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Dynamic behaviour of nano-sized voids in hexagonal close-packed materials Grégoire, Claire Marie 2018

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Dynamic behaviour of nano-sized voids in hexagonalclose-packed materialsbyClaire Marie Gre´goireM.Sc., Collegium Sciences et Techniques a` l’Universite´ d’Orle´ans, 2015A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)The University of British Columbia(Vancouver)April 2018c© Claire Marie Gre´goire 2018AbstractThe dynamic behaviour and failure mechanisms of nano-sized voids in single crystals is studiedfor three hexagonal close-packed materials by means of molecular dynamics simulations. Ourstudy reveals that in Magnesium the response is highly anisotropic leading to a brittle to ductiletransition in the failure modes under different load orientations. This transition is accompa-nied by different mechanisms of deformation and is associated with the anisotropic HCP latticestructure of Mg and the associated barrier for dislocation motion. Remarkably, brittle failure isobserved when external loads produce a high stress triaxiality while the response is more ductilewhen the stress triaxiality decreases.On the other hand, the failure in other two hexagonal close-packed materials studied inthis work, i.e, Titanium and Zirconium, is more ductile, in high contrast with the brittle failureobserved in Magnesium. We find that this difference is due to the fact that nano-sized voidsin Titanium and Zirconium emit substantially more dislocations than Magnesium, allowing forlarge displacements of the atoms and plastic work, including non-basal planes. Based on ourfindings, we postulate that this brittle failure in Magnesium is due to a competition betweendislocations emission in the basal plane and crack propagation in non-basal planes. Thus, wepropose to use the ratio between unstable stacking fault and surface energy in these materials toassess the tendency of hexagonal close-packed materials and alloys to fail under brittle or ductilemodes. Using this ratio, we critically identify the low surface energy of Mg as responsiblefor this brittle behaviour and recommend that Mg-based alloys with large surface energies canlead to better performance for dynamic applications. The fundamental mechanisms observed,therefore, explain the low spall strength of Mg and suggest the possibility of manipulating somemechanisms to increase ductility and spall strength of new lightweight Mg alloys.iiLay SummaryThis thesis presents the understanding of a metal called Magnesium (Mg) inspired by impactproblems. Lightweight engineering applied in the transportation sector is known to decreasepollutant emissions and improve fuel efficiency by reducing the weight of cars. Hence, Mgis a good candidate because of its low density however poor mechanical properties make itsusceptible to catastrophic failure in a brittle mode, which ultimately hinder the widespreadindustrial use of this material. This is due to its complex structure, but the comparison withtwo transition metals Titanium (Ti) and Zirconium (Zr), having this same structure, shows thata fundamental study of Mg could bring more insight to explain its dramatic failure. We haveobserved, using atomistic simulations, that a competition between two mechanisms has to betaken into account. We found out that the surface energy associated with the crack propagationwithin the specimen has a central role in this process. The data provided in this work can be usedto generate impact models and to design better Mg-based alloys developing superior dynamicsproperties.iiiPrefaceThe research presented in this Master of Applied Science thesis is an original work of the authorunder the supervision of Professor Mauricio Ponga. We gratefully acknowledge the financialsupport from the Natural Sciences and Engineering Research Council of Canada (NSERC) andthe support of Compute Canada through the Westgrid consortium and the Argonne LeadershipComputing Facility for giving access to their supercomputers.Chapter 2 was published in Acta Materialia. Claire Gre´goire, Mauricio Ponga (2017)Nanovoid failure in Magnesium under dynamic loads, Acta Materialia, 133, 1-15. https://doi.org/10.1016/j.actamat.2017.05.016.ivContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Hexagonal Close Packed lattice structure . . . . . . . . . . . . . . . . . . . . . 51.4 Defects in HCP materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.1 Miller Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Dislocations lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.3 Stacking fault defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.4 Slip systems in HCP lattice structure . . . . . . . . . . . . . . . . . . . 92 Nano-void failure in Magnesium under dynamic loads . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Nano-void simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Dislocation simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Frank-Read mechanism simulations . . . . . . . . . . . . . . . . . . . 182.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20v2.3.1 Dislocation barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Case 1: Hydrostatic tensile load . . . . . . . . . . . . . . . . . . . . . 202.3.2.1 Void size effect . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2.2 Cell size effect . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2.3 Temperature effect . . . . . . . . . . . . . . . . . . . . . . . 242.3.2.4 Ensemble choice . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2.5 Assessment of the interatomic potentials for the loading case 1 282.3.3 Loading case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.4 Loading case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.5 Loading case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.6 Loading case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Dynamic behaviour of nano-voids in Titanium and Zirconium . . . . . . . . . . 483.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.1 Nano-void simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.2 Dislocation simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.3 Stacking fault and surface calculations . . . . . . . . . . . . . . . . . . 523.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.1 Dislocation barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.2 Stacking fault energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.3 Nano-void simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.3.1 Stress evolution . . . . . . . . . . . . . . . . . . . . . . . . 553.3.3.2 Void evolution . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.3.3 Dislocation emission . . . . . . . . . . . . . . . . . . . . . 593.3.3.4 Strain rate effect . . . . . . . . . . . . . . . . . . . . . . . . 603.3.3.5 Void size effect and shape effects . . . . . . . . . . . . . . . 633.3.3.6 Temperature effect . . . . . . . . . . . . . . . . . . . . . . . 633.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4.1 Analytical two dimensional model . . . . . . . . . . . . . . . . . . . . 673.4.2 Two dimensional simulations . . . . . . . . . . . . . . . . . . . . . . . 693.4.3 Ratio between unstable stacking fault vs. surface energies . . . . . . . 714 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A LAMMPS input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82viList of tables2.1 Selected values of n for different loading cases. . . . . . . . . . . . . . . . . . 162.2 Values of the critical resolve shear stress needed to promote dislocation glidein different slip systems of relevance for HCP-Mg at T = 300 K. All reportedvalues are in MPa and were obtained with the EAM potential at a strain rate ofε˙ = 108 s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Peak stress (σp) and corresponding peak strain (εp) for different strain rates(ε˙) obtained with the loading case 1 using a Mg single crystal of dimensions38× 68× 68 nm3 with approximately 7 million atoms containing a nano-voidof radius R= 3.15 nm at T = 300 K for the EAM and MEAM potentials. Valuesbetween () correspond to the MEAM potential. . . . . . . . . . . . . . . . . . 222.4 Evolution of the peak stress with the cell size for the loading case 1 at a strainrate of ε˙ = 107 s−1 obtained with the EAM potential using a Mg single crystalcontaining one nano-sized void of radius R = 3.15 nm. . . . . . . . . . . . . . 262.5 Summary of most relevant data obtained in this work for different loading cases.σp is the averaged peak stress in GPa, εp are the values of deformation for thepeak stress. The strain values are reported in %. Dislocation density valuescorrespond to the maximum value during the whole simulation in [×1016 1/m2].Dislocation velocities reported in m/s. All values reported in this table wereobtained from simulations carried out at ε˙ = 108 s−1 with the EAM potential.Values in ( ) correspond to the MEAM potential. . . . . . . . . . . . . . . . . . 443.1 Values of the critical resolve shear stress needed to promote dislocation glidein different slip systems of relevance for Ti and Zr at T = 300 K. All reportedvalues are in MPa at a strain rate of ε˙ = 108 s−1 and compared to Mg obtainedfrom Gre´goire and Ponga work [1]. . . . . . . . . . . . . . . . . . . . . . . . . 53vii3.2 Summary of most relevant data obtained in this work for Mg, Ti, and Zr, wherethe parameters a0 and c0 are shown (in A˙). σp is the averaged peak stress (GPa),εp is the value of the deformation at the peak stress ([%]). σc is the criticalstress for the dislocation emission (GPa), εc is the value of the deformation atthe critical stress ([%]). Dislocation density values ρ correspond to the maxi-mum value during the whole simulation (×1016 m−2). Velocities for basal, vb,and pyramidal-II dislocations, vP−II , are reported in m·s−1. The stable and un-stable stacking fault energies γs(0001) and γu(0001), respectively, are measured in thebasal plane. The surface energy denoted γ(1122) is performed in the pyramidal-IIplane. All the energy are presented in mJ·m−2. All values reported in this tablewere obtained from simulations carried out at ε˙ = 108 s−1 with EAM potentials. 553.3 Values of the dimensionless ratio ξ between surface energy and unstable stack-ing fault energy for Mg, Ti and Zr. SFE and surface energy values reported in[mJ·m−2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71viiiList of figures1.1 Evolution of engineering materials with non linear timescale [2]. ”Relative im-portance” is emphasized where the rate of change is far faster today than at anyprevious time in history. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Evolution of the mine production per year for Al, Mg, Ti and Zr in thousandmetric tons. The data includes the metal and compounds used in several indus-tries and was compiled from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 HCP lattice and its unit vectors. a) Arrangement of the atoms in the HCP latticestructure and its lattice vectors e1, e2 and e3. The unit cell of the HCP latticestructure is highlighted in red. b) HCP structure and three crystallographic di-rections in the basal plane (a1, a2, a3) and one in the c−direction. Numbersbetween braces indicate the Miller indices. . . . . . . . . . . . . . . . . . . . . 61.4 Burgers circuit shown in a schematic crystal a) with and b) without a disloca-tion. In this example, the Burgers vector is shown with a red arrow. The Burgersvector is perpendicular to the dislocation line, which in this case is in the direc-tion out of the paper, indicating that the dislocation in the example is an edgedislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Screw dislocations in HCP-Mg in the centre of a single crystal. The atoms arecoloured according to the common neighbour analysis. Atoms in red are in HCPstructure, green indicates FCC structure and white other type of structure. Thescrew dislocation is dissociated into two partials (white) separated by a stackingfault defect (green) a) Perspective view of the dislocation. b) Side view of thedislocation showing the slip plane and the direction of the Burgers vector. . . . 81.6 Stacking fault in HCP lattice a) Schematic picture of the HCP structure with afaulted plane. It is evident from the figure that the ...ABAB... ordered sequenceof planes is broken and a defect appears. Such defect is called stacking fault.b) View of the previous screw dislocation (see Fig. 1.5(a)) removing all atomsin the HCP structure (only the defect is shown). We see that the atoms in thestacking fault in green, with the two partials in white. . . . . . . . . . . . . . . 91.7 Representation of the slip systems in the HCP lattice structure (a) basal, (b)prismatic, (c) pyramidal-I, and (d) pyramidal-II slip systems. . . . . . . . . . . 11ix1.8 Illustration of the two most common twin systems in the HCP lattice structure.a) compressive and b) tensile twin systems usually activated in HCP materialsto generate deformation along the [0001] direction. . . . . . . . . . . . . . . . 122.1 Schematic picture of the set up used to study the Frank-Read source in Mg.In the simulations, two cylindrical holes of radius R separated by a distance Lare generated. Between these two cylinders a prismatic 〈a〉 dislocation is alsointroduced. Then, a shear deformation is applied on the plane whose normalis [1100] along the [1120] direction. The maximum values of τyx are reportedas the critical stress to bow out and completely detach the dislocation from theobstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Evolution of the mean hydrostatic stress (σm) and Von Mises equivalent stress(σe) vs. strain at different strain rates for the loading case 1 using a Mg singlecrystal of dimensions 38× 68× 68 nm3 with approximately 7 million atomscontaining a nano-void of radius R= 3.15 nm at T = 300 K for the EAM potential. 212.3 Evolution of the dislocation density ρ vs. strain for the loading case 1 (hy-drostatic tensile load) at different strain rates using a Mg single crystal of di-mensions 38× 68× 68 nm3 with approximately 7 million atoms containing anano-void of radius R= 3.15 nm at T = 300 K. All results are obtained with theEAM potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 View of the sample towards the end of the simulation after cracks (labeled withletter C) have been propagated. Twins (labeled with letter T) have been emittedfrom the tip of the cracks (labeled with letter CT). The original or parent crystalis indicated by the letters PC. . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Evolution of the peak stress with the nano-void size for the loading case 1 ata strain rate of ε˙ = 107 s−1 obtained with the EAM potential using a Mg sin-gle crystal of dimension 38×68×68 nm3 with approximately 7 million atomscontaining one nano-sized void. Spherical nano-voids with radius between R =0-30a0 were studied. Predictions made with the model predicted by Lubarda etal. (Eqn. 2.6) with ω = 1 and ω = 2 . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Evolution of the peak stress with the temperature for the loading case 1 at a strainrate of ε˙ = 107 s−1 obtained with the EAM potential using a Mg single crystalof dimension 38×68×68 nm3 with approximately 7 million atoms containingone nano-sized void of radius R = 3.15 nm. . . . . . . . . . . . . . . . . . . . 262.7 a) Evolution of the mean hydrostatic stress (σm) vs. strain for the loading case1 obtained with EAM potential under the NVE and NVT ensembles using a Mgsingle crystal of dimensions 38× 68× 68 nm3 containing one nano-sized voidof radius R = 3.15 nm at T = 300 K. Solid lines correspond to a load applied atε˙ = 107 s−1 and dashed lines to a ε˙ = 108 s−1. b) Temperature evolution for theNVE ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27x2.8 Evolution of the virial stress vs. strain for the loading case 1 obtained with theEAM and MEAM potentials using a Mg single crystal of dimensions 38×68×68 nm3 with approximately 7 million atoms containing one nano-sized void ofradius R = 3.15 nm at T = 300 K. Solid lines correspond to a load applied atε˙ = 107 s−1 and dashed lines to a ε˙ = 108 s−1. . . . . . . . . . . . . . . . . . . 282.9 a) Evolution of the averaged virial stress (σxx+σzz2 ) and Von Mises equivalentstress vs. strain for the loading case 2 at a strain rate of ε˙ = 108 s−1 obtained withthe EAM potential using a Mg single crystal of dimensions 38× 68× 68 nm3with approximately 7 million atoms containing a nano-void of radius R = 3.15nm at T = 300 K. b) Evolution of the dislocation density. . . . . . . . . . . . . 302.10 Dislocation emission for the loading case 2 obtained with the EAM potential. (a)Dislocations at ε = 3.45%. Basal edge dislocations are observed (labeled withletter B). (b) Dislocations at ε = 3.6%. Pyramidal-II dislocations are emittedfrom the tip of basal edge dislocations (labeled with letter P-II). Dislocationsat (c) ε = 3.75% and (d) ε = 3.825%. Pyramidal-II dislocations move fasterthan basal and prismatic dislocations. After the peak stress is reached, multiplecracks are propagated along the pyramidal-II plane leading to a drop in the virialstress vs. strain plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.11 a) Evolution of the averaged virial stress (σxx+σyy2 ) and the Von Mises equivalentstress vs. strain for the loading case 3 at a strain rate of ε˙ = 108 s−1 obtained withthe EAM potential using a Mg single crystal of dimensions 38× 68× 68 nm3with approximately 7 million atoms containing a nano-void of radius R = 3.15nm at T = 300 K. b) Evolution of the dislocation density. . . . . . . . . . . . . 332.12 Emission of PDL from the void surface for the loading case 3 at a strain rateof ε˙ = 108 s−1 obtained with the EAM potential using a Mg single crystal ofdimensions 38×68×68 nm3 with approximately 7 million atoms containing anano-void of radius R = 3.15 nm at T = 300 K. a) ε = 3.375%, basal (labeledwith letter B) and prismatic (labeled with letter P) dislocations are emitted. b)ε = 3.45%, dislocations partially detached from the void surface. c) ε = 3.6%,first PDL completely detached from the void surface and d) ε = 3.75%, PDLsemitted in [21 10] directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.13 Square shape PDLs emitted from the void surface along the a) [2110] and b)[1210] directions. The PDLs have a Burgers vector of magnitude 〈a〉 and aregenerated with two partials (1/3[1010] + 1/3[1100]) separated by a stackingfault in the basal plane and two prismatic dislocations {1}/3[2−1−10](1010). 352.14 a) Evolution of the averaged virial stress (σxx) and the Von Mises equivalentstress vs. strain for the loading case 4 at a strain rate of ε˙ = 108 s−1 obtained withthe EAM potential using a Mg single crystal of dimensions 38× 68× 68 nm3with approximately 7 million atoms containing a nano-void of radius R = 3.15nm at T = 300 K. b) Evolution of the dislocation density. . . . . . . . . . . . . 37xi2.15 Dislocation emission from the void surface for the loading case 4 at a strain rateof ε˙ = 108 s−1 obtained with the EAM potential using a Mg single crystal ofdimensions 38×68×68 nm3 with approximately 7 million atoms containing anano-void of radius R = 3.15 nm at T = 300 K. Dislocations emitted from thevoid surface at a) ε = 5.125%, prismatic dislocation loop is generated by screw(labeled with letter S) and prismatic edge (labeled with letter P) segments, b)Loop at ε = 5.13% . The loop is bowed out by the Frank-Read mechanismincreasing its size multiple times. . . . . . . . . . . . . . . . . . . . . . . . . . 382.16 Dislocation emission from the void surface for the loading case 4 at a strainrate of ε˙ = 108 s−1 obtained with the EAM potential at range of strain of ε =5.25−6.125% using a Mg single crystal of dimensions 38×68×68 nm3 withapproximately 7 million atoms containing a nano-void of radius R = 3.15 nmat T = 300 K. The observed mechanism rapidly increases dislocations densitymaking a strain hardening indicated by the plateau observed in the equivalentVon Mises stress after the peak stress is reached. . . . . . . . . . . . . . . . . . 392.17 Frank-Read source mechanisms in Mg for a basal prismatic dislocation a0[1010]{0001}.Different snap shots of the dislocation bowing out from the obstacles (voids)showing a very similar behaviour than the one observed in void simulationsunder the loading case 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.18 Critical resolved shear stress for bowing out a prismatic 〈a〉 dislocation from thetwo cylinders using in the set up. The symbols correspond to MD simulationscarried out at T= 300 K and a strain rate of ε˙ = 108 s−1 with the EAM potential.R/b = 0, 4, 8 correspond to different radius selected to study the Frank-Readsource. The lines correspond to Eq. 2.7 for different values of R and L. . . . . 412.19 a) Evolution of the virial stress σzz and Von Mises equivalent stress vs. strain forthe loading case 5 at a strain rate of ε˙ = 108 s−1 obtained with the EAM potentialusing a Mg single crystal of dimensions 38× 68× 68 nm3 with approximately7 million atoms containing a nano-void of radius R = 3.15 nm at T = 300 K. b)Evolution of the dislocation density. . . . . . . . . . . . . . . . . . . . . . . . 422.20 Dislocation emission from the void surface for the loading case 5 at a strainrate of ε˙ = 108 s−1 obtained with the EAM potential at range of strain ofε = 4.2−4.35% using a Mg single crystal of dimensions 38×68×68 nm3 withapproximately 7 million atoms containing a nano-void of radius R = 3.15 nm atT = 300 K. a) Dislocation emission at ε = 4.2%, basal edge and pyramidal-IIdislocations are shown. b) Dislocation emission at ε = 4.275%, pyramidal-IIdislocations travel very fast through the sample reaching the end of the sim-ulation cell. c) Dislocation emission at ε = 4.35%, pyramidal-II dislocationsinteract with their periodic replica promoting strain hardening in Fig. 2.19. . . . 43xii2.21 Dislocation density vs. normalized strain λ = εε f for all loading cases studied inthis work. The stress triaxiality factor χi corresponds to the linear regime for thespecific loading cases. The dashed line indicates a transition in the behaviour ofthe material from brittle to ductile. . . . . . . . . . . . . . . . . . . . . . . . . 463.1 Schematic illustration of the shear deformation for the dislocation simulations.The picture shows the dislocation core, illustrated with a T, and two-skin layersused to apply the shear deformation through an imposed constant velocity. Theresult is a motion of the atoms that simulates a shear deformation. . . . . . . . 523.2 SFE computed in the basal plane, where the displacement deformation is per-formed on [1010]−direction for the HCP lattice structure. The square markershows the stable SFE (γs[0001]) with numerical value of 30.6, 7.0, and 36.5 mJ·m−2for Mg, Ti, and Zr, respectively. The circle marker represents the unstable faultenergy γu[0001] with numerical values of 46.5, 31.9, and 60.3 mJ·m−2 for Mg, Ti,and Zr, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 a) Evolution of the mean hydrostatic stress (solid line) and the void volumefraction (dashed line) vs. strain obtained for Mg, Ti, and Zr single crystals ofdimensions lx× ly× lz containing one nano-sized void of radius R = 10a0 atT = 300 K. The simulations are carried out at a strain rate of ε˙ = 108 s−1. b)Evolution of the dislocation density. . The numbers in the plot indicate the rateof change of dislocation density at the critical point, i.e., ρ˙(ε = εc). . . . . . . 563.4 Time evolution of the slided void surface for the simulation cells of lengthslx× ly× lz containing one nano-sized void of initial radius R = 10a0 at T = 300K carried out at a strain rate of ε˙ = 108 s−1. (a) Ti, (b) Zr, (c) Mg. . . . . . . . 583.5 Dislocation emission from the void surface in Ti for the simulations carried outat T = 300 K and at ε˙ = 108 s−1. The initial radius of the nano-void is R= 10a0and the snapshots correspond to different deformation levels. a) Basal disloca-tions (labeled with letter B) are emitted at ε = 2.7%. b) Prismatic dislocations(labeled with letter P) at ε = 2.85%. Dislocations at c) ε = 3.0% and d) ε =3.15%. Pyramidal-II dislocations are emitted from the tip of basal edge disloca-tions (labeled with letter P-II). . . . . . . . . . . . . . . . . . . . . . . . . . . 603.6 Dislocation emission from the void surface in Zr for the simulations carried outat T = 300 K and at ε˙ = 108 s−1. The initial radius of the nano-void is R= 10a0and the snapshots correspond to different deformation levels. a) Basal disloca-tions (labeled with letter B) are emitted at ε = 1.95%. b) Prismatic dislocations(labeled with letter P) at ε = 2.1%, dislocations partially detached from the voidsurface. c) ε = 2.4%, square-shape prismatic dislocation loops (labeled withletters PDL), PDLs completely detached from the void surface and are emittedin [1210] and [1120] directions. d) ε = 2.625%, Pyramidal-II dislocations areemitted from the tip of basal edge dislocations (labeled with letter P-II). . . . . 61xiii3.7 Detailed view of the square shape PDLs emitted from the void surface at a strain= 2.4 % traveling along the [1210] direction. The PDLs have a Burgers vector ofmagnitude 〈a〉 and are generated by two partials ( 13 [1100]+ 13 [0110]) separatedby a small stacking fault defect on the basal plane and two prismatic dislocations13 [2110](1010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.8 Strain rate effect for nano-sized voids in HCP materials. Strain rates rangingfrom ε˙ = 105 – 1010 s−1 for hydrostatic tensile loads in single crystal HCP ma-terials of dimensions lx× ly× lz with approximately 7 million atoms containingone nano-sized void of radius R = 10a0 at T = 300 K for Mg, Ti, and Zr. Thelines are obtained from a fitting law σm(ε˙) = σ0(1+aε˙b). . . . . . . . . . . . 623.9 Evolution of the mean hydrostatic stress vs. strain for various void sizes (R= 5−25a0) obtained for a single crystals of dimensions lx× ly× lz with approximately7 million atoms containing one nano-sized void at T = 300 K carried out at astrain rate of ε˙ = 108 s−1. a) Ti. b) Zr. . . . . . . . . . . . . . . . . . . . . . . 643.10 Evolution of the mean hydrostatic stress vs. strain obtained for Ti single crystalsof dimensions lx× ly× lz with approximately 7 million atoms containing onenano-sized void of radius R = 10a0 at T = 300 K and carried out at a strain rateof ε˙ = 108 s−1. Results obtained for the first and second set of simulations withrotation of the major axis of the ellipsoidal void along the a) [0001]−, and b)[1010]− directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.11 Evolution of the mean hydrostatic stress vs. strain for different temperaturesranging from 50 K to 600 K obtained for simulation cells containing one nano-sized void of radius R = 10a0 obtained at a strain rate of ε˙ = 108 s−1. (a) Ti, (b)Zr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.12 Schematic diagram of the stresses acting on the basal plane ((0001)) and on theglide plane of the pyramidal-II slip system ((1122)). Radial σr and tangentialσθ stresses are associated with the shear stress τxy and σy obtained using thestress transformation formula, where τxy along the horizontal slip plane is shownnext to the edge dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.13 View of the two-dimensional simulation in Mg carried out at a strain rate ofε˙ = 108 s−1 under biaxial stress. The void emits basal dislocations (green atoms)and tensile twins to accommodate non-basal deformation. Interestingly, when abasal dislocation intersects the twin boundary, a small surface is generated anda crack propagates in an unstable mode. . . . . . . . . . . . . . . . . . . . . . 70xivAcknowledgementsI would like to express my sincere gratitude to my advisor Dr. Mauricio Ponga for hiscontinuous support throughout my journey at UBC. I feel really lucky to have the chance towork with him. His encouragement and guidance were the keys for me to reach the purpose ofthis project. I acknowledge his transparent honesty that I believe has impacted positively on myresearch skills. I am grateful that I had the opportunity to fully fulfilled myself academically.I am thankful to the members of my dissertation committee: Prof. Anasavarapu SrikanthaPhani and Dr. Mattia Bacca, for generously offering their time, and a review of this document.I respect and thank Prof. Carl Ollivier-Gooch for helping to arrange a studious environmentin order to write my thesis.Thank you so much to my labmate David, and my friends for standing next to me. Alondra,Faisal, Harsh, Juuso, Miglu, Mike, Ratul, Sarah, Shahzaib, and Somatex, were definitively aninspiration for me and my energy was driven by your help.Last, but not the least, my parents were always in my thoughts and had an important role inthe achievement of this adventure.xvDedicationTo my family.xviChapter 1Introduction1.1 BackgroundHuman evolution is closely linked to technological innovations and developments in science.From stone age to the silicon era, technological evolution has played a major role in the de-velopment of many societies helping to create more advanced and sophisticated cities and in-frastructures, growing the population size, and the geographical range of humans among otheraspects. The evolution of materials usage by humans since the stone age up to today is shownin Fig. 1.1. Remarkably, we see a much faster evolution of a large variety of materials in thelast sixty years. This rapid increase in the spectrum of materials is fuelled by many aspects.One of these aspects is the need to develop stronger and yet lighter materials and structures toincrease the efficiency and performance of transportation vehicles, electronic devices, airplanes,human prosthesis, etc. In this sense, the fundamental study of materials and their mechanismsof deformation is of main interest in mechanical engineering applications as it provides a betterunderstanding of materials, but more importantly, it might provide new avenues to improve me-chanical properties of materials by manipulating these mechanisms of deformation. In addition,the development of new tools and technologies to study materials with great accuracy -fromatoms to components- plays an important role in this evolution.Specifically, due to their high specific ratio between strength and density, Hexagonal Close-Packed (HCP) materials offer tremendous possibilities to design light-weight materials and com-ponents and therefore, offer a technological alternative to replace iron and aluminum-basedalloys in transportation vehicles. For instance, Magnesium (Mg), one of the most abundantelements in the earth’s crust (according to the U.S. Geological Survey, Mineral CommoditySummaries [3] there are around 7,800,000 of thousand metric tons of reserves in the wholeworld), offers tremendous opportunities for modern lightweight applications in engineering. Itslow density and high specific strength (material’s strength divided by its density) makes Mgand its alloys recognized alternatives to steel and aluminum1. Indeed, weight reduction is oneof the most effective ways to increase fuel economy and energy efficiency of transportationvehicles needed to reduce Green House Gas (GHG) emissions and fuel consumption [4, 5, 6].This could positively impact the transportation sector responsible for pollutant emissions (CO2)1Mg density is ρMg = 1780Kg ·m−3 or one-fifth and one-half of the density of Fe and Al, respectively. ρAl =2700Kg ·m−3 and ρFe = 7874Kg ·m−3.1Figure 1.1: Evolution of engineering materials with non linear timescale [2]. ”Relative importance” isemphasized where the rate of change is far faster today than at any previous time in history.that contribute to 25% of total GHG emissions [4, 5]. As a result, the world production of Mghas experienced a rocketing ascend since 2010 where only 4990 of thousand metric tons wereproduced while in 2017 the world production reached 27,000 of thousand metric tons [3]. Thissignificant increase clearly indicates an interest of several industries to use more lightweightmaterials such as Mg, as can be seen in Fig. 1.2. Unfortunately, Mg and its alloys have poormechanical properties such as ductility, formability, and low yield strength, and they are moresusceptible to catastrophic failure which ultimately hinder the widespread industrial use of thesematerials.Titanium (Ti) and its inter-metallic alloys are attracting a great deal of interest in engineeringapplications for structural applications because of its unique high specific strength, low weight,excellent corrosion resistance even in aggressive environments, and good compatibility withcomposite materials. Moreover, their outstanding mechanical and thermal properties are helpfulfor medical, offshore drilling, chemical processing, nuclear fields, as well as in aeronautics andaerospace applications where high-performance materials are required. Unfortunately, Ti is notas abundant as Mg and therefore, it is substantially more expensive than other metallic materials.This ultimately restricts the usage of Ti and Ti-alloys to structures and components with high-performance requirements. The world production of Ti and TiO2 was 7,100 of thousand metrictons in 2017, with a total amount of reserves of 930,000 thousand metric tons in the world [3].Zirconium (Zr) and its alloys are a very promising material for atomic industry and power2Figure 1.2: Evolution of the mine production per year for Al, Mg, Ti and Zr in thousand metric tons. Thedata includes the metal and compounds used in several industries and was compiled from [3].engineering. Commonly used as the cladding of fuel rods in fission reactors, where the irradi-ation dose is extremely high, the advantageous properties of Zr are the low absorption cross-section for neutrons [7], the high resistance to corrosion in a high-temperature water environ-ment, and relatively high mechanical strength. One of the main reasons for selecting Zr alloysas nuclear material is its low thermal neutron capture cross-section which is about 30 times lessthan stainless steel [8]. Zr mechanical properties, tested in some studies [9, 10, 11, 12, 13],are greatly affected by the radiation damage, meaning that the thermo-mechanical processingof Zr components has a significant impact on components lifetime and reactor efficiency. Thedefects will cause material hardening, shift in the ductile to a brittle transition temperature andreduction of fracture toughness, presenting extreme rigorous challenges on the integrity of thesestructural components [14] for the safe operation of commercial light-water reactors. Similarlyto the previous case, Zr is not very abundant in the earth’s crust and therefore, it is used in veryspecific industrial applications. Consequently, the production is limited to 1600 thousand tonsper year on a world scale, representing up to 5 times less than either Ti [7,100] and around 17times less than Mg [27,000] [3].The aim of this work is to provide a fundamental understanding of the mechanisms of de-formation of these three HCP materials. However, due to its possibility to replace iron andaluminum alloys, the focus of this thesis is on Mg and its properties. The primary goal is to3understand the failure of nano-sized voids in Mg under dynamic loads. Nano-sized voids areimportant to understand the ductile failure of materials and are of extreme relevance in dynamicapplications such as impact problems and spall experiments [15, 16, 17, 18, 19, 20, 21, 22]. Un-der these circumstances, materials usually fail by nanovoid growth and coalescence that resultin the generation of macroscopic cracks, ultimately leading to the catastrophic failure of speci-mens. In the remaining of this chapter, we provide a basic introduction to molecular dynamics(MD) simulations and defects in crystalline materials.Chapter 2 presents the results of the dynamic behaviour of nanovoids in Mg obtained forfive loading cases. For all cases, the virial stress and the Von Mises equivalent stress vs. strainevolution, dislocations density, emission, topology, and velocities are studied. For the hydro-static tensile load, a meticulous analysis is provided including the effect of void size, cell size,temperature, and dislocation velocities. Chapter 3 shows a comparative study of the understand-ing of deformation mechanisms using Ti and Zr. Surprisingly, we found that the mechanismsof deformation are extremely different and observed a ductile failure with a large number ofdislocations. This is carefully analyzed in order to understand the differences between Mg, Ti,and Zr, and provide design parameters to improve Mg-based alloys with optimal dynamic resis-tance. Chapter 4 summarizes the main conclusions of this work and propose some future workand extensions.1.2 Molecular DynamicsIn this work, we make extensible use of Molecular Dynamics (MD) simulations to understandthe mechanisms of deformation of metals. MD is an excellent tool to study dislocation emissionfrom nano-sized voids however this technique suffers from the size and time limitations charac-teristic. Due to the need to track the thermal vibrations of the atoms, the time scales accessibleare restricted to typical time steps to the femtosecond range. Long-term phenomena are severelycurtailed from MD scope, where strain rates below ε˙ = 107 s−1 are studied with dynamic HotQuasi-Continuum (HotQC). Continuum models overcome these limitations by adapting the spa-tial resolution to the structure of the deformation field and facilitate the consideration of realisticsamples. The framework is shortened to equilibrium conditions and cannot be used to simulatethe rapid transient attendant to dynamic void growth. A non-equilibrium extension of QC, re-ferred to as HotQC approach construct effective thermodynamic potentials for systems of atomsaway from equilibrium.We use the Large-scale Atomic/Molecular Massively Parallel Simulator) (LAMMPS) code[23]. This is an open source code distributed by Sandia National Laboratories, originally de-veloped under a U.S. Department of Energy. The post-processing is carried out using the OpenVisualization Toolkit (Ovito) where dislocations and defects are visualized [24, 25].The atomic interactions are modelled with an interatomic potential. We mainly use theEmbedded Atom Method (EAM) proposed by Baskes et al. [26]. In the EAM model, the4energy of the ith atom isEi = F (ρi(ri j))+12N∑j 6=iφ(ri j) (1.1)ρi(ri j) =N∑j 6=if (ri j) (1.2)where F is called the embedding function which is a function of the electronic density ρi(ri j),and φ is the pair-potential interaction between atoms i and j. The embedding function representsthe energy to insert an atom i in the electron density ρi made by the surrounding j atoms. Theabove mentioned quantities are short ranged functions that are fit to experimentally measuredquantities or ab-initio data [27]. In order to model HCP materials, we use three different EAMpotentials for Mg [28], Ti [29] and Zr [30]. Additionally, in order to assess the accuracy of theEAM potentials, we use a more sophisticated and expensive second nearest neighbours modifiedembedded atom method (2NN-MEAM) where the electron density has an angular dependency[31]. The MEAM potential is ought to be more accurate, however, we will see that the 2NN-MEAM potential developed by [31] give the same defects than its more simplified counterpart,the EAM potentials. In addition, we also observed that the 2NN-MEAM suffers from a strongdependency of the elastic constant with respect to the hydrostatic deformation, leading to anunrealistic softening even for linear deformation.1.3 Hexagonal Close Packed lattice structureThe description of the HCP, shown in Fig. 1.3 starts with the definition of the atoms that areconsidered as hard spheres. A HCP crystal is a periodic arrangement of atoms or discrete pointsthat can be described by the following multi-lattice atomic collection of identical Bravais latticeswhich are translated to each other, i.e.,LHCP = {x : x3∑i=1νiei+K−1∑k=1ηkpk, νi and ηk are integers} (1.3)where the lattice vectors {ei}3i=1 are linearly independent and describe the periodicity of the unitcell and the shifts {p}K−1k=1 are vectors that describe the translation of the constituent Bravaislattices relative. Using cartesian coordinates, one possible choice of the lattice vectors that gen-erate the HCP lattice is given by e1 = a0{1,0,0}, e2 = a0{12 ,√32 ,0}, e3 = c0{0,0,1}, where a0and c0 are the lattice parameter in the x- and z- directions, respectively. Thus, in HCP materials,the ratio c0a0 is different to one. In order to generate the shift atom in the unit cell, we need to in-troduce one shift vector p = {a03 , a03 , c02 }. We also notice that the HCP structure is often referredas a stack of atoms that lie in two types of planes. For instance, Fig. 1.3 shows the atoms in5(a) (b)Figure 1.3: HCP lattice and its unit vectors. a) Arrangement of the atoms in the HCP lattice structure andits lattice vectors e1, e2 and e3. The unit cell of the HCP lattice structure is highlighted in red. b) HCPstructure and three crystallographic directions in the basal plane (a1, a2, a3) and one in the c−direction.Numbers between braces indicate the Miller indices.these two planes with blue and red. The atomic planes are usually called the basal plane whichhas sixfold symmetry, and the HCP structure is obtained by stacking one basal plane on top ofanother basal plane. However, we notice that two consecutive basal planes need to be shifted,as indicated by the shift vector p. Thus, it is customary to say that the HCP structure is made byan ordered sequence of basal planes, that we denote as . . .ABABABABABAB . . . noticing that theplanes A and B are the same, but they are shifted with respect to each other. The nomenclatureAB will be useful when we describe stacking fault defects.1.4 Defects in HCP materialsWe now briefly introduce some defects seen in materials, with a particular focus on defects inHCP crystals. For a detailed treatment of defects and dislocations in materials we refer thereader to Theory of dislocations by Hirth and Lothe [32].1.4.1 Miller IndicesIt is customary to express crystallographic directions and planes using a set of integers or indicesthat indicates the directions using the crystallographic directions shown in Fig. 1.3(b). Suchnotation is called Miller-Bravais notation. HCP lattice structures are designated by four indices6(hkil). The first three indices (h, k and i) are related to the basal plane and the forth one indicatesthe c−direction. Thus, the indices indicate the direction d = ha1+ ka2+ ia3+ lc.In HCP lattice, the three indices in the basal plane are redundant and they should satisfy thefollowing constrainth+ k+ i = 0. (1.4)The fourth index, (l), represents the c−axis ([0001]) perpendicular to the basal plane. Fig.1.3b illustrates these indices. There are also several related notations; the notation {hk i l} de-notes the set of all planes that are equivalent to (hk i l) by the symmetry of the lattice. In thecontext of crystal directions (not planes), the corresponding notations are:• [hk i l], with square instead of round brackets, denotes a direction in the basis of the directlattice vectors instead of the reciprocal lattice; and• similarly, the notation 〈hk i l〉 denotes the set of all directions that are equivalent to [hk i l]by symmetry.Thus, as indicated in Fig. 1.3b, the directions can be expressed using the Miller indices as:a1 = a03 [2110], a2 =a03 [1210], a3 =a03 [1120], and c = c0[0001].1.4.2 Dislocations linesWe now introduce the concept of dislocations in the context of HCP structures. By definition, anedge dislocation is a crystal line defect where an extra plane of atoms can be generated whereasa screw dislocation is a crystal line defect where the atoms are slipped along a plane (oftencalled the slip plane) and comprises a helical pattern around the dislocation line.We now introduce the concept of Burgers circuit which is helpful to identify the so-calledBurgers vector of the dislocation. Let us now take a perfect crystal –free of defects– and makean atom-to-atom closed path. We will call this path, the Burgers circuit. This can be seen in Fig.1.4(a) where the Burgers circuit has been highlighted with blue.Now, let us introduce a dislocation in the same crystal and make the same circuit atom-to-atom which encloses the dislocation line. This circuit is shown in Fig. 1.4(b). We immediatelysee that the path is open and an extra vector is needed to close this path. The vector neededto close the path is called the Burgers vector. Fig. 1.4(b) shows the Burgers vector, which ishighlighted in red, and it is perpendicular to the dislocation line that is pointing out of the planeof the paper. We call this an edge dislocation, whereas dislocations with the Burgers vectorpointing in the dislocation line are called screw dislocations, shown for instance in Fig 1.5. Wefinally notice that a slip system is fully characterized by a slip plane and a Burgers vector.7(a) (b)Figure 1.4: Burgers circuit shown in a schematic crystal a) with and b) without a dislocation. In thisexample, the Burgers vector is shown with a red arrow. The Burgers vector is perpendicular to thedislocation line, which in this case is in the direction out of the paper, indicating that the dislocationin the example is an edge dislocation.(a) Perspective view. (b) Side view.Figure 1.5: Screw dislocations in HCP-Mg in the centre of a single crystal. The atoms are colouredaccording to the common neighbour analysis. Atoms in red are in HCP structure, green indicates FCCstructure and white other type of structure. The screw dislocation is dissociated into two partials (white)separated by a stacking fault defect (green) a) Perspective view of the dislocation. b) Side view of thedislocation showing the slip plane and the direction of the Burgers vector.8(a) Atoms in a stacking fault defect. (b) Schematic view of the stacking faultFigure 1.6: Stacking fault in HCP lattice a) Schematic picture of the HCP structure with a faulted plane.It is evident from the figure that the ...ABAB... ordered sequence of planes is broken and a defect appears.Such defect is called stacking fault. b) View of the previous screw dislocation (see Fig. 1.5(a)) removingall atoms in the HCP structure (only the defect is shown). We see that the atoms in the stacking fault ingreen, with the two partials in white.1.4.3 Stacking fault defectsA stacking fault defect is a planar defect, that is characterized by a fault in the stacking se-quence previously detailed (see Section 1.3). As such, imagine that one of the layers is furthershifted with respect to the previous layer, such that the stacking sequence is . . .ABABCABAB . . ..This indicates a faulted stacking of basal planes leading to a stacking fault defect. Figures 1.6represent this stacking fault defect.1.4.4 Slip systems in HCP lattice structureHCP lattice structure is characterized by four independent slip systems, whereas cubic latticestructures such as Face-Centred Cubic (FCC) and Body-Centred Cubic (BCC) have five slipsystems [32]. A slip system is composed of a Burgers vector and a slip plane. Fig. 1.7 (a) showsthe basal slip system, with a Burgers vector of 13 [1120] and a slip plane of (0001). Similarly,a prismatic slip system, shown in Fig. 1.7 (b), includes a Burgers vector of 13 [1120], howeverthe slip plane is placed on the 〈1010〉 plane. Figures 1.7 (c) and (d) are named the pyramidal-I and pyramidal-II slip systems with the same Burgers vector of 13 [1123] and a slip plane of〈1011〉 and 〈1122〉, respectively. These slip systems result in the anisotropic behaviour ofHCP materials. For instance, the basal and prismatic slip systems are usually easily activated inMagnesium [33, 34, 35].9Hence, slip systems are responsible for the anisotropy and can create a profusion of twinningthat is activated in order to deform along the [0001]-crystallographic direction [36]. Compres-sive and tensile twins can be observed in Fig. 1.8 that describes the directions and the planes ofthese two twins: 1/3[1123] Burgers vector on (1011) plane and 1/3[1011] Burgers vector on(1012) plane for compressive and tensile twins, respectively.10(a) Basal. (b) Prismatic.(c) Pyramidal-I. (d) Pyramidal-II.Figure 1.7: Representation of the slip systems in the HCP lattice structure (a) basal, (b) prismatic, (c)pyramidal-I, and (d) pyramidal-II slip systems.11(a) Compressive. (b) Tensile.Figure 1.8: Illustration of the two most common twin systems in the HCP lattice structure. a) compressiveand b) tensile twin systems usually activated in HCP materials to generate deformation along the [0001]direction.12Chapter 2Nano-void failure in Magnesium underdynamic loadsAbstractThe dynamic behaviour of Magnesium (Mg) single crystals containing nano-sized voids and themechanisms responsible for failure under different loading conditions are studied by means ofMolecular Dynamics (MD) simulations. Our study reveals that the response is highly anisotropicleading to a brittle to ductile transition in the failure modes under different load orientations.This transition is accompanied by different mechanisms of deformation and is associated withthe anisotropic Hexagonal Close-Packed (HCP) lattice structure of Mg and the associated bar-rier for dislocation motion. Remarkably, brittle failure is observed when external loads pro-duce a high stress triaxiality while the response is more ductile when the stress triaxiality de-creases. The fundamental mechanisms observed in the simulations, therefore, explain the lowspall strength of Mg and suggest the possibility of manipulating some mechanisms to increaseductility and spall strength of new lightweight Mg alloys.2.1 IntroductionMagnesium (Mg), the lightest of all metallic materials and one of the most abundant elementsin the earth’s crust, offers tremendous opportunities for modern lightweight applications in en-gineering. Its low density and high specific strength (material’s strength divided by its density)make Mg and its alloys recognized alternatives to iron and aluminum based-alloys to satisfy thestrong demand for weight reduction. However, Mg and its alloys have poor mechanical proper-ties such as ductility and low yield strength, which hinder the widespread industrial use of thesematerials [37, 38, 39].This is largely a consequence of the HCP lattice structure of Mg. A key characteristic of theHCP lattice structure is that only four independent slip systems are available to produce plasticdeformation, which is in contrast with FCC and BCC lattice structures where five independentslip systems are available [32]. Additionally, in Mg, the basal and prismatic slip systems areusually easily activated whereas the critical stress for dislocation glide in the pyramidal systemsis at least one order of magnitude higher than in the basal plane [33, 34, 35, 36]. This anisotropy13in the slip systems results in a profusion of twinning that is activated in order to deform alongthe [0001]- crystallographic direction [36].In this chapter, numerical simulations are performed to understand the deformation mecha-nisms operative in Mg during dynamic failure. Dynamic failure is a complex phenomenon thatoccurs when materials are subject to high velocity impacts and is characterized by a stronglycoupled thermo-mechanical evolution of nano-voids. Under these conditions, materials usuallyfail by spallation [40, ?]. The spall failure happens due to the high tensile loads reached inthe spall plane, where nano-sized voids are quickly nucleated, leading to nano-void growth andcoalescence, and eventually, the propagation of cracks in the specimens [41, 42, 43]. Due tothe complex mechanisms that have place during dynamic failure and spallation, the experimen-tal characterization is extremely challenging and restricted only to indirect measures, i.e., freesurface velocity or post-mortem examination of specimens.In spite of the limitations of current experimental techniques to study nano-void growthunder dynamic loads, numerical simulations using MD provide a highly useful tool to probethese mechanisms. The main goal of this work is to shed light on the mechanisms that dominatedynamic failure mode by nano-void growth and thereby, provide insights that can be used toinform the design of new Mg alloys. MD has been used in the past by many researchers topredict nano-void growth in different materials including FCC [44, 45, 46, 47, 48, 49, 50, 51]and BCC [52, 53, 54] materials, to mention a few.While the literature is broad on either FCC or BCC materials, there is a clear lack of bib-liography for nano-void cavitation in HCP materials. In a recent study, Ponga et al. [55] havestudied the dynamic behaviour of nano-voids in Mg using the HotQC method and analyzed themechanisms of deformation for hydrostatic tensile loads and the transition regimes observed formoderate to high strain rates. While the study of Ponga et al. [55] provides valuable funda-mental understanding in the dynamic behaviour of Mg under hydrostatic tensile loads, the keymechanisms involved in the dynamic failure under different load orientations are not well under-stood and remain an open question. This work complements and expands the aforementionedstudy to multiple load conditions in an effort to answer some of these questions. Therefore, weseek to understand the mechanisms that have place during the dynamic failure of Mg under fivedifferent loading conditions using MD simulations. These loading conditions are used in orderto activate different slip systems to understand the main mechanisms active during nano-voidfailure.One of the most critical aspects of MD simulations is the choice of the interatomic potential.It is well known that the mechanisms of deformation may change from one potential to anotherduring nano-void growth [51] and for that reason, it is essential to assess the fidelity of theresults and mechanisms of deformation for different interatomic potentials. Thus, an ancillaryobjective of this work is to assess the fidelity of the mechanisms of deformation using differentpotentials for Mg. This task is not trivial since there is no standard or routine benchmark testfor interatomic potentials.This issue has been investigated by many researchers in recent years. Yasi et al. [33]14have studied the basal dislocations core in Mg using two Embedded Atom Method (EAM) [26]interatomic potentials proposed by Liu et al. [56] and Sun et al. [28], respectively. Morerecently, Ghazisaeidi et al. [57] have studied the edge and screw dislocation cores in Mg withthe aforementioned EAM potentials developed by Liu et al. and Sun et al. and a newly secondnearest neighbours Modified Embedded Atom Method (2NNMEAM) [58, 59] proposed by Kimet al. [60]. They have concluded that neither three potentials can capture all the details observedwith methods based on the Density Functional Theory. Motivated by these issues, Wu et al.[61] have developed an improved and promising Modified Embedded Atom Method (MEAM)based on the original potential of Kim et al. [60].It is important, therefore, to provide tools for future works where one can select the adequateinteratomic potential for the study of a particular problem of interest. Hence, we endeavour toassess the efficiency and fidelity of the mechanisms of void growth and dislocation emission fortwo different interatomic potentials. In addition, to test the potentials in single dislocations setup as in the previous works, we benchmark the potentials according to the peak stress, dislo-cations topology, and critical strain for dislocation emission under different loading conditions.Remarkably, only minor differences are encountered and the overall behaviour is essentially thesame for the two potentials.The chapter is organized as follows. Section 2.2 explains the methodology used in the sim-ulations carried out in this work. Section 2.3 presents the results obtained for five differentloading cases. For all cases, the virial stress and the Von Mises equivalent stress vs. strain evo-lution, dislocations density, emission, topology, and velocities are studied. For the hydrostatictensile load, a meticulous analysis is provided including the effect of void size, cell size, temper-ature, and dislocation velocities. Section 2.4 presents a discussion of the main results obtainedin this work. Finally, the document is summarized with the main conclusions.2.2 Methodology2.2.1 Nano-void simulationsNumerical simulations of nano-void cavitation under different tensile loads in pure Mg singlecrystals are carried out using MD at a range of strain rates between ε˙ = 107− 1010 s−1. Allsimulations are performed with the LAMMPS code [62] and the material (Mg) is modelledwith two different interatomic potentials developed by Sun et al. [28] and Wu et al. [61].These interatomic potentials will be referred to the remaining of this document as EAM andMEAM potentials, respectively. The first one is based on the EAM method [26] while theMEAM is based on the second nearest neighbours MEAM potential [58]. These potentials aresystematically used to analyze the fidelity of the mechanisms of deformation in Mg.A rectangular parallelepiped of dimensions lx× ly× lz is used to generate a Mg single crystal.The HCP lattice structure is generated with a0 = 0.319 nm and c0/a0 = 1.626, where a0 isthe lattice constant on the basal plane and c0/a0 is the ratio between the lattice constants in15the [0001] direction and the basal plane at 0K, respectively. Unless otherwise specified, theexternal dimensions of the computational cell are lx = 38 nm, ly = 68 nm and lz = 68 nm whichcorrespond to a selection of 1203 HCP units cells containing approximately 7 million atoms. Thecrystals are constructed in such a way that the crystallographic directions [2110], [0110] and[0001] of the HCP lattice structure are aligned with the x−, y− and z− axes of the computationalbox, respectively. Then, an initial void of radius R is generated by removing atoms from thecentre of the crystal. Unless otherwise specified R= 10a0 = 3.15 nm. Finally, periodic boundaryconditions are used in all simulations containing nano-voids.Unless otherwise specified, the temperature of the computational cell is initialized at a con-stant value of T = 300 K. The initial configuration is relaxed by integrating the equations ofmotion for 40 ps without applying any load. During this relaxation process the crystal is allowedto expand and/or contract independently in each orthogonal direction allowing to minimize theenergy and the pressure of the computational cell by using a NPT ensemble with a Nose´-Hooverthermostat and a time step of ∆t = 5 fs. Once the initial relaxation is completed, the individualcomponents of the stress tensor are zero within ±5 MPa.After the initial relaxation is achieved, a controlled displacement load is applied to the sim-ulation cell by using a homogeneous deformation gradient F at each time step. Let xin be theposition vector for an atom i at the time step n and the determinant F be the deformation gradi-ent, thus the new position of the atom i at the time step n+ 1 is computed as xin+1 = Fxin. Thedeformation gradient is:F =1+n1ε˙∆t 0 00 1+n2ε˙∆t 00 0 1+n3ε˙∆t (2.1)where n = {n1,n2,n3} is a vector whose components are either one or zero, ε˙ is the strainrate and ∆t is the time step. The deformation gradient F is changed for each loading caseby selecting different choices of n. The deformation gradient is modified in order to activatedifferent slip systems and understand the main mechanisms of nano-void growth. Table 2.1shows the individual values of the components of n for each loading case.Table 2.1: Selected values of n for different loading cases.Load x-component [2110] y-component [1100] z-component [0001]Case 1 1 1 1Case 2 1 0 1Case 3 1 1 0Case 4 1 0 0Case 5 0 0 116When boundary conditions are used in MD simulations, there is no need to provide a con-strained fixed layer near the vicinity of the boundary in order to follow the deformation gradient.This layer is automatically enforced by the boundary conditions.In order to quantify the strength of the simulation cell, virial stresses are reported. As MDcannot be assumed as continuum, the virial stress is a discrete equivalent of the Cauchy stress.Virial stresses are computed according to Allen and Tildesley [63]σαβ =−1V(∑ipiα pjβm+∑i∑jri jα fi jβ)α,β = x,y,z, (2.2)where σαβ are the components of the stress tensor, V is the total volume of the simulation cell,m is the mass, piα is the component α of the moment of atom i, ri jα = xiα − x jα is the componentα of the relative distance vector between atom i and j, and f i jβ is the component β of the forcevector on atom i due to atom j. Unless otherwise specified, the reported values correspond tothe mean or hydrostatic stress, i.e., σm = 1/3(σxx +σyy +σzz). Two independent methods areused to quantify the amount of plastic work generated in the simulation cell. The first quantityis the Von Mises equivalent stress σeσe =(12 ∑α>β(σαα −σββ )2+3 ∑α>βσ2αβ)1/2. (2.3)The Von Mises equivalent stress allows to introduce the stress triaxiality factor which is a directmeasure of the plastic deformation in the sampleχ =σmσe. (2.4)The second independent quantity is the dislocation density computed asρ =∑dldV, (2.5)where ld is the length of the individual dislocation segments in the simulation cell. Alldislocations are extracted with the Dislocation eXtraction Algorithm method and visualizedwith OVITO [24, 25]. When reported, dislocation velocities are computed using the proceduredescribed by Ponga et al. [51].2.2.2 Dislocation simulationsDislocation simulations have been performed for edge and screw component in the basal, pris-matic and pyramidal-II slip systems in Mg single crystals. All simulation cells are generatedsuch that the dislocation line coincides with the y−axis, and the z−axis is aligned with the17normal vector of the slip plane of the selected system. For all dislocation simulations, peri-odic boundary conditions are used along the dislocation line direction while free surfaces aresimulated in the other two perpendicular directions. In order to generate edge dislocations, theprocedure described by Osetsky and Bacon [64] has been followed and generated N unit cellsin the upper half part of the simulation cell while N− 1 unit cells are generated in the lowerhalf of the simulation cell. This generation procedure leads to a perfect edge dislocation in anotherwise perfect single crystal with free-surfaces at the boundary of the simulation cell. Screwdislocations were generated using the anisotropic elastic solution [32].In all simulations, the computational cell has approximately 300,000 atoms such that thelong-range elastic effect of the dislocation is taken into account. Once the initial dislocationgeometry is generated, the energy is minimized using a nonlinear conjugate gradient method.During the minimization, the atomic positions are re-scaled such that the total hydrostatic pres-sure is zero at the end of the simulation. Then, the temperature is initialized to T = 300 K anda shear deformation is applied along the x− direction at a strain rate of ε˙ = 108 s−1. The sheardeformation is applied by using a strain-controlled displacement in a top and bottom layers ofthe simulation cell. The thickness of these layers is set up to be around 5a0. The strain con-trolled displacement is applied by prescribing a constant velocity vx along the x− direction toall atoms in the top layer and −vx to all atoms in the bottom layer. The maximum shear stressσzx is reported as the critical stress required to move the dislocation.2.2.3 Frank-Read mechanism simulationsDue to the resemblance of one of the mechanisms of deformation observed during nano-voidgrowth simulations with the Frank-Read source mechanism of dislocation multiplication, wehave generated the following set up to better understand the critical stress required to fullydetach a prismatic 〈a〉 dislocation loop from two obstacles. The set up was inspired in thestudy proposed by Shimokawa and Kitada [65]. A single Mg crystal is generated such that itsdimensions are 200a0[2110]× 100c0[0001]× 40a0[1100]. Periodic boundary conditions areused in the x− and z− directions while free surface conditions are used in the y−direction.The simulation cell contained approximately 9 million atoms. Such large simulation cell isneeded to capture the full detachment of dislocation from the obstacles. Then, a prismatic〈a〉 edge dislocation was generated by removing one plane of atoms of dimensions L[0001]×20[1100] in an otherwise perfect crystal. L is an arbitrary length that is changed to study thecritical resolved shear stress needed to bow out the dislocation. This structure is also minimizedusing a nonlinear conjugate gradient and at the same time, the total hydrostatic pressure in theinitial configuration is reduced to zero. Two cylindrical voids with the principal axis alongthe [1100]−direction and centred at the end points of the missing plane are generated. Thecylindrical voids are generated to produce the immobile obstacle of the Frank-Read source. Thesimulations performed involved cylinders with different radius, i.e., R = 0,4a0 and 8a0. WhenR = 0, the voids are not generated at all and the prismatic dislocation is connected at its ending18points with two basal edge dislocations. Fig. 2.1 shows a picture of the sep up with a detail ofthe dislocation structure in the inset once the energy is minimized.z:[0001]y:[1100]x:[1120]L——Figure 2.1: Schematic picture of the set up used to study the Frank-Read source in Mg. In the simulations,two cylindrical holes of radius R separated by a distance L are generated. Between these two cylindersa prismatic 〈a〉 dislocation is also introduced. Then, a shear deformation is applied on the plane whosenormal is [1100] along the [1120] direction. The maximum values of τyx are reported as the criticalstress to bow out and completely detach the dislocation from the obstacles.Once the energy of the computational cell is minimized, a constant temperature of T = 300K is initialized. Then, the simulation cell is allowed to relax at a constant temperature using aNVT ensemble for approximately 10 ps. After the thermal equilibrium is reached, the simulationcell is subject to a shear deformation by using a displacement control of the sample in thex−direction. Similarly to the dislocation simulations, the displacement of the atoms at twolayers in the top and bottom of the simulation cell is controlled. The thickness of these layers isapproximately 5a0. Then, the shear deformation is generated by prescribing a constant velocityvx along the x−direction to all atoms in the top layer and −vx to all atoms in the bottom layer.The displacement control is applied at a strain rate ε˙ = 108 s−1 and the maximum value ofσyx is reported as the critical resolved shear stress necessary to detach the dislocation from theobstacles.192.3 Results2.3.1 Dislocation barriersThe study starts by analyzing the barriers for dislocation glide in different slip systems. Thebarrier for edge and screw dislocations in the basal, prismatic and pyramidal-II slip systems areshown in Table 2.2. The dislocation barrier is very small for both basal and prismatic plane. Inthe case of prismatic screw dislocations at T = 300 K, the dislocation can spontaneously crossslip due to the thermal vibrations before the critical stress for dislocation glide is reached andtherefore, it is not possible to measure the critical resolve shear stress at T = 300 K. On theother hand, the dislocation barrier for pyramidal-II for edge and screw components is at leastone order of magnitude higher than in the other two systems. These values are in agreementwith previous studies [61] carried out with the MEAM potential. This anisotropic response ofthe material places several restrictions to dislocation glide in the pyramidal-II slip system. Whilebasal and prismatic 〈a〉 dislocations are easy to move through glide, pyramidal-II dislocationsrequire stresses that are at least one order of magnitude more than in the other two systems andtherefore, it is more difficult to produce dislocation motion. Noteworthy, the screw componentof pyramidal-II dislocations has a barrier that is two orders of magnitude larger than for basaldislocations and almost one order of magnitude larger than prismatic and pyramidal-II disloca-tions with edge components. As detailed in the next section, basal, prismatic and pyramidal-IIdislocations can be emitted from the void surface. However, after the dislocations are emitted,the shear stresses decay very rapidly from the void surface, indeed proportional to ∼ 1/r2, withr the distance from dislocation to the void surface. Thus, even though dislocation emissionis possible in the three slip systems, the glide of pyramidal-II dislocations is more difficult toproduce.Table 2.2: Values of the critical resolve shear stress needed to promote dislocation glide in different slipsystems of relevance for HCP-Mg at T = 300 K. All reported values are in MPa and were obtained withthe EAM potential at a strain rate of ε˙ = 108 s−1.Slip System Edge ScrewBasal [2110]{0001} 3 3Prism [2110]{1100} 25 –Pyramidal-II [2110]{1100} 50 2802.3.2 Case 1: Hydrostatic tensile loadThe evolution of virial stress and the Von Mises equivalent stress vs. strain for the loading case 1obtained with the EAM potential for strain rates between ε˙ = 107−1010 s−1 is shown in Fig. 2.2.All simulations have shown the qualitatively same behaviour that is described next. An initiallinear elastic regime that extends until a critical stress for dislocation emission, σc, is reached.20Then, the void starts the process of cavitation by dislocation emission, increasing its volume ata high rate and basal dislocations are emitted from the void surface. Consequently, the virialstress slightly increases with deformation until it reaches the peak stress, σp (reported in Table2.3). Notably, the critical stress for dislocation emission, σc, and the peak stress, σp, are verysimilar due to the small dislocation interactions. As a result, all simulations under hydrostatictensile load lack of strain hardening. Once the peak stress is reached, the material rapidly failsdue to the emission of several cracks and deformation twins that are nucleated from the voidsurface and propagate through the computational cell. Table 2.3 summarizes the most relevantdata for the hydrostatic tensile load case. Noteworthy, the mean peak stress values shown in Fig.2.2 (and reported in Table 2.3) are in very close agreement for the two potentials in the rangeof strain rates simulated. The Von Mises equivalent stress shown in Fig. 2.2 indicates that thedeviatoric part of the stress is very small for the loading case 1. In a cubic material (like FCCor BCC) one would expect to have zero σe, however, due to the anisotropy of Mg and its HCPlattice structure σe is not zero. Nonetheless, the maximum values are approximately 20% of thepeak stress (σp) for all strain rates. The stress triaxiality, χ , defined in Eq. 2.4, is χ ≈ 6 duringthe linear regime for the different strain rates.Figure 2.2: Evolution of the mean hydrostatic stress (σm) and Von Mises equivalent stress (σe) vs. strainat different strain rates for the loading case 1 using a Mg single crystal of dimensions 38×68×68 nm3with approximately 7 million atoms containing a nano-void of radius R = 3.15 nm at T = 300 K for theEAM potential.The evolution of the dislocation density ρ computed with Eq. 2.5 is now described. The21Table 2.3: Peak stress (σp) and corresponding peak strain (εp) for different strain rates (ε˙) obtained withthe loading case 1 using a Mg single crystal of dimensions 38×68×68 nm3 with approximately 7 millionatoms containing a nano-void of radius R = 3.15 nm at T = 300 K for the EAM and MEAM potentials.Values between () correspond to the MEAM potential.Strain rate ε˙ [s−1]107 108 109 1010εp [%] 2.65 (3.45) 3.10 (3.90) 4.0 (5.25) 7.3 (10)σp [GPa] 2.52 (2.61) 2.88 (2.85) 3.47 (3.4) 4.77 (4.36)evolution of ρ obtained for the EAM potential for the different strain rates simulated in thiswork is depicted in Fig. 2.3. ρ starts at 1014 1/m2 for all strain rates. In the cases of ε˙ = 107and 108 s−1 the dislocation density is different to zero for only a short period of time, indicatingthat the failure occurs in a brittle fashion and very close to the first dislocation emission. Forthe remaining strain rates, ρ evolve for a longer period of time, however, the values are around5× 1014 1/m2 indicating that dislocations do not grow that much and are attached to the void.Only near the failure, ρ increases substantially reaching the 1016 1/m2.Remarkably, Ponga et al. [55] have studied the dislocation emission from nano-sized voidsunder hydrostatic tensile loads. Due to the similarity of the dislocation structures with theaforementioned work, we omit the description of dislocation emission. We finally describesome interesting aspects of the fracture of specimens under this loading case. We have observedthat 〈 c + a 〉 dislocations are emitted from the tip of basal dislocations followed by crackspropagating almost instantaneously after the emission of such dislocations. Table 2.2 indicatesthat the barrier to move pyramidal-II dislocations with screw components is at least two ordersof magnitude larger than in the basal plane. This large difference generates stress concentrationnear the nano-void surface leading to the propagation of cracks and pushes the material towardsmore brittle, or less ductile, failure mode. With the emission of cracks, the original crystal -orparent crystal- is allowed to rotate and tensile twins {1012} 〈 1011 〉 are emitted from the cracktip. This is clearly evident in Fig. 2.4 where several deformation twins (labeled with letter T)are emitted from the crack tip (letters CT). Notably, twins have only been nucleated when newsurfaces were generated and do not influence the dynamic response of the material.Dislocation velocities have been measured for the different potentials. For a load applied ata strain rate of ε˙ = 108 s−1, the velocities of the dislocations along the basal plane were 46.2 m/sand 42.45 m/s for the EAM and MEAM potential, respectively.2.3.2.1 Void size effectThe effect of the void size on the evolution of the peak stress obtained with the EAM potentialdepicted in Fig. 2.5, is described next. Simulations with a single Mg crystal of dimensions38× 68× 68 nm3 under a hydrostatic tensile load applied at a strain rate of 107 s−1 have been22Figure 2.3: Evolution of the dislocation density ρ vs. strain for the loading case 1 (hydrostatic tensileload) at different strain rates using a Mg single crystal of dimensions 38×68×68 nm3 with approximately7 million atoms containing a nano-void of radius R= 3.15 nm at T = 300 K. All results are obtained withthe EAM potential.performed changing the void radius from 0 to 30a0. Interestingly, the peak stress for defect-freeMg is σ∗p = 4.7 GPa. For samples with nano-sized voids, the peak stress reduces smoothly from4.7 to 1.9 GPa for a range of voids with radius between the range [2− 30a0]. It is interestingto compare these results with models for void growth. Lubarda et al. [66] have proposed aphenomenological two-dimensional model for void growth by dislocation emission under fortensile loads. The proposed model for void growth is based on dislocation emission from thevoid surface. Even though the simulations in this work are fully three-dimensional, the modelshould give a fairly good estimation for the critical stress for dislocations emission.Lubarda et al. [66] have estimated that the critical stress for dislocation emission shouldhappen whenσcG≥ b/R√2pi(1−ν)(1+√2ωb/R)4+1(1+√2ωb/R)4−1 , (2.6)where σc is the critical stress for dislocation emission, G = 25 GPa is the shear modulus, b= 0.32 nm the Burgers vector, R the void radius, ν = 0.22 is the Poisson coefficient and ω isthe coefficient between the dislocation dissociation distance and b, proposing that dislocations23Figure 2.4: View of the sample towards the end of the simulation after cracks (labeled with letter C) havebeen propagated. Twins (labeled with letter T) have been emitted from the tip of the cracks (labeled withletter CT). The original or parent crystal is indicated by the letters PC.will be likely emitted from the surface of the void if the equilibrium distance is less than thedislocation width. All constants have been taken from MD simulations using the EAM potential.Fig. 2.5 shows the data obtained with MD and the values predicted by the model of Lubarda etal. [66] with two values of ω = 1 and ω = 2. Noteworthy, the model captures the trends withvalues very close to the results obtained with MD.2.3.2.2 Cell size effectAfter investigating the behaviour of the material with the void size, the focus is now targetedon the effect of the cell size on the virial stress. Therefore, simulations for a fixed void size ofradius R = 3.15 nm with different computational cells containing 403, 603, 803, 1003, 1203 and1403 (from 256000 to 10,976,000 atoms) unit cells under hydrostatic tensile load applied at 107s−1 with the EAM interatomic potential have been carried out. Table 2.4 summarizes the peakstress values obtained for different simulations and indicates that the peak stress has only smallsensitivity to the cell size, showing a considerable reduction of the peak stress only for smallcells, i.e., for 403−803 unit cells. For cell sizes with more than 1003 unit cells, the peak stressis almost constant.2.3.2.3 Temperature effectFinally, the evolution of the peak stress under hydrostatic tensile load for a range of tempera-tures between 50−750 K has been investigated. All reported values have been obtained for the240 1 2 3 4 5 6 7ε [%]0123456σm [GPa]R=0R=5a0R=10a0R=20a0R=30a00 5 10 15 20 25 30R0123456Model ω = 1Model ω = 2MDFigure 2.5: Evolution of the peak stress with the nano-void size for the loading case 1 at a strain rate ofε˙ = 107 s−1 obtained with the EAM potential using a Mg single crystal of dimension 38×68×68 nm3with approximately 7 million atoms containing one nano-sized void. Spherical nano-voids with radiusbetween R = 0-30a0 were studied. Predictions made with the model predicted by Lubarda et al. (Eqn.2.6) with ω = 1 and ω = 2 .EAM potential a strain rate of ε˙ = 107 s−1. Fig. 2.6 shows the evolution of the peak stress asa function of temperature. For temperatures below TT = 150 K, the sample is relatively insen-sitive to changes in temperature and the peak stress, σ0T = 3.0 GPa, is approximately constant.For temperatures higher than TT , the peak stress shows a linear decay with the temperatureup to 750 K. TT indicates, therefore, the value of a transition temperature for Mg where thematerial is relatively insensitive to the effect of temperature for T ≤ TT . When T > TT thedata points obtained from MD simulations have been interpolated using a linear fitting func-tion, i.e., σp(T ) = aT (T −TT )+σ0T , with σp(T ) the peak stress at a given temperature T andaT =−2.6×10−3 GPa/K, a proportional coefficient.2.3.2.4 Ensemble choiceOne most important considerations one must take when carrying out MD simulations is to se-lect the right thermodynamic properties of the system, named ensemble. Two ensembles, the25Table 2.4: Evolution of the peak stress with the cell size for the loading case 1 at a strain rate of ε˙ = 107s−1 obtained with the EAM potential using a Mg single crystal containing one nano-sized void of radiusR = 3.15 nm.Unit Cells 403 603 803 1003 1203 1403Atoms ×106 0.25 0.86 2.04 4.0 6.9 11σp [GPa] 2.35 2.45 2.5 2.55 2.55 2.53Figure 2.6: Evolution of the peak stress with the temperature for the loading case 1 at a strain rate ofε˙ = 107 s−1 obtained with the EAM potential using a Mg single crystal of dimension 38×68×68 nm3with approximately 7 million atoms containing one nano-sized void of radius R = 3.15 nm.NVE (constant-number of particles, constant-volume, and constant-energy) and NVT (constant-number of particles, constant-volume, and constant-temperature) are the most common choicesin MD simulations. It is often not clear what choice is the most appropriate in dynamic sim-ulations at high strain rate. In order to clarify uncertainties, simulations have been performedusing the NVE and NVT ensembles for a Mg single crystal of dimensions 38× 68× 68 nm3containing one nano-sized void of radius R = 3.15 nm with the loading case 1 at T = 300 K.Simulations at two strain rates have been carried out, i.e., ε˙ = 107 and ε˙ = 108 s−1. Fig. 2.7(a)shows a comparison of the evolution of the mean stress for both ensembles at two strain rates.It is evident that there are only small differences (differences in the peak stress are less than3%). Fig. 2.7(b) shows the temperature evolution for the simulations carried out under the NVE26ensemble. Remarkably, the simulations show an initial thermoelastic effect and a subsequenttemperature increase due to the plastic work.(a) Virial stress vs. strain.(b) Temperature vs. strain.Figure 2.7: a) Evolution of the mean hydrostatic stress (σm) vs. strain for the loading case 1 obtained withEAM potential under the NVE and NVT ensembles using a Mg single crystal of dimensions 38×68×68nm3 containing one nano-sized void of radius R = 3.15 nm at T = 300 K. Solid lines correspond to aload applied at ε˙ = 107 s−1 and dashed lines to a ε˙ = 108 s−1. b) Temperature evolution for the NVEensemble.27As shown in the previous comparison, the behaviour of the material does not change signif-icantly under the choice of the ensemble, but the thermal softening might be overestimated withthe NVE ensemble if the local plasticity is very large. Consequently, the NVT ensemble is arbi-trarily chosen to control the simulation temperature for the remaining of this work. Therefore,the analysis is simplified since the goal is to focus on the deformation mechanisms.2.3.2.5 Assessment of the interatomic potentials for the loading case 1Using the loading case 1, a quantitative comparison is performed for the two different potentialsused in this work for strain rates between ε˙ = 107−1010 s−1. The evolution of the virial stress vs.strain for two strain rates, i.e., ε˙ = 107 and ε˙ = 108 s−1 is illustrated in Fig. 2.8 for comparison.Fig. 2.8 shows that the predicted peak stress values reported in Table 2.3 for the two potentialsare between a range of 100 MPa. The prediction of the peak stress in such a narrow intervalis quite remarkable and is due to the fact that the mechanisms of deformation observed in thesimulations are independent of the interatomic potential. Notably, for the two potentials, thereis no strain hardening after dislocations are emitted.Figure 2.8: Evolution of the virial stress vs. strain for the loading case 1 obtained with the EAM andMEAM potentials using a Mg single crystal of dimensions 38×68×68 nm3 with approximately 7 millionatoms containing one nano-sized void of radius R = 3.15 nm at T = 300 K. Solid lines correspond to aload applied at ε˙ = 107 s−1 and dashed lines to a ε˙ = 108 s−1.Despite these similarities, some differences are also evident in Fig. 2.8 between the EAMand the MEAM potentials. On the one hand, the slope of the stress vs. strain plot is almostconstant for the EAM potential up to the peak value, when the specimen fails. This is a clearindication that the tensor of elastic constants is almost independent of the deformation state. Onthe other hand, the slope of the stress vs. strain plot is slowly reduced as deformation increases28for the MEAM potential. The reduction of the slope for the MEAM potential is due to a strongdependence of the tensor of elastic constants with the hydrostatic deformation. For instance,for a ε = 1.5% the elastic constants change approximately 11% for the MEAM potential whilefor the EAM potential the change is around 2%. This indicates a strong dependence of thetensor of elastic constants for the MEAM potential, therefore, explaining the change in theslope. This behaviour was systematically observed for all loading cases studied in this work.The reduction of the slope makes the MEAM potential more ductile reaching the peak stress ataround ε = 3.5% and ε = 4% for ε˙ = 107 and ε˙ = 108 s−1, respectively.With the complete study for the loading case 1 (Hydrostatic tensile load), the study of thefailure mechanisms under different loading conditions using Mg single crystals is developed.Similarly to the previous case, simulations with the two potentials will be performed however,the reference strain rate is kept to ε˙ = 108 s−1. The change is merely due to computationalefficiency since the simulations carried out at ε˙ = 107 s−1 require one order of magnitude moreof time steps. We expect that the inertia effects to have a weak dependence for moderate strainrates, as depicted in Fig. 2.2.2.3.3 Loading case 2The evolution of the averaged virial stress ((σxx +σzz)/2) and the Von Mises equivalent stress(σe) vs. strain for the loading case 2 (see Table 2.1) at a strain rate of ε˙ = 108 s−1 obtained withthe EAM potential is shown in Fig. 2.9. The evolution of the virial stress is now described.An initial linear elastic regime without dislocation emission is observed until the deformationreaches the critical deformation for dislocations emission, εc = 3.4%, at a critical stress of σc =2.75 GPa. After this point, the evolution of the virial stress experiences an increment due to thedislocation emission and interaction reaching approximately a peak value of σp = 2.85 GPa fora deformation of εp = 3.6%. Once the strain has reached εp, the stress drops and the specimeneventually fails due to the propagation of cracks. This loading case shows very little strainhardening. Similar to the hydrostatic tensile loading case, the slope of the virial stress vs. strainplot is almost constant for the EAM potentials while for the MEAM potential a decrease in theslope is observed as deformation increases. Remarkably, regardless the difference in the slope ofthe virial stress vs. strain plot, the predicted peak stress is in a narrow range of 50 MPa becausethe mechanisms of deformation are independent of interatomic potential.The interest is now dedicated to the evolution of σe for this loading case. Remarkably, themaximum value is σe = 1.5 GPa leading to a stress triaxiality χ = 1.5 during the linear regime.Once the nano-void starts the dislocation emission, χ goes from 1.5 to 4 in a very short period ofdeformation (ε = 3.4−4.5%). This can be easily seen in Fig. 2.9(b) where ρ has been plotted.Remarkably, ρ = 1014 1/m2 for ε = 3.4% and increases very fast to a maximum of ρ = 1016 1/m2for ε = 4%. Thereafter, the dislocation density remains more and less constant until the momentthat cracks are completely propagated in the simulation cell. Next, the dislocation emission forthe EAM potential is described. The dislocation emission for a deformation between the range29(a) Virial stress vs. strain.(b) Dislocation density vs. strain.Figure 2.9: a) Evolution of the averaged virial stress (σxx+σzz2 ) and Von Mises equivalent stress vs. strainfor the loading case 2 at a strain rate of ε˙ = 108 s−1 obtained with the EAM potential using a Mg singlecrystal of dimensions 38× 68× 68 nm3 with approximately 7 million atoms containing a nano-void ofradius R = 3.15 nm at T = 300 K. b) Evolution of the dislocation density.30of ε = 3.45% and ε = 3.825% is illustrated in Fig. 2.10. Once the strain reaches εc, the voidstarts the process of cavitation by dislocations emission and basal dislocations are emitted fromthe void surface (and are indicated with letter B in Fig. 2.10(a)). As deformation increases,pyramidal-II dislocations are nucleated and emitted from the tip of basal dislocations. Thesedislocations are illustrated in Fig. 2.10 (b) and (c) with the letters P-II and have mainly edgecomponents. Notably, the emission of pyramidal-II dislocations leads to a small increase in thestress due to strain hardening. When the strain is in the neighbourhood of ε = 3.825%, cracksare propagated along the pyramidal-II plane leading to the failure of the specimen. Fig. 2.10(d)shows the dislocations just before cracks propagate in the sample. Noticeably, the deformationmechanisms observed for this loading case coincide for the two potentials.It is also important to remark that this case also shows a brittle type failure with slightlymore strain hardening than the hydrostatic tensile load case. The small strain hardening is dueto the fact that mainly pyramidal-II edge dislocations are emitted and propagated through thesimulation cell. As pointed out in Table 2.2, the critical stress required to glide pyramidal-IIdislocations with edge components is of the order of 50 MPa, around 6 times less than thebarrier required for screw components. Therefore, since the pyramidal-II dislocations emittedhave edge components, they can glide more easily and assist to produce plastic deformationalong the [0001]−crystallographic direction leading to larger dislocation density and more strainhardening than in case 1.Dislocation velocities have been measured for the two different potentials used in this work.For a load applied at a strain rate of ε˙ = 108 s−1, the velocities of the dislocations along thepyramidal-II plane were 1245 m/s and 1345 m/s for the EAM and MEAM potentials, respec-tively. These values represent a dislocation velocity of about ten to twenty times faster thandislocations on the basal plane clearly illustrating the anisotropic response of the material.2.3.4 Loading case 3The evolution of the averaged virial stress ((σxx +σyy)/2) and the Von Mises equivalent stress(σe) vs. strain for the loading case 3 (see Table 2.1) at a strain rate of ε˙ = 108 s−1 obtainedwith the EAM potential is shown in Fig. 2.11. The peak stress is in the neighbourhood of σp =2.75 GPa and is reached at a deformation of approximately εp = 4.1% for the EAM potential.Fig. 2.11 also shows the evolution of σe for this loading case. The maximum σe value isapproximately 1 GPa leading to a stress triaxiality factor χ ≈ 1.5. Similarly to case 2, oncethe dislocations are emitted from the void surface, χ quickly increases to 5 indicating a largeamount of plastic deformation. It is also interesting to understand the evolution of ρ for thisloading case which is shown in Fig. 2.11(b). Initially, for a ε = 3% the dislocation density startswith a value of ρ = 1013 1/m2. After this point, ρ is increased very quickly, and at approximatelyε = 4.4% reaches its maximum of ρ = 5×1016 1/m2, a value that will be retained until the endof the simulation indicating a much longer period of plastic deformation in comparison with theprevious loading cases.31(a) ε = 3.45% (b) ε = 3.6%(c) ε = 3.75% (d) ε = 3.825%Figure 2.10: Dislocation emission for the loading case 2 obtained with the EAM potential. (a) Dislo-cations at ε = 3.45%. Basal edge dislocations are observed (labeled with letter B). (b) Dislocations atε = 3.6%. Pyramidal-II dislocations are emitted from the tip of basal edge dislocations (labeled with let-ter P-II). Dislocations at (c) ε = 3.75% and (d) ε = 3.825%. Pyramidal-II dislocations move faster thanbasal and prismatic dislocations. After the peak stress is reached, multiple cracks are propagated alongthe pyramidal-II plane leading to a drop in the virial stress vs. strain plot.Next, the dislocations emission obtained for the EAM potential are described. The time evo-lution of the dislocation emission for the strain range between ε = 3.375−3.825% is illustratedin Fig. 2.12. The first dislocation is emitted at around a deformation of εc = 3.375% and a stressof about σc = 2.1 GPa. Fig. 2.12(a) shows the dislocations at this level of deformation. Similarto previous cases, basal and prismatic dislocations as well as staking faults are observed.Since no external loads are applied in the [0001] direction the nano-void emits more 〈a〉dislocations leading to a more ductile behaviour than the previous two cases. This is illustratedin Fig. 2.12(b) and (c) which correspond to a deformation of ε = 3.45% and ε = 3.6%, respec-32(a) Virial stress vs. strain.(b) Dislocation density vs. strain.Figure 2.11: a) Evolution of the averaged virial stress (σxx+σyy2 ) and the Von Mises equivalent stress vs.strain for the loading case 3 at a strain rate of ε˙ = 108 s−1 obtained with the EAM potential using a Mgsingle crystal of dimensions 38×68×68 nm3 with approximately 7 million atoms containing a nano-voidof radius R = 3.15 nm at T = 300 K. b) Evolution of the dislocation density.33tively. At this level of deformation, partials lying on the basal plane (indicated with letter Bin Fig. 2.12(b)) interact with prismatic dislocations (indicated with letter P in Fig. 2.12(b)) onthe {0110} plane creating a square-shape Prismatic Dislocation Loop (PDL) that is completelydetached from the void surface (see Fig. 2.12(c)) and travel along the [2110] direction. Even-tually, the same mechanism is repeated in four directions leading to an interaction of the PDLsas is shown in Fig. 2.12(d).(a) ε = 3.375% (b) ε = 3.45%(c) ε = 3.6% (d) ε = 3.75%Figure 2.12: Emission of PDL from the void surface for the loading case 3 at a strain rate of ε˙ = 108s−1 obtained with the EAM potential using a Mg single crystal of dimensions 38× 68× 68 nm3 withapproximately 7 million atoms containing a nano-void of radius R= 3.15 nm at T= 300 K. a) ε = 3.375%,basal (labeled with letter B) and prismatic (labeled with letter P) dislocations are emitted. b) ε = 3.45%,dislocations partially detached from the void surface. c) ε = 3.6%, first PDL completely detached fromthe void surface and d) ε = 3.75%, PDLs emitted in [21 10] directions.The geometry and dislocations that generate the PDLs are shown in Fig. 2.13 for two loopstraveling in the [2110] and [1210], respectively. The emission of PDLs along with the inter-action and annihilation promotes a short strain hardening period leading to a small increase in34the virial stress. As a consequence, the peak stress is about σp = 2.75 GPa reached at aboutεp = 4.05%. After this point, the virial stress vs. strain plot drops due to several cracks propa-gating through the crystal.(a) (b)Figure 2.13: Square shape PDLs emitted from the void surface along the a) [2110] and b) [1210]directions. The PDLs have a Burgers vector of magnitude 〈a〉 and are generated with two partials(1/3[1010] + 1/3[1100]) separated by a stacking fault in the basal plane and two prismatic dislocations{1}/3[2−1−10](1010).The behaviour observed for the MEAM potentials coincides with the one previously de-scribed. As pointed out in the previous cases, MEAM potential predicts a reduction of the slopein the stress vs. strain plot, leading to a delay of the initiation of plasticity. Nevertheless, themain aspects coincide with the EAM potential and the peak stress values are in close agreementfor all two potentials.Dislocation velocities have been measured for this loading condition. For a load applied astrain rate of ε˙ = 108 s−1, the dislocation velocities of the PDLs were 487 m/s and 259 m/s for theEAM and MEAM potentials, respectively. Notably, the aforementioned dislocation velocitiescorrespond to basal and prismatic 〈a〉 dislocations, which are the same as in the loading case 1.For the loading case 3, however, the velocity achieved by the dislocation loops is at least oneorder of magnitude faster than in the loading case 1 since they are completely detached from thevoid surface.2.3.5 Loading case 4The response of the material for the loading case 4 (uniaxial load applied along the [2110] di-rection) is now presented. Fig. 2.14(a) shows the evolution of σxx and the Von Mises equivalentstress (σe) vs. strain for the loading case 4 (see Table 2.1) at a strain rate of ε˙ = 108 s−1 obtainedwith the EAM potential. Let us now describe the evolution of the virial stress σxx for the EAMpotential. The linear elastic regime extends up to the peak value, σp = 2.45 GPa at εp = 5.15%of deformation. After reaching the peak value, the virial stress drops suddenly and a plastic35regime where the material emits dislocations is followed. This is indicated in Fig. 2.14 with thesawtooth profile observed after the peak value has been reached. This period extends approxi-mately from 5.15% to 6.5% for the EAM potential. During this period, the void emits multipledislocations leading to a ductile behaviour of the material promoting strain hardening. Finally,at around ε = 7%, the sample fails due to the emission of several cracks propagating throughthe material. Fig. 2.14(a) also shows the evolution of σe with the deformation. σe mimics thesame behaviour than σx and the maximum value for σe ≈ 1.25 GPa. For this loading case, χ ≈ 1during the linear regime and increases to χ ≈ 2 when the peak stress is reached. Fig. 2.14(b)shows the evolution of ρ as a function of the deformation. Remarkably, ρ = 1013 1/m2 for aε = 4.6% indicating the emission of dislocations. After this point, ρ increases up to a maximumvalue of ρ = 1016 1/m2, the moment where the peak stress is reached. Afterwards, ρ remainsapproximately constant as indicated in Fig. 2.14(b). For the MEAM potential, a very similarbehaviour is observed.Let us now describe the dislocation emission from the void surface obtained with the EAMinteratomic potential. Fig. 2.15 shows the emission of dislocations between ε = 5.125−5.135%for a strain rate of ε˙ = 108 s−1. The Burgers vector of the emitted loops is aligned with the[1210]-direction which is at sixty degrees with respect to the load direction. This slip system is,therefore, the one with a larger Schmidt factor since the maximum shear stresses are at forty-fivedegree with respect to the [2110]. Fig. 2.15 indicates that the loop is a prismatic dislocationwith both edge (labeled with the letter P) and screw (labeled with the letter S) components. Asthe stress increases with the deformation, the loop reaches a critical point where spontaneouslybows out increasing its size multiple times (see Fig. 2.15(b)). The observed mechanism hastwo stages, an initial dislocation emission from the void surface followed by a second stageof where the dislocation bows out and spontaneous detaches from the void. It is important toremark the resemblance of the observed mechanisms with the Frank-Read source mechanism ofdislocations multiplication. In this case, however, the nano-void acts as the nucleation sourceand obstacle at the same time. The detachment from the void occurs due to the fact that only oneprismatic 〈a〉 dislocation is emitted. Thus, the dislocation does not find any resistance to growthand eventually, it reaches a critical point where completely detaches from the void surface. InFCC or BCC materials this mechanism is not observed since many dislocations are emitted indifferent slip systems even under uniaxial deformation. Once the loop is completely detachedfrom the void surface, the dislocation velocity is around vd = 2200 m/sec. After the loop isdetached, the mechanism is repeated again, increasing ρ until it reaches the maximum value ofρ = 1016 1/m2.Fig. 2.16 shows the dislocations for a strain in the range of ε = 5.25− 6.125% whichcorrespond to the maximum value of ρ . At this stage, we have observed basal and prismaticdislocations. This is due to the fact that at T = 300 K, prismatic dislocation with screw compo-nents can spontaneously cross slip to the basal plane. This is consistent with our result obtainedfor single dislocation set up.In order to gain more insights into the aforementioned mechanism of dislocation detach-36(a) Virial stress vs. strain.(b) Dislocation density vs. strain.Figure 2.14: a) Evolution of the averaged virial stress (σxx) and the Von Mises equivalent stress vs. strainfor the loading case 4 at a strain rate of ε˙ = 108 s−1 obtained with the EAM potential using a Mg singlecrystal of dimensions 38× 68× 68 nm3 with approximately 7 million atoms containing a nano-void ofradius R = 3.15 nm at T = 300 K. b) Evolution of the dislocation density.ment from the void surface, we have carried out simulations to better understand and quantifythe critical stress for bowing out and detaching a prismatic 〈a〉 dislocation loop in Mg. Using37(a) ε = 5.125% (b) ε = 5.13%Figure 2.15: Dislocation emission from the void surface for the loading case 4 at a strain rate of ε˙ = 108s−1 obtained with the EAM potential using a Mg single crystal of dimensions 38× 68× 68 nm3 withapproximately 7 million atoms containing a nano-void of radius R = 3.15 nm at T = 300 K. Dislocationsemitted from the void surface at a) ε = 5.125%, prismatic dislocation loop is generated by screw (labeledwith letter S) and prismatic edge (labeled with letter P) segments, b) Loop at ε = 5.13% . The loop isbowed out by the Frank-Read mechanism increasing its size multiple times.the simulation set up explained in Section 2.2.3, we have carried out a number of simulationswith different void radius R separated by a distance L. Then, a shear deformation was applied onthe plane normal to the y−axis along the z−direction at a strain rate of ε˙yz = 108 s−1. The initialdislocation geometry is shown in Fig. 2.17(a) where the two cylindrical voids can be seen. Asdeformation increases, the dislocation loop starts to bow out, as depicted in Fig. 2.17(b). Whenthe shear stress is large enough, the loop fully detaches from the obstacles -voids- and the mech-anism is repeated again. Similarly, to the study of dislocation barrier, the screw component ofthe prismatic loop spontaneously crosses slip to basal plane, splitting into two partials separatedwith a stacking fault. Nevertheless, the loop is able to fully detach independently of the type ofdislocations involved. Fig. 2.18 shows the evolution of the virial stress with the deformation forvarious combination of R and L obtained with the described set up.The prediction of the critical stress for the Frank-Read source mechanism obtained with theanalytic model proposed by Scatergood and Bacon [67]τSBcr =µb2piL[ln(r0D+r0L)−1+1.52](2.7)is also illustrated in Fig. 2.18 for the same combinations of R and L used in MD simulations.38(a) ε = 5.25%(b) ε = 6.125%Figure 2.16: Dislocation emission from the void surface for the loading case 4 at a strain rate of ε˙ = 108s−1 obtained with the EAM potential at range of strain of ε = 5.25− 6.125% using a Mg single crystalof dimensions 38× 68× 68 nm3 with approximately 7 million atoms containing a nano-void of radiusR = 3.15 nm at T = 300 K. The observed mechanism rapidly increases dislocations density making astrain hardening indicated by the plateau observed in the equivalent Von Mises stress after the peak stressis reached.39(a) Initial dislocation loop. (b) Loop before detachment.(c) Loop fully detached.Figure 2.17: Frank-Read source mechanisms in Mg for a basal prismatic dislocation a0[1010]{0001}.Different snap shots of the dislocation bowing out from the obstacles (voids) showing a very similarbehaviour than the one observed in void simulations under the loading case 4.In Eq. 2.7, µ = 17 GPa is the shear modulus of the material, b = 0.319 nm the magnitude ofthe Burgers vector, L the distance between the two source points, D is the diameter of the voidsand r0 is the dislocation core radius (usually r0 = b). Taking L/b = 22, Eq. 2.7 gives τSBcr = 260MPa which agrees well with the value of τMDcr = 264 MPa obtained with MD. However, as oneincreases the ratio L/b, the critical values predicted by Eq. 2.7 are below the values obtainedwith MD with errors that can be of the order of 80%.From Fig. 2.14, it is easy to see that the deviatoric part of the stress denoted by σe exceedsthe critical value for dislocation bow out (τcr ≈ 260 MPa) and therefore, explaining why theloop detaches very quickly from the nano-void surface.40Figure 2.18: Critical resolved shear stress for bowing out a prismatic 〈a〉 dislocation from the two cylin-ders using in the set up. The symbols correspond to MD simulations carried out at T = 300 K and a strainrate of ε˙ = 108 s−1 with the EAM potential. R/b = 0, 4, 8 correspond to different radius selected tostudy the Frank-Read source. The lines correspond to Eq. 2.7 for different values of R and L.2.3.6 Loading case 5Fig. 2.19(a) shows the evolution of σzz and the Von Mises equivalent stress (σe) vs. strain for theloading case 5 (see Table 2.1) at a strain rate of ε˙ = 108 s−1 obtained with the EAM potential.Let us now describe the evolution of the virial stress σzz. An initial linear regime is observeduntil the virial stress reaches the peak stress in the neighbourhood of σp = 3 GPa. After reachingthe peak stress the specimen has shown a secondary regime of plastic deformation that extendsapproximately from ε = 3.8% up to ε = 5%. According to this period of plastic deformation,a strain hardening is observed in the virial stress vs. strain plot. Similarly to previous cases, σeshows approximately the same evolution than σzz. The maximum value is σe = 2.3 GPa reachedapproximately at ε = 3.8%. The stress triaxiality is χ = 0.5 during the elastic regime and risesto χ = 3.6 when dislocations are emitted. Fig. 2.19(b) shows the evolution of ρ with the strain.Similarly to the previous cases, ρ increases once the dislocation has been emitted and reaches aconstant value in the neighbourhood of ρ = 5×1015 1/m2. Similarly to other loading cases, thebehaviour observed for the MEAM potential is very similar to the EAM potential.We now turn our attention to the dislocation emission and interaction for the EAM potential.The time evolution of the dislocations emitted for this loading case is shown in Fig. 2.20.The increase in the peak stress in comparison with other loading cases is attributed to the factthat the deformation is aligned with the [0001] direction and pyramidal-II (〈c+ a〉) dislocation41(a) Virial stress vs. strain.(b) Dislocation density vs. strain.Figure 2.19: a) Evolution of the virial stress σzz and Von Mises equivalent stress vs. strain for theloading case 5 at a strain rate of ε˙ = 108 s−1 obtained with the EAM potential using a Mg single crystalof dimensions 38× 68× 68 nm3 with approximately 7 million atoms containing a nano-void of radiusR = 3.15 nm at T = 300 K. b) Evolution of the dislocation density.shear loops are emitted. Pyramidal-II dislocations travel in a corrugated plane as described ina previous work of Li and Ma [68]. These dislocations have mainly edge components, and therequired shear stress needed to produce dislocation glide is around 50 MPa (See Table 2.2).Since the deviatoric part of the stress (σe) exceeds this value, the pyramidal-II dislocation cantravel very quickly far away from the void. However, when ε = 5%, σe decreases considerablydue to inelastic effects and the driven force to glide dislocation decreases. At this point, cracksare nucleated promoting the failure of the specimen.For a strain rate of ε˙ = 108 s−1, the velocity of the dislocations along the basal plane were135 m/s and 123 m/s for the EAM and MEAM potentials, respectively. Along the pyramidal-II42(a) ε = 4.2% (b) ε = 4.275%(c) ε = 4.35%Figure 2.20: Dislocation emission from the void surface for the loading case 5 at a strain rate of ε˙ = 108s−1 obtained with the EAM potential at range of strain of ε = 4.2− 4.35% using a Mg single crystalof dimensions 38× 68× 68 nm3 with approximately 7 million atoms containing a nano-void of radiusR= 3.15 nm at T = 300 K. a) Dislocation emission at ε = 4.2%, basal edge and pyramidal-II dislocationsare shown. b) Dislocation emission at ε = 4.275%, pyramidal-II dislocations travel very fast throughthe sample reaching the end of the simulation cell. c) Dislocation emission at ε = 4.35%, pyramidal-IIdislocations interact with their periodic replica promoting strain hardening in Fig. 2.19.slip system, the dislocation velocities were 895 m/s and 946.5 m/s for the EAM and MEAM po-tentials, respectively. This clearly illustrates the anisotropic response of the HCP lattice structureand the challenges associated with this type of materials.432.4 DiscussionAfter describing the results for different loading cases, we proceed to discuss the most impor-tant outcomes obtained in this work (See Table 2.5 for comparison between EAM and MEAMpotential). We start the discussion with the transition failure mode observed in the simulations.Then, the fidelity of the mechanisms obtained with the two different interatomic potentials isanalyzed followed by other key insights learned in this work.Table 2.5: Summary of most relevant data obtained in this work for different loading cases. σp is theaveraged peak stress in GPa, εp are the values of deformation for the peak stress. The strain values arereported in %. Dislocation density values correspond to the maximum value during the whole simulationin [×1016 1/m2]. Dislocation velocities reported in m/s. All values reported in this table were obtainedfrom simulations carried out at ε˙ = 108 s−1 with the EAM potential. Values in ( ) correspond to theMEAM potential.Load Failure σp [GPa] εp [%] χ = σmσe ρmax vd [m/sec]Case 1 Brittle 2.88 (2.85) 3.10 (3.52) 6.0 0.03 46.2 (42.5)Case 2 Brittle 2.85 (2.90) 3.60 (4.85) 1.5 1.0 1245 (1345)Case 3 Ductile 2.75 (2.55) 4.10 (5.20) 1.5 2.5 487 (259)Case 4 Ductile 2.45 (2.63) 5.15 (5.60) 1.0 2.0 2200 (2159)Case 5 Ductile 3.00 (2.90) 3.80 (5.40) 0.5 0.5 895 (946)Our simulations showed a transition in the failure mode of Mg single crystals specimens con-taining nano-voids under different loading conditions. For the loading case 1 (Hydrostatic ten-sile load), our simulations showed a brittle behaviour of the material with the smaller amount ofdislocation density among all loading cases studied in this work (see Fig. 2.3). Under this load-ing case, the nano-void emitted basal, prismatic and pyramidal-II dislocations. However, crackswere nucleated right after dislocations were emitted. This is due to the fact that pyramidal-IIdislocations require a large critical resolved shear stress to glide as indicated in Table 2.2. Inparticular, screw components require an exceedingly large shear stress to glide, of the order of300 MPa. While this shear stress might be available near the void surface -see σe in Fig. 2.2-,the stresses decay with the distance to the void, indeed at ∼ 1/r2. As pyramidal-II disloca-tions move away from the void surface, the driving force needed to promote glide decays veryquickly until the stress is below the critical resolved shear stress. Thus, pyramidal-II disloca-tions stop their motion, generating stress concentration, nucleation, and propagation of cracksin the simulation cell.Unlike other materials like FCC or BCC where strain hardening plays a major role in theevolution of the virial stress and its increment after cavitation [45, 53, 51], Mg does not showsecondary strain-hardening near the nano-void for hydrostatic tensile loads and the maximumstrain and stress that can afford before failing is very limited. The lack of strain hardening isunusual in nano-void simulations and one of the main observations of this work. Due to the44lack of strain hardening, the material shows a brittle failure, indicating that the capacity of thematerial to emit dislocations and generate plastic deformation is small. This observation -whichis independent of the choice of the interatomic potential- is of fundamental importance in thedesign of new lightweight Mg-alloys with optimal dynamic resistance since, as suggested bythe simulations, the spall strength of pure Mg is very low due to this phenomenon.Additionally, using the loading case 1, systematic studies were performed in order to under-stand the effect of the strain rate, cell size, void size, temperature effect and ensemble choice.Our study provides valuable data that can be used in atomic-informed continuum spall modelsfor Mg.On the other hand, for the loading case 4, a more ductile behaviour was observed. Underthis loading case, the nano-void emitted a large amount of dislocations thanks to a mechanismwhich is similar to the Frank-Read source. This mechanism generates a large amount of 〈a〉dislocations that interact and generate a period of plastic deformation with strain hardening (in-dicated by a plateau in Fig. 2.14). This high contrast between the mechanisms of deformationunder different loading conditions is a clear evidence of the anisotropic demeanour of the mate-rial and clearly manifests a transition between brittle to ductile behaviour. The brittle behaviourwas observed when the stress triaxiality χ was large while for smaller values of χ (i.e., χ ≈ 1)a more ductile behaviour was observed.Other loading cases showed interesting behaviours. In particular, case 3 showed the emis-sion of square PDLs. These loops were emitted along the [2110] crystallographic directions.PDLs have been experimentally observed in others HCP materials using HR-TEM (i.e., Zirco-nium [69] and references therein) but have never been observed nor predicted in Mg to the bestof our knowledge. This rich variety of mechanisms of deformation observed in Mg specimenscontaining nano-sized voids is noteworthy and illustrates the difficulties to develop continuumscale models that can accurately predict deformation and failure in Mg retaining the full levelof details observed in atomic simulations.To better understand the transition from brittle to ductile failure, we compare the evolution ofthe dislocation density for all loading cases studied in this work. Fig. 2.21 shows the evolution ofthe dislocation density as computed by Eqn. 2.5 with respect to the normalized strain computedas λ = ε/ε f , where ε f is the strain to which the computational cell emits the first crack. Fig.2.21 shows remarkable differences in the evolution of the dislocation density, ρ , for the differentloading cases that are evident for their high contrast. For instance, the case 1 (hydrostatic tensileload) and the case 2 show a very small value of dislocation density, on the order of ρ = 10141/m2. This is not surprising at all since both cases showed a brittle type failure with a verysmall dislocation emission and interaction. Moreover, the critical point for dislocation emissionoccurs approximately at εc ≈ 0.85ε f , which indicates that the first dislocation emission happensvery close to the failure of the material.On the contrary, cases 3, 4 and 5 showed a more ductile behaviour with a gradual incrementof the dislocation density. For instance, the case 3 shows a more and less linear increment ofρ in the plot between λ = 0.7 to λ = 0.9. This period is characterized by the emission of45Figure 2.21: Dislocation density vs. normalized strain λ = εε f for all loading cases studied in this work.The stress triaxiality factor χi corresponds to the linear regime for the specific loading cases. The dashedline indicates a transition in the behaviour of the material from brittle to ductile.shear loops from the void surface shown in Fig. 2.13(a) and (b). After λ ≥ 0.9, ρ experiencesa sudden increment due to the detachment of the PDLs. The maximum value of dislocationdensity is ρ = 2.5×1016 1/m2. This value is approximately twice than peak value for the case 2and around two orders of magnitude larger than the maximum dislocation density for the case 1(hydrostatic tensile load). Similarly, cases 4 and 5 showed values of ρ that are in the same orderof magnitude that case 3.The evolution of dislocation density shown in Fig. 2.21 manifests large differences in theplastic behaviour of the material under different loading conditions. Remarkably, Fig. 2.21shows a clear gap between cases 1 and 2, and cases 3, 4 and 5, highlighted with the dashedline in the plot. Once again, this is an evidence of the transition from brittle to ductile for thedifferent loading cases.The fidelity of the mechanisms of deformation was also investigated using two differentinteratomic potentials. Table 2.5 shows the most relevant data obtained in this work for theEAM and MEAM potentials. Remarkably, the peak stress values, dislocation velocities and themechanisms of deformation obtained were in close agreement for the two potentials. It wasobserved, however, that the MEAM potential consistently predicted a much softer behaviour-denoted by the slope of the virial stress vs. strain curve- leading to values of εc and εp largerthan for the EAM potential. This is due to the fact that the tensor of elastic constants stronglydepends on the deformation. Our study indicated that the peak values, dislocation emission,46and void growth are very similar between the two tested potentials. Therefore, for systematicstudies of temperature, cell size and void effects, the potential proposed by Sun et al. [28] wasutilized to determine peak values and mechanisms of deformation since it is computationallymore efficient.47Chapter 3Dynamic behaviour of nano-voids inTitanium and ZirconiumAbstractWe study the dynamic behaviour of nano-sized voids in two different HCP materials, Titanium(Ti) and Zirconium (Zr), subject to hydrostatic tensile loads by means of molecular dynamicssimulations. Surprisingly, we find that these materials have a ductile failure when subject to hy-drostatic tensile loads, in high contrast with the brittle failure observed in previous chapter fornano-sized voids in Magnesium (Mg). This difference is due to the fact that nano-sized voidsin Ti and Zr emit substantially more dislocations than Mg, allowing for large displacements ofthe atoms and plastic work, including non-basal planes. We postulate that this opposite failurebehaviour of Mg in comparison to other HCP materials is due to a competition between dislo-cation emission in the basal plane and crack propagation in non-basal planes. This competitionbetween two different mechanisms of deformation explains why Mg fails under brittle mode,while Ti and Zr fail under ductile mode. Thus, we propose to use the ratio between unstablestacking fault energy and surface energy in these materials to assess the tendency of HCP mate-rials and alloys to fail under brittle or ductile modes. Using this ratio, we critically identify thelow surface energy of Mg as responsible for this brittle behaviour and recommend that Mg-basedalloys with large surface energies can lead to better performance for dynamic applications.3.1 IntroductionMetallic alloys play an important role in the design of structural components due to their manydesirable mechanical properties such as high strength, large plastic deformation, resistance tocorrosion and fatigue life, among others. However, the need to develop more sustainable andenergy efficient structures, in particular in transportation vehicles, is pushing scientist and en-gineers to use less traditional design materials, such as polymers, composites and last but notleast, less conventional alloys such as those based on hexagonal close-packed materials (HCP).Among all these choices, HCP-based alloys offer a wide spectrum of interesting mechanicalproperties which are in many cases superior to traditional metallic materials based on FCC orBCC materials. For instance, Magnesium (Mg), Titanium (Ti) and Zirconium (Zr), are three48HCP materials that have been used for structural applications in many applications, such aslightweight structures in transportation vehicles and airplanes, human implants, nuclear powerplants [7, 8, 13, 70, 71, 72, 73, 74, 75]. Yet, besides their many appealing mechanical prop-erties, HCP materials are generally characterized by being less ductile than traditional metallicmaterials, such as Aluminum (Al) and Iron (Fe) based alloys. Generally speaking, HCP ma-terials suffer from shear localization and deformation locking mechanisms that generate stressconcentration, crack nucleation and propagation and, eventually, the catastrophic failure of com-ponents. This behaviour is less understood and thus, hinders widespread industrial utilization ofHCP alloys due to safety reasons among other factors.The low ductility of HCP materials is related to their atomic lattice structure. In the HCPlattice, only four independent slip systems are available to generate plastic deformation, in con-trast with the five independent slip systems available in FCC and BCC materials [32]. Thisrestriction is also accompanied by the fact that the HCP unit cell is not cubic, and the length ofthe Burgers vectors changes depending on the slip system considered. Thus, while it is relativelyeasy to activate the basal and the prismatic slip system in all HCP materials, the pyramidal slipsystems (in its two variant, pyramidal-I, and pyramidal-II) are more rarely activated due to factthat the energy required to nucleate dislocation and move them along this slip system is muchlarger than the basal or prismatic slip planes. This anisotropic behaviour, which is inherent tothe HCP lattice, favours deformation in the basal or prismatic plane but makes more difficultthe plastic deformation in any other slip system with [0001] components [33, 34, 35, 36]. Asa result, HCP materials circumvent this difficulty by allowing deformation by twinning, whichis more infrequently seen in FCC or BCC materials. Deformation by twinning is by far lessunderstood; complicating the basic understanding of ductile failure in HCP materials [32, 36].Indeed, while two twinning modes are widely accepted to exist, recent studies reveal that thedeformation by twinning is more complex and rich than previously thought [76].The need for developing Mg alloys with superior mechanical properties have attracted theattention of many scientists in material science and several works were dedicated to character-ize the mechanical properties of Mg-alloys and to understand the pyramidal-I and II dislocationin pure Mg providing valuable information. For instance, in order to increase the ductility ofHCP materials, many alloys have been developed, especially in the case of Mg-based alloys.Sandlo¨bes et al. [77] have studied the effect of rare-earth elements alloys in Mg empiricallyand numerically with ab-initio simulations. The general trend observed in experiments is thatthe alloys become more ductile than the pure material, at the cost of reducing the yield strength.Additionally, the simulations performed by Sandlo¨bes et al. [77] showed that the stacking faultenergy is significantly reduced. More recently, Sandlo¨bes et al. [78] developed new Mg-alloysusing Al and Calcium (Ca) that show good ductility while retaining a high yield strength. Simi-larly to previous studies, the main observation from atomic simulations is that the stacking faultenergy in the basal plane is considerably decreased. Unfortunately, the fact that the SFE de-creases in ductile Mg-alloys does not explains the increment of plastic deformation since basaldislocations do not generate non-basal deformation. However, it is expected that some corre-49lations between basal and non-basal deformations exist, that are hidden in the mechanisms ofdeformation of such alloys.In order to understand the dynamic failure of Mg, Ponga et al. [55] and Gre´goire et al.[1] studied the response of nano-voids under several loading conditions and strain rates. Thesestudies have revealed that indeed Mg has a brittle failure when subject to hydrostatic tensileload. However, when subject to loads with low-stress triaxiality, this behaviour is removed andthe material becomes more ductile. Thus, a question of remarkable interest is to understand ifthis transition from brittle to ductile is inherent to HCP materials. In this work, we endeavour toanswer that question by carrying out a systematic comparison between Mg and other two com-mon structural HCP materials, i.e., Ti and Zr. Surprisingly, we find out that both Ti and Zr havea more ductile behaviour when subject to hydrostatic tensile load, independently of having thesimilar atomic structure, and Peierls barriers for dislocation glide. In order to understand thisunexpected result, we carefully examine the surface energy, the crack propagation behaviour inthese three materials, and perform a simple two-dimensional analytical analysis of the stressesnear the void. Interestingly, we found that the brittle behaviour in Mg is due to a low surfaceenergy that leads to unstable cracks propagation from the nano-sized voids. In order to charac-terize the tendency to fail under brittle mode, we propose to use a very simple scalar parameterthat takes into account the propensity of the material to either emit dislocations and emit cracks.This new finding, which has not been identified before as far as we are concerned, explains thebrittle behaviour of Mg specimens and suggest new avenues to develop ultra light Mg-basedalloys for optimal dynamic behaviour.The chapter is organized as follows. Section 3.2 explains briefly the methodology used in thesimulations carried out in this work. The procedure is based on Gre´goire and Ponga previouswork [1] used to investigate the dynamic behaviour of nano-voids in Mg. For all the threeelements Mg, Ti and Zr, the dislocation barriers, the mean hydrostatic stress, the dislocationdensity, the dislocation emission as well as the stacking fault energy and surface energy arecomputed in Section 3.3. The analysis of the effect of void size, void shape, temperature, andstrain rate complements the understanding. Section 3.4 presents a discussion of the main resultsobtained in this chapter.3.2 Methodology3.2.1 Nano-void simulationsThe dynamic behaviour of nano-voids under hydrostatic tensile load in Ti and Zr single crystalsis investigated by means of molecular dynamic (MD) simulations. All simulations are per-formed with the large-scale atomic/molecular massively parallel simulator (LAMMPS) code[62]. These numerical simulations are performed at a strain rate ranging from ε˙ = 105−108s−1.We systematically study the dislocation Peierls stress, stress vs. strain relation, dislocation den-sity, and dislocation emissions during nano-void cavitation. The atomic interactions are simu-50lated by means of the Embedded Atom Method (EAM) interatomic potential. In particular, weuse the potentials developed by Ackland [29] and Mendelev et al. [30] to model Ti, and Zr,respectively.Nano-void simulations are carried out in a single HCP crystal. The setup is the same used inour previous work [1] in order to make quantitative comparisons between three different HCPmaterials. These single crystals are generated with a computational cell of dimensions lx× ly× lz,where lx = 120a0, ly = 120√3a0 and lz = 120c0, where a0 and c0 are the lattice constant on thebasal plane and in the [0001]-direction, respectively. c0a0 =√8/3 is the theoretical ratio betweenthe lattice constants in the [0001]-direction and the basal plane at T = 0 K. These externaldimensions correspond to a selection of 1203 unit cells containing approximately 7 millionatoms. The crystals are constructed in such a way that the crystallographic directions [2110],[0110] and [0001] of the HCP lattice structure are aligned with the x−, y− and z− axes ofthe computational box, respectively. Then, an initial void of radius R is generated by removingatoms from the centre of the crystal. Unless otherwise specified, R = 10a0. The parameters a0and c0 are dependent on the metallic element, and are reported in Table 3.2. Periodic boundaryconditions are used in all simulations containing nano-voids and all simulations are carried outusing the canonical ensemble (NVT).At each time step, a homogeneous deformation is applied to the atoms in the simulation cellusing a homogeneous deformation gradient F ∈ R3×3 that produces a hydrostatic tensile stressstate, i.e.,F =1+λ 0 00 1+λ 00 0 1+λ , (3.1)where λ is a deformation parameter that is adjusted to achieve the desired strain rate. Thestrength is quantified with the virial stress (Eqn. 2.2), and the reported values correspond tothe mean hydrostatic stress (σm = 1/3(σxx +σyy +σzz)) [63] unless otherwise specified. Otherparameters are also reported, such as dislocation density (ρ) Eqn. 2.5, extracted with the Dislo-cation eXtraction Algorithm method and visualized with OVITO [24, 25]. Void volume fraction(v f ), defined as the ratio between the void volume (Vv) and the volume of the simulation cell (V )and shape are reported by using a Delaunay triangulation over the atoms in the surface of thevoid.3.2.2 Dislocation simulationsIn order to understand the anisotropy in the dislocation glide of HCP materials, we compute thePeierls barrier for edge dislocations in Ti and Zr and compare these values with our previouswork [1]. Edge components in the basal, prismatic and pyramidal-II slip systems have beensimulated following the same technique in [1]. The edge dislocations are generated with Oset-sky and Bacon procedure [64]. The computational cell has approximately 300,000 atoms and51considers the long-range elastic effect of the dislocations. The strain controlled displacement isapplied by prescribing a constant velocity vx along the x− direction to all atoms in the top layerand−vx to all atoms in the bottom layer and the shear deformation is illustrated in Fig. 3.1. Thethickness of these layers is set up to be around 5a0. The maximum shear stress σzx is reportedas the critical stress required to move the dislocation.Skin LayerSkin LayerSkin LayerSkin LayerxyTTSkin LayerSkin LayerSkin LayerSkin LayerxyTTFigure 3.1: Schematic illustration of the shear deformation for the dislocation simulations. The pictureshows the dislocation core, illustrated with a T, and two-skin layers used to apply the shear deforma-tion through an imposed constant velocity. The result is a motion of the atoms that simulates a sheardeformation.3.2.3 Stacking fault and surface calculationsA key parameter that defines the material’s ability to emit dislocations on the basal plane is thestacking fault energy (SFE), which can be directly linked to the amount of energy needed toovercome to perform slip in a material. Thus, the SFE becomes a central parameter that will de-termine the propensity of the material to emit basal dislocations and whether these dislocationswill be dissociated or not [32]. Thus, SFE calculations are carried out using Molecular Statics(MS) simulations for Mg, Ti, and Zr. The computational cell consists of a single crystal withthe [2110] and [1010] crystallographic directions of the HCP lattice structure aligned with thex− and y− axis of the cell. The simulation cell was taken to have the following dimensionslx× ly× lz, where lx = 100a0, ly = 100√3a0 and lz = 40c0 with approximately 60,000 atoms.We are interested in obtaining two main parameters, the stable and unstable stacking fault en-ergy γs(0001) and γu(0001), respectively. These values are obtained by moving the upper half ofthe simulation cell along the [1010]− direction. γs(0001) correspond to the local minima in thissurface plane while γu(0001) correspond to the local maximum on this surface.Similarly, we perform surface energy calculations for specific crystallographic directions.The goal of these simulations is to provide insights into the ability of the material to emit cracksin a brittle fashion.Both stacking fault energy and surface energy are computed as the excess of energy in the52sample with respect to the reference crystalline structure per unit of area, i.e.,γi =Edefect(N)−NEcoh2A, (3.2)where γi represents the SFE or surface energy, Edefect(N) is the energy in the sample containingN−atoms with the stacking fault or surface, Ecoh is the cohesive energy per atom in the referencecrystalline structure, and A is the cross-sectional area of the simulation cell. The factor of twoappears to take into account that there are two surfaces generated. The values are reported inTable 3.2.3.3 Results3.3.1 Dislocation barriersWe now analyze the barriers for dislocation glide in the basal, prismatic and pyramidal-II slipsystems for edge dislocations, and the values of the critical resolve shear stress needed to pro-mote dislocation glide the corresponding slip systems of relevance for the HCP lattice structure.Previous results focusing on Mg have shown a very small dislocation barrier for both basal andprismatic plane, whereas the same components in pyramidal-II are at least one order of magni-tude higher [1].Slip system Edge Ti Edge Zr Edge MgBasal [1120]{0001} 15 3.5 3Prism [1120]{1010} 3.5 6 25Pyramidal-II [1123]{1122} 195 300 50Table 3.1: Values of the critical resolve shear stress needed to promote dislocation glide in different slipsystems of relevance for Ti and Zr at T = 300 K. All reported values are in MPa at a strain rate of ε˙ = 108s−1 and compared to Mg obtained from Gre´goire and Ponga work [1].Table 3.1 shows the values of the Peierls barrier for edge dislocations in HCP materialsfor different slip systems. Interestingly, both Ti and Zr have an anisotropic behaviour whichfollows the same trend than Mg. For instance, Table 3.1 shows that the Peierls barrier for basaland prismatic system is relatively low for all three materials, while the pyramidal-II slip systemis about one or two orders of magnitude higher than the basal and prismatic slip system. Thisanalysis shows that HCP materials favours basal deformation, and the barriers for non-basaldeformation are unusually high.533.3.2 Stacking fault energyThe SFE is computed in the basal plane, where the displacement deformation is performed on[1010]−direction for the three metallic elements Mg, Ti, and Zr. Fig. 3.2 plots the SFE of thematerials with respect to the normalized displacement deformation. Two important points are ofinterest while applying the movement. On the one hand, the square marker represents the stableSFE γs[0001] with numerical values of 30.6, 7.0, and 36.5 mJ·m−2 for Mg, Ti, and Zr, respectively.On the other hand, the circle marker represents the unstable SFE γu(0001) with numerical valuesof 46.5, 31.9, and 60.3 mJ·m−2 for Mg, Ti, and Zr, respectively. For dislocation emission, wenow focus on the unstable SFE, that is the key parameter to emit dislocations. Ti is showing alower unstable SFE, whereas Zr and Mg experience a remarkable increase (90% for Zr, and 45%for Mg) that should influence the general behaviour of the materials when subject to dynamicloads. It ultimately means that the capability to emit basal dislocations is increased for Ti withrespect to Mg and Zr.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35[1010]-direction [Unit cell]0102030405060StackingFaultEnergy[mJ/m2]  Mg Ti ZrFigure 3.2: SFE computed in the basal plane, where the displacement deformation is performed on[1010]−direction for the HCP lattice structure. The square marker shows the stable SFE (γs[0001]) withnumerical value of 30.6, 7.0, and 36.5 mJ·m−2 for Mg, Ti, and Zr, respectively. The circle markerrepresents the unstable fault energy γu[0001] with numerical values of 46.5, 31.9, and 60.3 mJ·m−2 for Mg,Ti, and Zr, respectively.3.3.3 Nano-void simulationsWe now describe the results of the simulations under hydrostatic tensile loads for a strain rateof ε˙ = 108 s−1 for Ti and Zr and include for reference non-reported results for Mg obtained54Table 3.2: Summary of most relevant data obtained in this work for Mg, Ti, and Zr, where the parametersa0 and c0 are shown (in A˙). σp is the averaged peak stress (GPa), εp is the value of the deformation atthe peak stress ([%]). σc is the critical stress for the dislocation emission (GPa), εc is the value of thedeformation at the critical stress ([%]). Dislocation density values ρ correspond to the maximum valueduring the whole simulation (×1016 m−2). Velocities for basal, vb, and pyramidal-II dislocations, vP−II ,are reported in m·s−1. The stable and unstable stacking fault energies γs(0001) and γu(0001), respectively, aremeasured in the basal plane. The surface energy denoted γ(1122) is performed in the pyramidal-II plane.All the energy are presented in mJ·m−2. All values reported in this table were obtained from simulationscarried out at ε˙ = 108 s−1 with EAM potentials.a0 c0 σc εc σp εp σ f low ε f low ρmax vb vP−II γs(0001) γu(0001) γ(1122)Mg 3.2 5.22 2.41 2.4 2.9 2.95 - - 0.31 322 4500 30.6 46.5 350.5Ti 2.97 4.85 5.6 2.25 7.0 3.1 2.3 3.67 8.56 227 1370 7.0 31.9 1195.0Zr 3.23 5.17 6.0 1.8 8.4 2.6 3.6 3.13 3.59 733 702 36.5 60.3 1700.4from our previous work [1]. The most relevant data obtained from our simulations have beensummarized in Table 3.2.3.3.3.1 Stress evolutionWe now describe the stress evolution for the simulations containing one single nano-void in Tiand Zr subject to hydrostatic tensile loads. Fig. 3.3 shows the evolution of the mean hydrostaticstress (σm), void fraction (v f ), and dislocation density (ρ) for Ti and Zr vs. the strain of thesample. We also include the behaviour for Mg obtained under the same conditions by Gre´goireand Ponga [1] in order to establish quantitative comparisons with these three different HCPmaterials.Let us now focus our attention on the evolution of the mean hydrostatic stress for Ti andZr, depicted in Fig. 3.3(a). Both materials show a very similar behaviour that is described next.First, an initial linear elastic regime without plastic deformation is observed. Next, a secondstage starts and it is characterized by the onset of plasticity, where dislocations are emitted fromthe void surface. The onset of plasticity starts at the critical point (εc,σc) in the stress vs. strainplot. These values are reported in Table 3.2 and highlighted in the plot with a square marker forTi and Zr. Finally, a third stage begins when the stress drops off and reaches an approximatelyconstant value that it is usually known as flow stress. This stage starts at the flow strain andstress (ε f low,σ f low) reported in Table 3.2 and it is shown in Fig. 3.3(a) with a triangle marker.The flow stress is usually related to the amount of work needed to move the dislocation forestin the sample. Interestingly, Fig. 3.3(a) also shows the stress vs. strain evolution for Mg asobtained by Gre´goire and Ponga [1]. We see that while stage I and II are essentially the same,there is a high contrast between the behaviour observed for Ti and Zr once the stress drops off.Remarkably, the behaviour of Mg is brittle, where the stress drop indicates a stage of crackpropagation and catastrophic failure of the specimen. This is very different in Ti and Zr, wherea large amount of plastic deformation is seen.550 1 2 3 4ε [%]0123456789σm[GPa]TiZrMg0 1 2 3 400.511.5vf[%](a) Virial stress and volume fraction vs. strain.2 2.5 3 3.5 4ε [%]10131014101510161017DislocationDensityρ[1/m2] Failure 5×1013   2×1015   6×1014  Ti Zr Mg(b) Dislocation density vs. strain.Figure 3.3: a) Evolution of the mean hydrostatic stress (solid line) and the void volume fraction (dashedline) vs. strain obtained for Mg, Ti, and Zr single crystals of dimensions lx× ly× lz containing one nano-sized void of radius R = 10a0 at T = 300 K. The simulations are carried out at a strain rate of ε˙ = 108s−1. b) Evolution of the dislocation density. . The numbers in the plot indicate the rate of change ofdislocation density at the critical point, i.e., ρ˙(ε = εc).56In order to better understand this difference, we investigate the evolution of the dislocationdensity in the simulation cells. Fig. 3.3(b) shows the evolution of the dislocation density vs. thestrain for the three different materials considered in this work. Additionally, the initial rate ofchange of dislocation density (ρ˙(ε = εc)) is indicated in the plot as well. Fig. 3.3(b) indicatesthat the dislocation density for Ti and Zr quickly increases after the critical point for dislocationsemission has been reached and, eventually, the dislocation density is saturated in the sample.This saturation point is directly related to the flow stress needed to move the dislocations inthe sample. We also include the evolution of the dislocation density for Mg at the same strainrate. Clearly, Fig. 3.3(b) indicates that the rate of dislocation emission is much slower in Mg incomparison with the other two HCP materials.3.3.3.2 Void evolutionWe now turn our attention to the evolution of the void fraction and shape as depicted in Fig.3.3(a). Clearly, the void fraction shows a nonlinear behaviour that is related to the evolutionof the virial stress. As such, there is an initial linear regime where the void fraction increaseslinearly with the strain, as expected. This linear behaviour extends until the critical stress fordislocation emission, indicated by the point (εc,σc) in the stress vs. strain plot (see also Table3.2 for specific material). Shortly after the critical point, the void experiences an abrupt changeof volume where the void fraction increases up to vTif = 150% and vZrf = 50%, for Ti and Zr,respectively. This rapid increment of the volume fraction has a place in a very short timescaleand can be approximated by an exponential growth law. As we will explain later, during thisstage, multiple dislocations are emitted from the void surface and this mechanism is usuallyknown as void cavitation by dislocation emission. Remarkably, the exponential void fractiongrowth quickly stops when the sample reaches the point (ε f ,σ f ) in the stress vs. strain plot (seeTable 3.2). After this point, the void fraction grows approximately linearly with the deformationas depicted in Fig. 3.3(a).Interestingly, the slope of the void fraction plot after the sample has reached the flow stressis very close to the elastic one, indicating that the void growth is driven by the imposed ho-mogenous displacement field rather than by dislocation emission. This effect can be seen byvisualizing the surface of the void as a function of the strain. This has been done by identifyingall atoms on the void surface and used to reconstruct the void surface. Fig. 3.4 shows one sliceof that surface over the x−z plane centred at (x= 0, z= 0) for multiple strain values for Mg, Ti,and Zr. Focusing our attention on Ti and Zr, we see that the void surface is drastically increasedafter the cavitation point (or critical point) as shown in the Fig. 3.4(a) and (b). However, whenthe stress in the simulation cell reaches the flow stress, we see that the void surface is increased,but at a much slower rate than before (c.f. Fig. 3.4(a) at ε = 3.6− 4.2%). As previously ex-plained, at this level of deformation, the dislocation density in the simulation cell has reacheda critical value and it is saturated, meaning that it is not possible to emit more dislocations inorder to accelerate the void growth. Thus, the growth is driven by the homogeneous deformation570 20 40 60 80 100 120[1010]-direction [Unit cell]2030405060708090100110[0001]-direction[Unitcell]ε = 0%ε = 3.15%ε = 3.3%ε = 3.6%ε = 4.2%(a) Ti.0 20 40 60 80 100 120[1010]-direction [Unit cell]2030405060708090100110[0001]-direction[Unitcell]ε = 0%ε = 2.55%ε = 2.85%ε = 3.0%ε = 4.2%(b) Zr.0 20 40 60 80 100 120[1010]-direction [Unit cell]2030405060708090100110[0001]-direction[Unitcell]ε = 0%ε = 2.925%ε = 3.0%ε = 3.075%ε = 3.15%(c) Mg.Figure 3.4: Time evolution of the slided void surface for the simulation cells of lengths lx × ly × lzcontaining one nano-sized void of initial radius R = 10a0 at T = 300 K carried out at a strain rate ofε˙ = 108 s−1. (a) Ti, (b) Zr, (c) Mg.58gradient imposed in the simulation.Of remarkable interest is to compare the evolution of the void surface between Mg, Ti, andZr. Fig. 3.4(c) shows the same slice of the void surface obtained for Mg at different strains.Clearly, a very dissimilar behaviour is seen for Mg, where the void is clearly growing by crackpropagation, as explained by Gre´goire and Ponga [1]. The crack propagation happens in a veryshort time scale, i.e., between ε = 3− 3.15% or ∆t = 15 ps, leading to a crack propagationspeed of vcrack = 480 m·s−1. This is a clear indication that Mg is prone to brittle failure underhydrostatic tensile loads.3.3.3.3 Dislocation emissionFig. 3.5 shows the dislocation emission from the void surface for a strain range between ε = 2.7- 3.15 % at a strain rate of ε˙ = 108 s−1 for Ti. The cavitation process is started by the emissionof partial dislocations in the basal plane (indicated with letter B in Fig. 3.5(a)). These partialsare followed by large stacking faults indicated by the atoms in green. This is expected as Ti hasa very low stacking fault energy. Fig. 3.5(b) shows the emission of prismatic 1/3[2110](1010)dislocations indicated with the letter P. As deformation increases, we observe that the amountof basal dislocations is increased considerably, as shown in Fig. 3.5(c). Eventually, we observeemission of pyramidal-II dislocations in the sample, as shown in Fig. 3.5(d). Interestingly, thesepyramidal-II dislocations are emitted from the basal dislocations, allowing for non-basal defor-mation of the void. After reaching this point, the number of dislocations remains approximatelythe same, and the dislocation density reaches its saturation point. Dislocation velocities havebeen measured and the values are reported in Table 3.2. Interestingly, the pyramidal-II dislo-cations show a value six times more than the velocity of basal dislocations, indicating a highlyanisotropic response of the material.Let us now turn our attention to the dislocation emission in Zr. Fig. 3.6 shows the time evo-lution of the dislocation emission for a strain range between ε = 1.95 - 2.4 % obtained at a strainrate of ε˙ = 108 s−1 for Zr. Similarly to Ti, the void cavitation starts with the emission of severalpartial dislocations, indicated in Fig. 3.6(a) with the letter B. We notice that, due to the largestacking fault energy of Zr, these partials are separated with a very small stacking fault. This is inhigh contrast with Ti, where the partials are separated by very large stacking fault defects. Next,prismatic 1/3[2110](1010) dislocations are emitted from the void surface and are indicatedwith the letter P in Fig. 3.6(b). However, in contrast to Ti, the basal and prismatic dislocationscombine to generate a square shape prismatic dislocation loop (PDL) highlighted in Fig. 3.6(c).We see the emission of multiple PDL in four [2110] directions. Remarkably, square shape PDLsof this type have been reported in Zr specimens subject to radiation in nuclear reactors, wherethe concentrations of nanovoids is large [69]. Interestingly, similarly to Ti, our simulations alsopredict the emission of pyramidal-II dislocations from the tip of basal dislocations, as denoted inFig. 3.6(d) with the letters P-II. Focusing our attention in the PDL loops emitted from the voidsurface, we find that they are generated by two prismatic dislocations 1/3[1210](1010) and two59(a) ε = 2.7% (b) ε = 2.85%(c) ε = 3.0% (d) ε = 3.15%Figure 3.5: Dislocation emission from the void surface in Ti for the simulations carried out at T = 300K and at ε˙ = 108 s−1. The initial radius of the nano-void is R = 10a0 and the snapshots correspond todifferent deformation levels. a) Basal dislocations (labeled with letter B) are emitted at ε = 2.7%. b)Prismatic dislocations (labeled with letter P) at ε = 2.85%. Dislocations at c) ε = 3.0% and d) ε = 3.15%.Pyramidal-II dislocations are emitted from the tip of basal edge dislocations (labeled with letter P-II).basal dislocations that are dissociated in two partials, 13 [1100]+13 [0110]), as indicated in Fig.3.7. This geometry is in clear agreement to experimental studies using high-resolution transitionelectron microscope (HR-TEM) [69].3.3.3.4 Strain rate effectIt is interesting to analyze the effect of the strain rate and its implications in the mechanismsof deformation. Unfortunately, MD simulations are only limited to high strain rates due to thelimitation in the time step which restricts the strain rates to values equal or above of ε˙ ≥ 107s−1. In this work, we use the dynamic HotQC method [49, 50, 51, 55, 79] in order to accessto moderate strain rates, this allows us to simulate a wide range of strain rates between ε˙ =60(a) ε = 1.95% (b) ε = 2.1%(c) ε = 2.4%(d) ε = 2.625%Figure 3.6: Dislocation emission from the void surface in Zr for the simulations carried out at T = 300K and at ε˙ = 108 s−1. The initial radius of the nano-void is R = 10a0 and the snapshots correspondto different deformation levels. a) Basal dislocations (labeled with letter B) are emitted at ε = 1.95%.b) Prismatic dislocations (labeled with letter P) at ε = 2.1%, dislocations partially detached from thevoid surface. c) ε = 2.4%, square-shape prismatic dislocation loops (labeled with letters PDL), PDLscompletely detached from the void surface and are emitted in [1210] and [1120] directions. d) ε =2.625%, Pyramidal-II dislocations are emitted from the tip of basal edge dislocations (labeled with letterP-II).105−1010 s−1 for both Ti and Zr.Fig. 3.8 summarizes the most relevant output including results for Mg obtained by Gre´goireand Ponga [1] and by Ponga et al. [55]. The data obtained from MD simulations is shownwith discrete points while the solid lines are obtained from a power fitting law of the typeσm(ε˙) = σ0(1+aε˙b). Interestingly, the three materials show the same type of behaviour, wherethe peak stress increases with the strain rate. Remarkably, for very low strain rates the peakstress tends to go to a constant value σ0. This is in agreement with continuum scale theory ofvoid growth under dynamic conditions [80]. However, we see large differences in the absolute61Figure 3.7: Detailed view of the square shape PDLs emitted from the void surface at a strain = 2.4 %traveling along the [1210] direction. The PDLs have a Burgers vector of magnitude 〈a〉 and are generatedby two partials ( 13 [1100] +13 [0110]) separated by a small stacking fault defect on the basal plane andtwo prismatic dislocations 13 [2110](1010).105 106 107 108 109 1010ε˙ [s−1]246810121416σm[GPa]MgTiZrσm(ε˙) = σ0(1 + aε˙b )Figure 3.8: Strain rate effect for nano-sized voids in HCP materials. Strain rates ranging from ε˙ = 105– 1010 s−1 for hydrostatic tensile loads in single crystal HCP materials of dimensions lx× ly× lz withapproximately 7 million atoms containing one nano-sized void of radius R = 10a0 at T = 300 K for Mg,Ti, and Zr. The lines are obtained from a fitting law σm(ε˙) = σ0(1+aε˙b).values of the peak stress between Mg, and Ti and Zr. Generally speaking, the peak stress in Mgis about three-times smaller than Ti and Zr.623.3.3.5 Void size effect and shape effectsWe now investigate the effect of the void size and shape in the evolution of the virial stress.The purpose of this study is two-fold. First, we want to understand if the void size influencesthe ductile behaviour observed in Ti and Zr. Second, we also want to understand the effects ofchange of shape of the void, as the spherical void used throughout this work might be a verysimplified case that does not represent most of the cavities seen in experiments. The virial stressvs. strain for various void radius between R = 5a0− 25a0 obtained at a strain rate of ε˙ = 108s−1 is shown in Figs. 3.9(a) and (b) for Ti and Zr, respectively. As expected, the peak stressis reduced as the radius of the sample increases. However, Figs. 3.9(a) and (b) show that theductile behaviour seen for smaller voids is also seen for larger voids. Moreover, we see that theflow stress remains approximately the same for the various void sizes used in the experiments,i.e., σTif = 2.25−2.75 GPa, and σZrf = 3−3.5 GPa. Fig. 3.9 also shows the critical stress valuespredicted with the model proposed by Lubarda et al. and described in Eq. 2.6. The valuesobtained with the model for ω = b is shown in the figure with dashed lines. While the trendsare captured, we see a large discrepancy in the values illustrating the complexity of nanovoidfailure in HCP materials and the need to use high accurate simulations tools such as MD.Next, we turn our attention to the change of void shape in the effect of the virial stress vs.strain evolution. We restrict our study to ellipsoidal shapes that are generated by removing allatoms inside the ellipsoid defined as(x′Rx)2+(y′Ry)2+(z′Rz)2< 1, (3.3)where xp = {x′,y′,z′} (xp = R(θ ,γ)x) is obtained from a rotation matrix R(θ ,γ) that is func-tion of the rotation angles θ and γ around the [0001]− direction and around the [1010]−direction in the HCP lattice structure. Two set of simulations were carried out, one withθ = {0,pi/6,pi/3,pi/2}, γ = 0 and Rx = 1.5Ry, Ry = Rz = 10a0; and a second set with withθ = 0, γ = {0,pi/6,pi/3,pi/2} and Rx = 1.5Ry, Ry = Rz = 10a0. These results for Ti at a strainrate of ε˙ = 108 s−1 are shown in Fig. 3.10. The main observation is that there are no significantchanges to the virial stress evolution with only minor changes in the peak stress and the flowstress remains unaffected by this shape variation. Noteworthy, the ductile behaviour denoted bya large amount of plasticity and the flow behaviour after the nano-void cavitation is independentof the void shape. Similar results are obtained for Zr but are omitted here.3.3.3.6 Temperature effectLastly, the effect of temperature is studied for Ti and Zr for a temperature range between T =50−600 K. The simulations were carried out under a hydrostatic tensile load at a strain rate ofε˙ = 108 s−1 containing one spherical nano-void of radius R = 10a0. The results indicate thatthere is a linear decrease of the peak stress and flow stress with the increment of the temperature630 1 2 3 4ε [%]0123456789σm[GPa]R = 5a0R = 10a0R = 15a0R = 20a0R = 25a00 5 10 15 20 25 30R0123456789ModelMD(a) Ti.0 1 2 3 4ε [%]0123456789σm[GPa]R = 10a0R = 15a0R = 20a0R = 25a0R = 30a00 5 10 15 20 25 30R0123456789ModelMD(b) Zr.Figure 3.9: Evolution of the mean hydrostatic stress vs. strain for various void sizes (R = 5− 25a0)obtained for a single crystals of dimensions lx× ly× lz with approximately 7 million atoms containingone nano-sized void at T = 300 K carried out at a strain rate of ε˙ = 108 s−1. a) Ti. b) Zr.of the sample. The response of both materials can be interpolated with linear laws. The lineardecay when T > TT can be interpolated using a linear fitting function, indicated on the graph inFig. 3.11, i.e., σp(T ) = aT (T −TT )+σ0T , with σp(T ) the peak stress at a given temperature Tand aT = -4.7 × 10−3 GPa/K, a proportional coefficient.640 1 2 3 4ε [%]01234567σm[GPa] Sphere θ  = 0 θ  =  pi/6 θ  =  pi/3 θ  =  pi/2(a) Void effect in Ti. First set of simulations.0 1 2 3 4ε [%]01234567σm[GPa] Sphere γ  = 0 γ  =  pi/6 γ  =  pi/3 γ  =  pi/2(b) Void effect in Ti. Second set of simulations.Figure 3.10: Evolution of the mean hydrostatic stress vs. strain obtained for Ti single crystals of dimen-sions lx× ly× lz with approximately 7 million atoms containing one nano-sized void of radius R = 10a0at T = 300 K and carried out at a strain rate of ε˙ = 108 s−1. Results obtained for the first and second set ofsimulations with rotation of the major axis of the ellipsoidal void along the a) [0001]−, and b) [1010]−directions.3.4 DiscussionThroughout this work, we systematically compared the failure of single HCP crystals contain-ing one nano-void under hydrostatic tensile loads for three materials, namely Mg, Ti, and Zr.Besides to have very similar atomic geometry, relative low stable stacking fault energy and650 1 2 3 4ε [%]02468σm[GPa]50K100K200K300K400K500K600K0 200 400 600Temperature [K]02468(a) Ti.0 1 2 3 4ε [%]0246810σm[GPa]50K100K200K300K400K500K600K0 200 400 600Temperature [K]0246810 σp = aT(T - TT) + σ0T(b) Zr.Figure 3.11: Evolution of the mean hydrostatic stress vs. strain for different temperatures ranging from50 K to 600 K obtained for simulation cells containing one nano-sized void of radius R = 10a0 obtainedat a strain rate of ε˙ = 108 s−1. (a) Ti, (b) Zr.comparable Peierls barrier for multiple slip systems, c.f. Table 3.2, the failure in Mg underhydrostatic tensile load is brittle while their other HCP counterpart structural materials studiedin this work showed a ductile failure with a large amount of dislocation emission and plasticdeformation. From previous results, our analysis indicate that Mg fails due to crack propagationemitted from the void. Interestingly, these cracks propagate along specific directions, as can66clearly be seen in Fig. 3.4(c). These directions seem to be along the {1122} crystallographicplanes, which are the glide plane of the pyramidal-II slip system. On the other hand, our simu-lations have shown that for Ti and Zr these cracks are never seen, at least under the conditionsimposed in our simulations. Clearly, there is a competition between dislocation emission in thebasal plane and crack propagation in Mg that deserves more attention.3.4.1 Analytical two dimensional modelFigure 3.12: Schematic diagram of the stresses acting on the basal plane ((0001)) and on the glide planeof the pyramidal-II slip system ((1122)). Radial σr and tangential σθ stresses are associated with theshear stress τxy and σy obtained using the stress transformation formula, where τxy along the horizontalslip plane is shown next to the edge dislocation.We now wish to analyze this failure situation and the competition between dislocation emis-sion and crack propagation using a simplified model that allows us to do simple analytical anal-ysis and still provide some comparison with MD simulations. Thus, we reduce the complexityof the problem and consider a two-dimensional cylindrical void subject to biaxial hydrostaticstrain as shown in Fig. 3.12. Let us now analyze the stresses acting on the basal plane, (0001),and on the (1122) plane. Fig. 3.12 shows a cylindrical void of radius R under a biaxial stressstate along the vertical and horizontal directions of the plot. The value of the remote stress isσ0. The stresses in these directions are principal values. We want to analyze the stress near thesurface of the void, i.e., r = R, as a function of the angle θ . The elastic solution for this problemstates that the radial σr and tangential σθ stresses are given by [81]67σr = σ0(1− R2r2), σθ = σ0(1+R2r2). (3.4)Clearly, the normal stress acting at any plane centred on the circle is given by σθ valid forθ ∈ [0,2pi). Thus, the normal stress acting on the (1122) is σθ . Now, we want to compute thestresses acting on the basal plane as a function of σr and σθ , as shown at the right of Fig. 3.12.Using the stress transformation formula, we find thatσy =σθ2(1+ cos2θ) , τxy =σθ2sin2θ , (3.5)valid for r = R (where σr = 0). Interestingly, we can now associate τxy as the stress acting onthe basal plane that will nucleate dislocations from the void surface, while σθ is the stress actingon any radial plane that will propagate cracks; and it is the stress acting normal to the (1122)crystallographic plane.In the three dimensional setup, Gre´goire and Ponga [1] have shown that the critical stressto nucleate dislocations in samples containing one nano-sized void of radius R = 10a0 is ap-proximately σc = 2.5 GPa for a strain rate of ε˙ = 107 s−1. As shown by Gre´goire and Ponga[1], these values also seem to be in close agreement with the model proposed by Lubarda et al.[66]. Taking σ0 = 2.5 GPa, we obtain that the stress acting on the (1122) plane is σθ ≈ 5.0GPa, while the shear stress acting on the basal plane is τxy = 2.25 GPa. On the other hand, wewant to estimate the critical stress for unstable crack propagation in the (1122) plane, that isthe one observed by Gre´goire and Ponga [1]. This critical stress for unstable crack propagationis computed by using Griffith’s approach for brittle materials and the stress intensity factor atthe crack tipKIc = σ fβ (a,R)√piac (3.6)where KIc is the critical stress intensity factor, σ f is the remote failure stress, β (a,R) is a geo-metrical factor that depends on the crack size and radius, and ac is the critical crack size neededto propagate the crack in unstable mode. For brittle materials, the critical stress intensity factoris related to the critical strain energy release rate (GIc) asK2Ic = EGc(1−ν2) (3.7)and Gc = 2γ(1122). The critical crack size includes the radius of the void and the crack in thevoid surface, i.e., ac = R+ acrack. β is a geometrical factor that accounts for the initial crackconfiguration. A very crude approach is to compute β by considering a circular hole with twocracks, c.f. [82] page 259. Taking the dimensions of the cell, we can estimate the critical cracksize acr that is needed to generate an unstable crack propagation from the void surface. Sinceβ = β (acr) the process needs to be done in an iterative way. Taking E = 45 MPa, γ(1122) = 350mJ·m−2, σ f = 2.5 GPa, and ν = 0.3 we found that after two iterations, the geometrical factor68converges to β ≈ 0.4 and the critical crack size excluding the radius is acrack ≈ 1.5a0.Remarkably, this small critical crack size, needed to develop an unstable crack propagationseems to be in very close agreement with our previous MD simulations as can be seen in Fig.3.4(c) besides the fact that our analysis is two-dimensional and inertial effects are not beingincluded. Surprisingly, we see that when the shear stress acting on the basal plane reachesits critical value (τcxy = 2.25 GPa), the critical crack size needed to generate an unstable crackpropagation in the simulation cell is only acrack ≈ 1.5a0. This simple calculation shows that thetwo mechanisms, dislocation emission in the basal plane and crack propagation on the (1122)plane, are indeed competing with each other and it is possible to have a mixed failure nearthe nano-void. Thus, the surface energy plays an important role in the ductile failure of HCPmaterials, as indicated by the previous analysis.3.4.2 Two dimensional simulationsTo test this hypothesis, we performed two-dimensional simulations of a cylindrical void underbiaxial stress in order to simulate the conditions of our analytical model. In all two-dimensionalsimulations, the void diameter was kept to be 10% of the simulation length, which was changedbetween lx = 100−300a0. Interestingly, we observe that when the x− direction is aligned withthe [2110]−crystallographic direction of the hcp lattice, basal and 1/3[1011](1012) tensiletwins (see Fig. ??(b)) are emitted as can clearly be seen in Fig. 3.13(a). However, the two-dimensional simulations reveal something unexpected, as it is nicely depicted in Fig. 3.13(b).Interestingly, our simulations indicated that after a basal and the twin boundary intersect, anew surface is generated and it has a length of approximately ∼ 4a0, which can be seen in theinset of Fig. 3.13(b). The newly generated surface, serves as the initiation point for unsta-ble crack propagation along the (1122)−plane. This critical crack size is in good agreementwith the critical crack size found in the analytical approach, which gives us more confidenceof the analytical analysis. Moreover, our simulations have also shown that new surfaces nu-cleated when basal dislocations intersected the twin boundary at other locations, which is reas-suring. This simple set of simulations provide a critical evidence of our hypothesis and shedlight into the mechanisms of brittle failure of Mg under hydrostatic tensile stress. Our simula-tions also indicated that crack propagation occurred when the x− direction was aligned with the[0110]−crystallographic direction of the hcp lattice, even though tensile twin did not nucleateand pyramidal-II dislocations were emitted.69(a)(b)Figure 3.13: View of the two-dimensional simulation in Mg carried out at a strain rate of ε˙ = 108 s−1under biaxial stress. The void emits basal dislocations (green atoms) and tensile twins to accommodatenon-basal deformation. Interestingly, when a basal dislocation intersects the twin boundary, a smallsurface is generated and a crack propagates in an unstable mode.70We now notice the following facts. Under high hydrostatic tensile loads, the vacancy for-mation energy reduces significantly, until eventually reaches values close to zero. Indeed, us-ing large-scale ab-initio simulations, Ponga et al. [83] have shown that under εvol = 10% ofvolumetric deformation in Mg, the vacancy formation energy reduces from 0.768 to 0.45 eV,indicating that the number of vacancy in equilibrium increases five orders of magnitude from1.25× 10−13 to 2.75× 10−8. This, in addition to the low surface energy of Mg in the (1122)plane, makes possible the generation of cracks near the void surface. Indeed, we have observedmultiple vacancy nucleations for all three materials.3.4.3 Ratio between unstable stacking fault vs. surface energiesTable 3.3: Values of the dimensionless ratio ξ between surface energy and unstable stacking fault energyfor Mg, Ti and Zr. SFE and surface energy values reported in [mJ·m−2].Material γu(0001) γ(1122) ξ =γ(1122)γu(0001)Mg 46.5 350.5 7.5Ti 31.9 1195.0 37.5Zr 60.3 1700.4 28.2Based on the previous analytical analysis, we conclude that the brittle behaviour experiencedby Mg under hydrostatic tensile loads is directly associated to its inability to emit non-basaldislocations, and to its low surface energy that is the main mechanism that the material has todissipate its stored strain energy density under brittle behaviour. Among all material parametersanalyzed in this work for the three HCP materials, the most large difference is seen in the surfaceenergy of Mg (γ = 350 mJ·m−2) in comparison with Ti (γ = 1195 mJ·m−2) and Zr (γ = 1700mJ·m−2). In order to assess these two competing mechanisms, we propose to analyze the ratiobetween unstable stacking fault energy and surface energy. A key parameter for dislocationemission on the basal plane is the unstable stacking fault energy, which is related to the energybarrier needed to overcome to generate slip. Similarly, under brittle crack propagation, Griffith’stheory establishes that the critical material values are the surface energy. As such, we proposeto analyze the ratio between unstable stacking fault energy and the surface energy to assess thetendency of the materials to either slip or crack along these two planes. Let us now introducethe following factorξ =γ(1122)γu(0001), (3.8)which is a direct measure of the tendency of a material to slip or to crack along the basal orpyramidal-II plane, respectively. Table 3.3 shows the values of the ratio for the three-differentmaterials using the EAM potentials. We clearly see that Ti has the largest value followed by Zr.71Interestingly, Mg has the smallest ratio which is about 80% and 74% smaller than Ti and Zr,respectively. Using our back-of-the-envelope calculation, we see that increment of the surfaceenergy in Mg will lead to ratios ξ comparable with Ti and Zr.It is now interesting to analyze the behavior of Mg-based alloys with large ductility re-ported by Sandlo¨bes et al. [77]. As explained in the introduction, the simulations carried outby Sandlo¨bes et al. have shown a clear decrease of the stable stacking fault energy in thesealloys. Using the values reported by Sandlo¨bes et al. we observe that a reduction of the unstablestacking fault energy of 50% will means values of ξ = 8−101. This means that the low stackingfault energy Mg-based alloys should behave much like Ti and Zr, explaining the switch betweenbrittle and ductile failure.1If we consider that the surface energy remains unaffected.72Chapter 4Conclusions and future workBefore closing, we present the main conclusions elucidated from this work and state possibledirections to continue it. In Chapter 2, the dynamic response of Mg single crystal specimenswas studied using MD for strain rates between ε˙ = 107− 1010 s−1 at different temperaturesand multiple loading conditions. This study revealed that the response was highly anisotropicleading to a transition in the failure modes from brittle to ductile behaviour. The anisotropicresponse was accompanied by different mechanisms of deformation, topological dislocationstructures, and virial stress vs. strain relations. Interestingly, very little strain hardening wasobserved for simulations cells containing nano-sized voids in Mg under hydrostatic tensile loads.This small strain hardening explained the relatively low spall strength of Mg (around 1 GPa)and its alloys (see Ponga et al. [55] for a summary of spall strength of Mg and Mg-alloys) andsuggested the possibility of control and manipulate some mechanisms of deformation to designnew lightweight Mg-alloys with optimal dynamic resistance.In Chapter 3, the dynamic failure of nano-voids under hydrostatic tensile loads in Ti and Zrwas systematically investigated by means of molecular dynamics simulations. Our simulationsshowed that under these loads, ductile failure occurs due to nano-void cavitation by dislocationemission with a large amount of dislocation density and plastic deformation. Interestingly, inboth materials, shortly after the critical stress for dislocation emission was reached, the voidgrows at an exponential rate quickly reaching a saturation point for the dislocation density.This saturation delayed the failure of the specimen and avoided the propagation of cracks inthe sample even though plastic deformation in non-basal planes is rarely seen. Throughoutsystematic variations of the void size and shape, strain rate and temperature, our simulationsshowed that the ductile failure is intrinsic of Ti and Zr and is unaffected by these changes.Surprisingly, this behaviour is in clear contrast with the failure in Mg, where under sameloading conditions a brittle failure was observed. Using a simple two-dimensional model tounderstand this puzzling behaviour of Mg, we analyzed the stresses acting on the basal and non-basal planes and conclude that two mechanisms, basal slip and crack propagation in non-basalplanes, are likely to be active during the failure of Mg under hydrostatic tensile loads. In light ofthis analysis, we concluded that the surface energy in non-basal planes, a factor that it is usuallynot taken into account in most atomic scale simulations, plays a central role in the brittle failureof Mg. Using these insights, we introduced a non-dimensional material parameter to assessthe propensity of the material to either emit cracks or deform through dislocations in the basalplane. Therefore, our work gives new insights to design a route map to develop new Mg-based73alloys with optimal dynamic resistance.We conclude by summarizing that the insights provided by simulations carried out in thiswork at a wide range of strain rates provide a mean to understand, manipulate and -ultimately-control the anisotropic and complex dynamic behaviour of Mg. The data provided in this workcan be used to generate atomic-based spall models of Mg, thereby aiding in the design of newand better Mg-based alloys with superior dynamics properties. In particular, the effect of alloy-ing elements on the dynamic behaviour of Mg and the development of multi-scale models forspall strength of Mg alloys is an avenue that can be explored in order to elucidate new mecha-nisms of deformation.74Bibliography[1] C. 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Ortiz, “A sublinear-scaling approach todensity-functional-theory analysis of crystal defects,” Journal of the Mechanics andPhysics of Solids, vol. 95, pp. 530 – 556, 2016.8182Appendix ALAMMPS input file# Input file for triaxial tensile load of single crystal Magnesium# Settingsunits metal → Distance [A˙], time [ps], pressure [bar]dimension 3boundary p p p → Periodic boundary conditionsatom style atomicvariable lattice equal 3.196 → Lattice parametervariable length equal 120.0 → Box lengthvariable radius equal 10.0 → Void radiusvariable center equal 60.0 → Position void center# Initialization/Regions and atoms definitionlattice hcp ${lattice} orient x 1 0 0 orient y 0 1 0 orient z 0 0 1region whole block 0 ${length} 0 ${length} 0 ${length}region 2 sphere ${center} ${center} ${center} ${radius} side increate box 1 wholecreate atoms 1 region whole# Potential/Force fieldspair style eam/fspair coeff ∗ ∗ Mg mm.eam.fs Mgneighbor 0.8 binneigh modify delay 10 check yes# Equilibrium/System stabilizationtimestep 0.001velocity all create 300 12345 mom yes rot nofix 1 all npt temp 300 300 0.1 aniso 0 0 0.1 drag 1# Create voiddelete atoms region 2 compress no# Thermal outputthermo 100thermo style custom step lx ly lz press pxx pyy pzz pe temprun 1500083# Deformation rununfix 1reset timestep 0timestep 0.003fix 1 all nvt temp 300.0 300.0 0.01variable srate1 equal 1.0e8 → Strain rate [s−1]# Conversion in picosecondsvariable srate equal ”v srate1 / 1.0e12”fix 2 all deform 1 x erate ${srate} y erate ${srate} z erate ${srate}# Output strain and stress calculationsvariable tmp1 equal ”lx”variable L1 equal ${tmp1}variable tmp2 equal ”ly”variable L2 equal ${tmp2}variable tmp3 equal ”lz”variable L3 equal ${tmp3}variable strainxx equal ”(lx - v L1)/v L1”variable strainyy equal ”(ly - v L2)/v L2”variable strainzz equal ”(lz - v L3)/v L3”variable p1 equal ”v strainxx”variable p2 equal ”v strainyy”variable p3 equal ”v strainzz”# Conversion in GPavariable p4 equal ”-pxx/10000”variable p5 equal ”-pyy/10000”variable p6 equal ”-pzz/10000”variable p7 equal ”-pxy/10000”variable p8 equal ”-pxz/10000”variable p9 equal ”-pyz/10000”fix def1 all print 1000 ”${p1} ${p2} ${p3} ${p4} ${p5}”${p6} ${p7} ${p8} ${p9} file Mg Uni virial-10e8.txt screen no# Creation of Ovito visualization filedump 1 all custom 2500 dump.comp.∗ mass type xs ys zsc peratom fx fy fzdump modify 1 element Mg84# Display thermothermo 100thermo style custom step v strain temp v p1 v p2 v p3 v p4 v p5 v p6 ke pe pressrun 14000085

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