro-pillarcompression experiments at room temperature [11], both{11 �22} h�1 �123i and {10 �11} h�1012i twin systems activate, but{11 �22} h�1 �123i twin system dominates over the {10 �11} h�1012 itwin system. Recently, Morrow et al. [36] reported the activationof all variants of {11 �22} h�1 �123i twin system at room temperaturefor single crystal Ti compressed along the c-axis at 103 s�1 strainrate using split-Hopkinson pressure bar. This suggests that activa-tion of twin systems following the Schmid criterion and dominanceof {11 �22} twinned volume for the Hennig potential is in goodagreement with that reported in micro-pillar and dynamic com-

Fig. 5. Structure factor as a function of time

pression experiments at room temperature [11,36]. Based onobservations from numerical simulations and correlations withexperimental observations at lower strain rates, it may be apparentthat the Hennig potential performs better than the Kim. However,it should also be noted that the numerical experiments carried outin this manuscript at ultra high strain rates in comparison to strainrates in experimental investigations [11,36] are yet to be validatedthrough experimental observations.

To get the quantitative information of the twinned volume, wecompute the total twin volume fraction and twin variant volumefraction for each activated twin system. The total twin volume frac-tion at any given instant of time is defined as the ratio of number oftwinned atoms (Ntw) at that time point to the total number ofatoms (N) in the simulation domain. The twin variant volume frac-tion at any given time point is defined as the ratio of number ofatoms belonging to bth twin variant (Nb) to the total number ofatoms in the simulation domain. Fig. 8 shows the time evolutionof the total twin volume fraction and twin variant volume fractionfor {10 �11} h�1012 i twin system with the Kim potential and for{10 �11} h�1012 i and {11 �22} h�1 �123i twin systems with the Hennigpotential.

As observed from Fig. 8 that for both the Kim and the Hennigpotentials, the total twin volume fraction once appeared increasesexponentially due to the rapid growth of the twins. After that, itincreases slowly due to the interaction of twins belonging to othervariants which obstruct the growth of the twins. It can also beobserved from Fig. 8 that the exponential regime of twin volumefraction is different for each twin variant (which can be clearlyseen for variants of {11 �22} twins). This gives an idea about thespatial distribution of nucleated twins belonging to different twinvariants. Note that the spatial distribution of nucleated twinsbelonging to different variants plays an important role on the dom-inance of a twin variant over the others. For the Hennig potential, aclear dominance of one variant over the others for both {10 �11}h�1012 i and {11 �22} h�1 �123i twin systems is observed, while thevariants of {10 �11} h�1012 i twin system compete with each otherfor the Kim potential (Fig. 8). The dominance of C26 variant of{11 �22} twins and C14 variant of {10 �11} twins for the Hennigpotential (Fig. 8) indicates that the twins belonging to C26 andC14 experience less influence due to other variants of {10 �11}h�1012 i and {11 �22} h�1 �123i twin systems.

Although all variants of {10 �11} h�1012 i twin system for theKim potential and all variants of {10 �11} h�1012i and {11 �22}h�1 �123i twin systems for the Hennig potential get activated, butthey do not contribute equally to the total twin volume fractionin either case even with equal Schmid factor (c.f. Appendix A). It

for the Kim and the Hennig potentials.

Fig. 6. Distribution of c-axis misorientation at different time points for the Kim and the Hennig potentials.

Fig. 7. Activated variants of {10 �11} h�1012 i twin system for the Kim potential (206 ps) and variants activated for {10 �11} h�1012 i and {11 �22} h�1 �123i twin systems for theHennig potential (102 ps). The color of a twin refers to the variant type of a twin system. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

24 S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29

is also interesting to note that for both {10 �11} h�1012 i and {11 �22}h�1 �123i twin systems activated with the Hennig potential, one ofthe twin variants dominates over the others even with equal Sch-mid factor for all variants. This suggests that the evolution of twinvariants and dominance of one variant over the others do notdepend on the Schmid factor. Recently, Rawat et al. [15] in theirmolecular dynamics simulations on the evolution of f10 �12g ten-sion twins in magnesium single crystals under multiaxial loadingconditions also observed that the activation of twin systems andtheir variants followed the Schmid factor criterion in all simulated

loading conditions but the evolution and dominance of twin vari-ants were not dictated by the Schmid factor criterion.

3.3. Dislocation analysis

To understand the activation of slip systems and evolution ofdislocation density, we perform dislocation analysis using Disloca-tion Extraction Algorithm (DXA) [32] implemented in Ovito [33].The DXA [32] divides the crystal into good and bad crystal regionsusing Common Neighbour Analysis [37]. The good crystal region

Fig. 8. Time evolution of total twin volume fraction and twin variant volume fraction for {10 �11} h�1012i twin system with the Kim potential and for {10 �11} h�1012 i and{11 �22} h�1 �123i twin systems with the Hennig potential.

S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29 25

refers to those parts of the crystal where atoms belong to a perfectlattice structure (may have slight deviations due to elastic distor-tions), stacking faults and coherent grain boundaries. While badcrystal region refers to those parts of the crystal where atoms donot belong to any perfect lattice structure as well as stacking faultsand coherent grain boundaries. The dislocation cores belong to abad crystal region. To extract the dislocation information, the algo-rithm uses the Delaunay tessellation where the edges of the tessel-lation (which define a set of elementary atom-to-atom steps) areused to construct the Burgers circuit. As it is not possible to con-struct all possible Burgers circuits to find the dislocations in a crys-tal, the algorithm extracts the information for a boundary surfaceseparating the good and bad regions. This boundary known asinterface mesh includes those triangular Delaunay facets whichhave a good tetrahedral element on one side and a bad elementon the other. Once interface meshes are identified, the algorithmconstructs all possible Burgers circuits on the interface meshes tofind out all dislocations.

Fig. 9 shows the time evolution of dislocation density for theKim and the Hennig potentials.

Burgers vector analysis shows that there are three types of dis-locations present in the system: (a) 1

3h1 �210i prismatic dislocations

(b) 13h1 �100i partial dislocations on the basal plane [38] and (c)

13h11 �23i pyramidal dislocations. In addition to these dislocations,some other dislocations which are not identifiable by DXA are alsopresent. For the Kim potential, 1

3h1 �100i partial dislocations and13h1 �210i prismatic dislocations show significant contribution tothe total dislocation density. In addition to these basal dislocations,13h11 �23i non-basal pyramidal dislocations also contribute to thetotal dislocation density. While for the Hennig potential, only13h1 �100i partial dislocations contribute to the total dislocation den-sity and prismatic and pyramidal dislocations do not contribute tothe total dislocation density. It is interesting to note that the Hen-nig potential which shows contribution of only partial dislocationsto the total dislocation density lead to the activation of {10 �11}h�1012 i and {11 �22} h�1 �123i twin systems, while only {10 �11}h�1012 i twin system activate for the Kim potential for which pris-

matic and pyramidal dislocations also contribute to the total dislo-cation density in addition to the partial dislocations. Note that thepyramidal hc þ ai dislocations can accommodate the c-axis defor-mation in addition to the compression twins. This means that theactivation of {11 �22} h�1 �123i twin system is suppressed by theactivation of pyramidal dislocations for the Kim potential. In addi-tion to this, the evolution trend of the total dislocation density isquite similar to that for the evolution of total twin volume fraction(c.f. Fig. 8) indicating a correlation between the creation of twinsand dislocation density. The Kim potential which shows lesstwinned volume with activation of only one twin system (Fig. 8)shows highest total dislocation density compared to that obtainedwith the Hennig potential which shows large twinned volume withactivation of two twin systems. For both potentials, the other dis-locations dominates over the partial dislocations.

3.4. Effect of strain rate on the results

So far, we have discussed the results obtained at 109 s�1 strainrate. To see the role of applied strain rate on these results (yieldpoint, twin evolution and dislocation density), we perform addi-tional simulations at 108 s�1 strain rate with both the Kim andthe Hennig potentials.

3.4.1. Stress-strain responseFig. 10 shows the comparison of stress generated as a function

of applied strain at 109 s�1 and 108 s�1 strain rates for the Kim andthe Hennig potentials.

For both the Kim and the Hennig potentials, the overall stress at108 s�1 strain rate follows the same trend as obtained at 109 s�1

strain rate. However, quantitatively there are some differences.For the Kim potential, the yield stress is almost same at both strainrates but the strain corresponding to the yield point is lower for

108 s�1 strain rate (De109�108y = 0.0034). While for the Hennig poten-

tial, the strain corresponding to the yield point is almost same butthe yield stress is lower for 108 s�1 strain rate

(Dr109�108y = 0.15 GPa). For both potentials, the rapid drop in stress

Fig. 9. The evolution of dislocation density for the Kim and the Hennig potentials.

Fig. 10. Comparison of stress developed as a function of applied strain at 109 s�1 and 108 s�1 strain rates for the Kim and the Hennig potentials.

26 S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29

occurs at 108 s�1 strain rate in comparison to that at 109 s�1 strainrate.

3.4.2. Twin volume fractionFig. 11 shows the comparison of total twin volume fraction and

twin variant volume fraction obtained at 109 s�1 and 108 s�1 strainrates as a function of applied strain with the Kim and the Hennigpotentials.

For the Kim potential, all variants of {10 �11} twin activate andthe total {10 �11} twin volume fraction is larger at 108 s�1 strainrate compared to that obtained at 109 s�1 strain rate. In addition

to the differences in total twin volume fraction at both strain rates,there are also some differences at twin variant level. For 108 s�1

strain rate, two variants of {10 �11} twins dominate over the othersin comparison to 109 s�1 strain rate where all six variants competeeach other and no clear dominance of a variant is observed. For theHennig potential in Fig. 11, no {10 �11} twins activate at 108 s�1

strain rate in comparison to 109 s�1 strain rate where they activateand contribute significantly to the overall twinned volume. This isconsistent with the experimental observations where {10 �11}twins are not commonly observed at ambient temperature exceptfew experiments such as [11] (where experiments have been car-

Fig. 11. Comparison of total twin volume fraction and twin variant volume fraction evolutions at 109 s�1 and 108 s�1 strain rates as a function of applied strain for {10 �11}h�1012 i twin system with the Kim potential and {10 �11} h�1012 i and {11 �22} h�1 �123i twin systems with the Hennig potential.

S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29 27

ried out at lower strain rates). All variants of {11 �22} twins activateat 108 s�1 strain rate which is consistent with 109 s�1 strain rate.However, twin variant level dynamics is quite different at 108 s�1

strain rate. In Fig. 11, the twin variant (C26) which is the most dom-inant at 109 s�1 contributes negligibly to the total twin volumefraction at 108 s�1 strain rate. In addition to this, twin variants(C21,C22 and C24) which have very small contribution to the totaltwin volume fraction at 109 s�1 strain rate are the ones which dom-inate at 108 s�1 strain rate. Moreover, the total {11 �22} twin vol-ume fraction at 108 s�1 strain rate is large compared to thatobtained at 109 s�1 strain rate.

Fig. 12. Comparison of dislocation densities at 109 s�1 and 10

3.4.3. Dislocation densityFig. 12 shows the comparison of dislocation densities obtained

at 109 s�1 and 108 s�1 strain rates as a function of applied strainwith the Kim and the Hennig potentials.

The Kim potential shows slightly high dislocation density at108 s�1 strain rate compared to that at 109 s�1 strain rates. The13h11 �23i pyramidal dislocations have small contribution to thetotal dislocation density at 108 s�1 strain rate compared to thatat 109 s�1 strain rate. In addition to this, 1

3h1 �100i partial disloca-tions and other dislocations contribute more to the total disloca-tion density at 108 s�1 strain rate compared to that at 109 s�1

8 s�1 strain rates with the Kim and the Hennig potentials.

28 S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29

strain rate. For the Hennig potential, the overall dislocation densityis lower at 108 s�1 strain rate compared to that at 109 s�1 strainrate. At both strain rates, only 1

3h1 �100i partial dislocations andother dislocations have significant contribution to the total disloca-tion density.

4. Discussion

We simulate the c-axis compression of perfect single crystaltitanium under uniaxial stress condition at room temperature withtwo inter-atomic potentials at 109 s�1 strain rate. The objective isto understand the deformation behaviour of a single crystal tita-nium subjected to c-axis uniaxial compressive stress loading situ-ation with respect to the potential used for the study. Thedeformation behaviour is characterized in terms of the stress-time response, structural evolution, activation of compression twinsystems and the twin variants of the activated twin systems, twinvolume fraction, activation of dislocation slip and dislocation den-sity. To check the sensitivity of the results obtained at 109 s�1

strain rate, we also perform simulations at 108 s�1 strain rate bothwith the Kim and the Hennig potentials.

For the Kim potential, yielding at 108 s�1 strain rate occurs atlower strain compared to that at 109 s�1 strain rate while yieldstress is almost same at both strain rates. For the Hennig potential,yielding at 108 s�1 strain rate occurs at lower stress compared tothat at 109 s�1 strain rate but the strain corresponding to yield isalmost same. All variants of only {10 �11} h�1012 i twin system acti-vate both at 109 s�1 and 108 s�1 strain rates. This means that theapplied strain rate does not affect the activation of twin systemsand twin variants for the Kim potential. The activation of twin sys-tem at both strain rates does not follow the Schmid factor criterionbut activation of twin variants follows. The applied strain rateaffects the evolution and dominance of twin variants. For deforma-tion at 108 s�1 strain rate, a clear dominance of twin variantsoccurs over the others which is not observed at 109 s�1 strain rate.For the Hennig potential, all variants of {10 �11} h�1012 i and{11 �22} h�1 �123i twin systems activate at 109 s�1 strain rate andall variants of only {11 �22} h�1 �123i twin system activate at108 s�1 strain rate. This indicates that the applied strain rate affectsthe activation of twin systems but not the twin variants for theHennig potential. The activation of twin systems and twin variantsfollow the Schmid factor criterion both at 109 s�1 and 108 s�1 strainrates. Note that the activation of {10 �11} h�1012 i twin system isnot commonly reported in experiments at ambient temperatureexcept the few experiments such as [11]. The variants activateddo not contribute equally to the total twin volume fraction evenwith equal Schmid factor suggesting that the evolution and domi-nance of twin variants are not dictated by the Schmid factor crite-rion. The twin variants which contribute most to the total twinvolume fraction at 109 s�1 strain rate are not the same at 108 s�1

strain rate. The large amount of dislocation density is observedfor the Kim in comparison to the Hennig potential both at109 s�1 and 108 s�1 strain rates. The slip systems which activateat 109 s�1 strain rate also activate at 108 s�1 strain rate for theKim as well as the Hennig potentials. However, the dislocationdensity is not same at both strain rates. For the Kim potential,the total dislocation density at 108 s�1 strain rate is slightly highcompared to that at 109 s�1 strain rate. While for the Hennigpotential, it is significantly low at 108 s�1 strain rate compared tothat at 109 s�1 strain rate. The presence of hc þ ai pyramidal dislo-cations for the Kim potential demonstrates that the slip mecha-nism accommodates the c-axis deformation rather than thetwinning mechanism (note 11�22h�1 �123i twin system does not acti-vate for the Kim potential). However, the presence of compressiontwins along with high dislocation densities for both the potentials

suggest that the c-axis deformation is accommodated by the com-bined action of twinning and slip for the Kim and the Hennigpotentials.

In order to comment on which potential performs better incomparison to the other, experimental investigations at ultra-high strain rates should be carried out under similar uniaxial stressconditions (as has been carried out in these simulations). It may benoted that apparently there are some studies in literature [9,36](under uniaxial stress conditions) which demonstrate formationof f11 �22g twins only. The Hennig potential demonstrates forma-tion of f11 �22g twins at a strain rate of 108 s�1 strain rate but bothf10 �11g and f11 �22g twins at a strain rate of 109 s�1. This is in con-tradiction to the Kim potential which demonstrates f10 �11g twinsfor both 109 s�1 and 108 s�1 strain rates. As per the Schmid crite-rion also, the formation of f11 �22g twins is favored compared tothat of f10 �11g twins. However, it should also be noted that priorto making a statement regarding the Hennig is better than the Kimfor this type of loading situation, the experimental investigationsrevealing this type of behaviour are based on the experiments car-ried out at maximum strain rates of 103 s�1 [9,36] which are signif-icantly lower than what is reported in this manuscript and also thetemperature rise in those experiments is not same as what we havein these simulations. Thereby without experimental investigationsat similar strain rates (as carried out in this manuscript), it is diffi-cult to comment on which potential is better for the simulationunder uniaxial stress conditions.

5. Summary and conclusions

In summary, we simulate the c-axis compression of single crys-tal Ti under uniaxial stress conditions using two inter-atomicpotentials to characterize the response of single crystal titaniumin terms of the compression twins and dislocation activity. Forthe Kim potential, only {10 �11} h�1012 i twin system activates bothat 109 s�1 and 108 s�1 strain rates, while for the Hennig potentialboth {10 �11} h�1012 i and {11 �22} h�1 �123i twin systems activateat 109 s�1 strain rate and only {11 �22} h�1 �123i twin system acti-vates at 108 s�1 strain rate. For the Kim potential, the applied strainrate affects the evolution of twin variants and dislocations density.However, it does not affect the activation of twin systems and twinvariants. While for the Hennig potential, it not only affects the acti-vation of twin systems but also affects significantly the dislocationdensity as well as the twin variant evolution. The main conclusionsof this study are as follows:

� The activation of twin systems and their variants with the Hen-nig potential is consistent with the Schmid factor while activa-tion of twin systems with the Kim potential does not follow theSchmid criterion.

� The twin variants activated do not contribute equally to thetotal twin volume fraction even with equal Schmid factor forboth potentials indicating that the Schmid factor cannot dictatethe evolution and dominance of twin variants.

� The presence of compression twins and high dislocation densityfor the Kim and the Hennig potentials suggest that both twin-ning and dislocation slip accommodate the c-axis deformation.

� The presence of hc + ai dislocations for the Kim potential sug-gests that the activation of {11 �22} h�1 �123i twin system is sup-pressed by the activation of pyramidal dislocations toaccommodate the c-axis compression.

The foregoing observations can be useful to improve the twin-ning evolution laws in crystal plasticity finite element simulations.However, the conclusions drawn in this study are limited to this

Table A-1Schmid factor and misorientation angle for compression twin systems.

Twin type Twin variants Schmid factor C-axis misorientation(h0001i) (degree)

f10 �11g C11 f10 �11g h�1012i 0.42 57.22

C12 f�1011g h10 �12i 0.42

C13 f01 �11g h0 �112i 0.42

C14 f0 �111g h01 �12i 0.42

C15 f�1101g h1 �102i 0.42

C16 f1 �101g h�1102i 0.42

f11 �22g C21 f11 �22g h11 �2 �3i 0.45 64.40

C22 f�1 �122g h�1 �12 �3i 0.45

C23 f�12 �12g h�12 �1 �3i 0.45

C24 f1 �212g h1 �21 �3i 0.45

C25 f�2112g h�211 �3i 0.45

C26 f2 �1 �12g h2 �1 �1 �3i 0.45

S. Rawat, N. Mitra / Computational Materials Science 141 (2018) 19–29 29

particular type of loading condition. To gain further understandingon twinning evolution and compare the response of single crystalTi with the potentials, the simulations should be performed underdifferent loading conditions. In order to comment on the suitabilityof one force potential over the other, experimental investigationsare needed by the research community.

This manuscript focuses on the activation and evolution of com-pression twins which is important to understand the evolution anddominance of twin variants under some particular loading condi-tion and does not investigate the nucleation mechanism of com-pression twins. It would be interesting to explore the nucleationmechanism of compression twins with these potentials under dif-ferent loading conditions.

Acknowledgments

We acknowledge the Centre for Theoretical Studies for the useof computational facility where simulations have been carried out.

Appendix A. Schmid factor for compression twins

Table A-1 shows the possible six twin variants of {10 �11}h�1012 i and {11 �22} h�1 �123i compression twin systems.

To distinguish the twin variants from each other for two twinsystems, each variant is represented by Cij, where ‘i’ refers to twinsystem type and ‘j’ refers to variant type of that twin system. TheSchmid factor calculations have been performed for the loadingdirection and are tabulated in Table A-1. The c-axis misorientationsfor two twin systems are also presented in Table A-1.

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