Towards the ultimate strength of iron: spalling through laser shock

The ul ti mate strength of ma te ri als is reached at strain rates ap proach ing the De bye fre quency, when the de for‐ ma tion time at the atomic scale ap proaches the time for atoms to move away from the equi lib rium to their ex‐ treme sep a ra tion po si tion (~5.5 × 10 s for iron). We con ducted high-power pulsed laser ex per i ments on sin gle, poly-, and nanocrys talline iron, gen er at ing ten sile pulses with strain rates ap proach ing the De bye fre‐ quency, 10 s – 10 s , and nanosec ond time du ra tions. We find iron strengths vary ing be tween 5 and 10 GPa, a fac tor of ten higher than the sta tic ten sile strength. Ul tra fine-grained iron sam ples ex hibit a lower ten sile strength, ~4-6 GPa, than sin gle crys tal iron, ~10 GPa. MD sim u la tions show that this is due to dif fer ences in the ini ti a tion sites for voids, pri mar ily at grain bound aries for the nanoand poly crys talline con di tions. Sparse run away voids (~5 μm di am e ter) and ev i dence of sur face melt ing are ob served for the sin gle crys tal iron and are likely due to strain-in duced melt ing when suf fi cient de for ma tion oc curs. The process of sep a ra tion lead ing to spalling is mod eled by mol e c u lar dy nam ics, and the mech a nisms ob served in the ex per i men tally re cov ered spec i mens are de ter mined: in sin gle crys tals voids nu cle ate at the in ter sec tion of twins, while in nanocrys talline spec i mens grain bound aries are the prin ci pal sources of void nu cle ation. An a lyt i cal cal cu la tions are ap plied to the dis lo ca tions gen er ated by the emis sion of shear loops from the void sur faces and the geo met ri cally nec es sary dis lo ca tion den si ties are found to be con sis tent with pre dic tions from mol e c u lar dy nam ics cal cu la tions. © 2021


Introduction
Spalling plays an im por tant role in high-ve loc ity im pact, for ex ample in bal lis tics, ge o log i cal events, and aero space de bris. Dam ag ing spall can be caused by the im pact of a pro jec tile or de bris onto a surface, or from ejec tion of ma te r ial af ter the for ma tion of an im pact crater [1][2][3]. This is due to the in ter ac tion of stress waves dur ing shock com pres sion and re lease; spall frac ture oc curs due to the ten sile stress gen er ated by a re flect ing shock at a ma te r ial bound ary [4,5]. The spall strength of ma te ri als un der dy namic ten sile load ing con di tions has been found to have a " re verse" Hall-Petch re la tion ship, in creas ing with in creas ing grain size [6][7][8][9].
The spall strength of iron (Fe) is of in ter est to geo physics and structural en gi neer ing, as it is a ma jor com po nent of me te orites and rocky plan e tary cores, and of steels and al loys. Spal la tion in iron has been stud ied in the past with flyer Corresponding author. E-mail address: mameyers@ ucsd. edu (M.A. Meyers) plate and gas gun meth ods at strain rates up to 10 s [6, [10][11][12] and through mol e c u lar dy nam ics (MD) at strain rates above 10 s [13-15], but lit tle work has been done us ing lasers to spall iron [9,[16][17][18]. Iron un der goes the α to ϵ phase trans for ma tion at ~13 GPa dur ing com pres sion [19] and the re verse (upon un load ing) at ~10 GPa [20][21][22]. Laser shock-in duced spall ex per i ments in poly crys talline iron revealed that dam age is strongly af fected by the α to ϵ phase trans for mation; smooth spall and a dense twin dis tri b u tion are the re sult of spall oc cur ring af ter the re verse phase trans for ma tion upon un load ing, sim ilar to ob ser va tions at lower strain rates [18][19][20][21][22]. Thin sam ples (250 μ m) that were trans formed to ϵ -Fe were found to have a smooth spall with dense twin dis tri b u tion, while thicker sam ples (400 μ m) that remained mostly in the α phase showed brit tle spall and a lack of twins [18]. This was fur ther in ves ti gated with MD sim u la tions of nanocrystalline iron, where a phase-trans form ing po ten tial shows that af ter the ϵ -α phase trans for ma tion there is a much higher den sity of twins and a smoother spall sur face, while a phase-sta ble po ten tial shows many cracks and rough spall [14]. The  a b c c c d tion pro vides an in creased con cen tra tion of void nu cle ation sites, eventu ally lead ing to duc tile spall. Other laser shock-in duced spall ex per iments show that sin gle crys tal iron has a higher spall strength and a duc tile frac ture sur face while poly crys talline iron is softer and spalls along grain bound aries [9], and that spall strength in creases with increas ing strain rate [16,17]. Clearly, be hav ior of iron spal la tion changes as a func tion of ap plied high strain rate and ini tial mi crostruc ture, but there has never been a sys tem atic study to un der stand the in ter de pen dence of these ef fects. In this work we aim to de ter mine spall strength de pen dence on grain size (sin gle, poly, and nanocrys talline) and strain rate (10 s -10 s ) in pure iron.

Materials
Iron sam ples con sisted of sin gle crys tal iron foils of 250 μ m (Ac cumet Ma te ri als Co, 99.94+%) or 100 μ m thick ness (Sur face Prepa ration Lab o ra tory, ~99.98%), poly crys talline foils of 100 μ m and 250 μ m thick ness (Good fel low, ~99.5%), and nanocrys talline foils that were pro duced from 1 mm thick sin gle crys tal sam ples (Ac c umet Ma te ri als Co, 99.94+%) us ing high-pres sure tor sion (HPT). The HPT process was con ducted for 20 turns at 2 GPa to pro duce an av er age grain size of 100 nm. All foils were then laser cut to 2.5 × 2.5 mm squares and me chan i cally pol ished to an op ti cal fin ish us ing 30, 12, 9, and 5 μ m sili con car bide and alu minum ox ide pa per, fol lowed by 1 μ m di a mond sus pen sion. Un shocked sam ples were fur ther me chan i cally pol ished to 0.04 μ m fin ish us ing a col loidal sil ica sus pen sion. Fi nal thick nesses are re ported in Table 1.
Grain sizes were de ter mined through Elec tron Backscat ter Dif fraction (EBSD) on a FEI Apreo Field Emis sion Scan ning Elec tron Mi croscope (FE SEM). EBSD con firmed that the sin gle crys tal had no grain bound aries and [001] was ori ented in the shock di rec tion. Grain sizes were found to be 100-250 μ m elon gated and 100 nm equiaxed to moder ately elon gated for poly-and nanocrys talline iron, re spec tively ( Fig.  1).

Experimental Design
A 30 μ m thick poly styrene (PS) ab la tor was ad hered to one side of each sam ple foil us ing Hard man Dou ble Bub ble epoxy. All glue was tacked on the edges to min i mize the gap be tween the Val ues in paren the ses rep re sent stan dard de vi a tion due to vari a tion in sam ple thickness. EBSD maps of (a) sin gle crys tal, (b) poly crys tal, and (c) nanocrys tal iron. The sin gle crys tal map shows no grain bound aries and strong [001] ori en ta tion where unindexed black spots are rem nant pol ish ing me dia. The poly crys tal map shows grain sizes rang ing from 100-250 μ m with a slight pre ferred ori en ta tion of [111] where unin dexed pix els are due to scratches. The nanocrys tal map shows grain size av er ag ing around 100 nm with no pre ferred ori en ta tion where unin dexed pix els around grain bound aries are due to the high strain as so ci ated with HPT. Sin gle and poly crys tal maps taken with 0.5 μ m step size and nanocrys tal map taken with 10 nm step size. sam ple sur face and ab la tor. This as sem bly was then glued to a stainless-steel washer with 10 mm outer di am e ter and 2 mm in ner di am e ter. Re cov ery gel (Gelly brand can dle wax) was molten onto a de bris shield, and upon so lid i fy ing was placed ap prox i mately 15 cm be hind the target in the ex pected di rec tion of mo tion (Fig. 2a).
Sam ples were laser shocked in Tar get Area 1 of the Jupiter Laser Fa cil ity at Lawrence Liv er more Na tional Lab o ra tory. The 100 J 2ω laser had a nom i nal square pulse shape with 10 ns du ra tion and 1 mm spot size, re sult ing in peak power of ap prox i mately 1 TW/ cm . VISAR (veloc ity in ter fer om e try sys tem for any re flec tor) [23,24] was used to record the ve loc ity of the sam ple rear free sur face (Fig. 2b) from which spall strength, peak pres sure, and strain rates are cal cu lated. Two in depen dent in ter fer om e ters were used to en sure the data is con clu sive and ap pro pri ately cal i brated (etalon thick nesses: d = 49.968 mm and d = 100.21 mm). 1-D ra di a tion hy dro dy namic sim u la tions us ing (a) Ex per i men tal set-up. The iron tar get is glued onto poly styrene ab la tor, which is then glued onto a stain less steel washer. Com pos ite is placed in the tar get mount and the de bris shield and mo men tum catch (poly mer gel) is placed be hind tar get. VISAR laser is si mul ta ne ously used with the drive laser to cap ture in ter fer ence fringes from the free sur face. (b) 1-D ra di a tion-hy dro dy namic sim u la tions pre dict free sur face ve loc ity sim i lar to ex per i men tal mea sure ment.
HY DRA [25] were run to pre dict peak pres sure and spall strength (Fig.  2b).

Computational Modelling
Non-equi lib rium MD sim u la tions of shock com pres sion were performed, fol low ing widely used meth ods [14, [26][27][28][29][30]. Ex per i men tal laser shock com pres sion was mod eled by in tro duc ing a time-de pen dent pis ton ve loc ity pro file which dic tates the pre scribed shock den sity and strain rate [26]. The con trolled ac cel er a tion and de cel er a tion pro files mimic the stress pro file in tro duced dur ing shock com pres sion [14,27,31]. The pis ton was lin early ac cel er ated to 800 m/ s over 5 ps, main tained at that ve loc ity for 20 ps, then de cel er ated to sta tion ary over 20 ps. As a re sult, and by con sid er ing the strain rate to be ap prox imated by the spa tial de riv a tive of the par ti cle ve loc ity, the lon gi tu di nal strain rate at the point of max i mum ten sion is ap prox i mately 10 s . Al though higher than the ex per i men tally ap plied strain rates, it will be shown in Sec tion 3.3 that it is ad e quate to cap ture the de for ma tion mech a nisms tak ing place in the ex per i ments. Four Fe con fig u ra tions were mod eled, sin gle crys tal with [001] ori ented in the shock di rection, and three nanocrys talline sam ples with av er age grain sizes of 14, 12, and 10 nm. The nanocrys talline sam ples were pre pared us ing Atomsk [32] with sam ple di men sions of 50 × 50 × 150 nm , com pris ing ap prox i mately 32 mil lion atoms. Pe ri odic bound ary con di tions trans verse to the shock di rec tion (z) were em ployed, al low ing for un con strained ex pan sion of the free sur face in the shock di rec tion. An Em bed ded Atom Model (EAM) po ten tial was fit to ad e quately re pro duce the α -ϵ phase tran sition of Fe [14, [28][29][30][33][34][35][36]. Sim u la tions were run us ing LAMMPS [37] and vi su al iza tion was per formed us ing OVITO [38] post-pro cessing al go rithms (adap tive com mon neigh bor analy sis [39], poly he dral tem plate match ing [40], con struct sur face mesh [41] and dis lo ca tion ex trac tion al go rithm (DXA) [42]). The nanocrys talline sam ples were min i mized and ther mally an nealed at 0.7T for 0.5 ns. All sam ples were ther mal ized at 300 K and zero pres sure prior to load ing.
In ad di tion to non-equi lib rium MD, a nanome ter-sized void in iron was mod eled us ing the same po ten tial as de scribed above. The sim u lation do main was ini tially set up as a cu bic sin gle crys tal sam ple contain ing 56 unit cells with one spher i cal void (r = 1.5 nm) at the center of the sam ple. Pe ri odic bound ary con di tions were im posed in all direc tions and the sam ple was equi li brated to zero pres sure at an ini tial tem per a ture of 300 K. A uni ax ial ten sile strain rate of 10 s was applied in the [001] di rec tion for 120 ps, re sult ing in a to tal of 12% volume strain with lat eral strains im peded. A 1 fs time step was cho sen, and the sim u la tion was run with a con stant NVE (num ber, vol ume, energy) in te gra tion con sis tent with the mi cro-canon i cal en sem ble such that no tem per a ture con trol is im posed, and tem per a ture ef fects due to plas tic ity could be mea sured. DXA was used to iden tify line de fects.

Microstructural characterization
Frac tured sur faces can be de scribed as ei ther duc tile or brit tle depend ing on the mech a nisms that oc cur dur ing spal la tion. Duc tile fracture is char ac ter ized by dim pled sur face mor phol ogy that is the re sult of the nu cle ation, growth, and co a les cence of voids, pro duc ing a contin u ous frac ture sur face. In con trast, brit tle frac ture is char ac ter ized by smooth facets that are the re sult of the sep a ra tion of atomic bonds along spe cific crys tal lo graphic planes. The spalled sur face of re cov ered sam ples was char ac ter ized by SEM, and EBSD was used to char ac ter ize cross-sec tions to un der stand the fail ure mech a nism. Shocked sin gle crys tal iron (Fig. 3a, d, e) shows dim pling at the nanome ter scale that is char ac ter is tic of duc tile fail ure, while poly-and nanocrys tal iron sam ples (Fig. 3b-c) show flak ing that could be a re sult of fail ure via sep a ra tion along grain bound aries (Fig. 3e). The high-pres sure tor sion process used to fab ri cate the nanocrys tal sam ples pro duces thin, elongated grains along which the spall oc curs. Ev i dence of these elon gated grains can be seen in Fig. 3f. The dim ple size in sin gle and poly crystalline sam ples and the shear-like bands in nanocrys talline sam ples are on the or der of 1 nm ( Fig. 3g-i). In sin gle crys tal sam ples, a rem nant of the spall plane can be seen near the edge of the sam ple (Fig. 4a), in which twin bound aries can be found (Fig. 4b). The un mapped re gion around the twin bound aries can be at trib uted to voids/ cracks or pos sibly a highly strained re gion that EBSD is un able to map. The mis ori enta tion an gle for the bound aries seen in Fig. 4b was mea sured to bẽ 60°, sug gest ing the for ma tion of {001}/ { 112} twins. Ad di tion ally, these are found to be Σ 3 bound aries (Fig. S1), which are known to result from twin ning [43]. Ran domly dis trib uted voids are also seen through the cross sec tion of the sam ple ( Fig. 4c). For poly crys talline Fe, mul ti ple spall lay ers can be seen in cross-sec tional SEM im ages ( Fig. 5a) with sep a ra tion oc cur ring along grain bound aries where there is a large den sity of voids (Fig. 5b). The den sity of geo met ri cally nec es sary dis lo ca tions (GNDs) was mapped us ing Mat lab tool box MTEX ac cord ing to the method of Pantleon [44]. In both cases, GND den sity is ap prox i mately 10 m (Figs. 4d-e, 5d-e), which is con sis tent with heav ily work-hard ened met als [43] as well as shock com pressed tan ta lum [45,46]. Fur thermore, this tech nique un der es ti mates the GND den sity as 3D maps are needed for a more ac cu rate mea sure ment. The cir cu lar re gions with high dis lo ca tion den si ties ( Fig. 4e) are due to sub-sur face voids that emit shear dis lo ca tion loops (Fig. 4f). When these loops in ter sect the pol ished sur face, they gen er ate the cir cu lar re gions of high GND density. These are ev i dence for the shear loop emis sion mech a nisms for void growth de scribed in Sec tion 3.4. Ev i dence of in tra gran u lar plastic ity can be seen in the poly crys talline sam ples ( Fig. 5d-e) where there is higher GND den sity within grains. This ob ser va tion is also ev ident in sim u la tions as de scribed in Sec tion 3.3.
Ves tiges of a brit tle frac ture mech a nism were ob served at the edges of the spall plane ( Fig. 6), where pres sure and strain rate are lower than at the cen ter of the sam ple. This can be ex plained by a brit tle-toduc tile spall tran si tion oc cur ring at a crit i cal strain rate, as de scribed by Grady [47]. For iron, this crit i cal strain rate is cal cu lated to be on the or der of 10 s , a rea son able value for the edge of the spalled region where the shock has de cayed. Smaller grained sam ples, how ever, do not dis play this be hav ior near the spall edge, be cause their spall behav ior is dom i nated by grain bound aries. This shift in spall mor phol ogy is con sis tent with pre vi ous ob ser va tions sum ma rized by Mey ers and Aimone [48].
Voids (~10 μ m di am e ter) on the spalled sur face were seen ex clusively in sin gle crys tal sam ples (Fig. 7), with smooth in ner sur faces which are in dica tive of melt ing and re-so lid i fi ca tion of the ma te r ial. Dur ing shock com pres sion the tem per a ture is ex pected to rise [4], but not enough to melt the en tire sam ple. Sec tion 3.3 will dis cuss mol e c ular dy namic cal cu la tions that de ter mine the tem per a ture both of the over all sam ple and lo cally around a void.
The for ma tion of voids can oc cur through a va ri ety of mech a nisms op er at ing at dif fer ent length scales ( Fig. 8) [49][50][51][52][53]. At the atomic scale, in trin sic va cancy com plexes can form nanoscale voids [50], although larger scale de fects would dom i nate over such small-scale imper fec tions. De for ma tion-in duced dis lo ca tion cell walls with a crit i cal mis ori en ta tion have large strain en ergy that can be re lieved through void nu cle ation [54,55]. At larger length scales twin and grain boundaries are com mon void ini ti a tion sites (Sec tion 3.3), be cause their high in ter fa cial en ergy and weak bond ing al low for the pref er en tial nucle ation of voids [51,52]. Sec ond-phase par ti cles or in clu sions can cause cracks to prop a gate into the ma trix ma te r ial, which in turn can cause debond ing of in ter faces [49].

Spall strength determination
The ex per i men tal strain rate and spall strength were cal cu lated from the free sur face ve loc ity pro files us ing the peak free sur face ve locity, , and first min i mum free sur face ve loc ity, (also known as the spall pull back sig nal). The sim pli fied acoustic ap proach yields the fol low ing lin ear ap prox i ma tion re la tion ship be tween spall strength and the par ti cle ve loc ity drop (u -u ) [5]: where is the ini tial den sity and is the sound ve loc ity of iron [4]. The strain rates were cal cu lated by ap ply ing the fol low ing acoustic approx i ma tion: (2) where is the time dif fer ence be tween and . The ap prox i mate strain rate for the 250 μ m thick sam ples was ~10 s , and a re duc tion in thick ness to 100 μ m in creased the strain rate ten-fold, to ~10 s . The peak pres sure achieved dur ing the spalla tion event was cal cu lated us ing the Hugo niot re la tion ship be tween pres sure and par ti cle ve loc ity, as sum ing par ti cle ve loc ity is ap prox imately half of the free sur face ve loc ity. Peak pres sures ranged from 60 GPa -100 GPa, well above the the o ret i cal 13 GPa α -ϵ phase trans forma tion pres sure. Rep re sen ta tive VISAR traces of free sur face ve loc ity de pict ing strain rate and grain size de pen dence can be seen in Fig. 9. The re sult ing strain rate and spall strengths for 50 shots are sum marized in Table 1.  The spall strength is ex pected to be high est in sin gle crys tal be cause of the scarcity of frac ture nu cle ation sites. Grain bound aries are a common nu cle ation site for voids un der dy namic ten sile load ing, and for dis lo ca tion pile-up lo ca tions un der shear load ing [56]. The ten sile strength of iron by slip is highly strainrate de pen dent; in con trast, the strength of the grain bound aries can be con sid ered to be strain-rate in sen si tive, to a first ap prox i ma tion. Thus, a crit i cal strain rate is reached be yond which the slip stress is higher than the grain-bound ary co he sion. At this strain rate, the fail ure mecha nism changes from in tra gran u lar to in ter gran u lar fail ure. This re sult is in agree ment with lit er a ture and spall the ory that dates back three decades [5,[9][10][11][13][14][15][16][17][18]47,56,57] (Fig. 10). The sin gle crys tal samples con sis tently show the high est spall strength, whereas the nanocrystalline sam ples have the low est. Poly crys talline sam ples show much larger vari a tion in spall strength, due to sev eral fac tors: (1) large vari ation in grain size (5 -250 μ m), (2) sam ple pu rity (99.5 -99.999%), and (3) sam ple pro cess ing con di tions. Grain size, which is fre quently un re ported in the lit er a ture, will clearly af fect spall strength, as smaller grains will have a larger grain bound ary area per unit vol ume and there fore be weaker. Sam ple pu rity will also af fect the spall strength as solute atoms tend to mi grate to wards grain bound aries to de crease strain en ergy. Lower pu rity iron will have more grain-bound ary solutes which, in turn, de crease the grain bound ary co a les cence strength and stress for nu cleating voids. Dif fer ences in pro cess ing meth ods used in sam ple prepa ra tion can also in tro duce vari a tions in the lev els of strain in the struc ture. We fit the spall strengths by us ing power laws (Fig.  10), which con verge at a high strain rate close to the De bye fre quency, or the fre quency of atomic vi bra tions. At this ex treme strain rate the spall strength can be con sid ered the ul ti mate ten sile strength, 35 GPa [47], at which point the in ter atomic forces can no longer hold the struc ture to gether. The fit for poly crys talline iron does not cur rently con verge be cause of the lack of ex per i ments or sim u la tions at strain rates higher than ~10 s and in con sis ten cies in the grain struc ture. Sim i lar fail ure be hav ior has been seen in spalled tan ta lum [7], vanadium [8], and iron at lower strain rate [18].

Molecular Dynamics Simulation of Spalling
As the ini tial shock runs across the sam ple, an α to ϵ phase tran sition takes place. This is ex pected af ter the re sults by Gunkel mann et al. [34] and Amadou et al. [28] for sim i lar shock con di tions. For the single crys tal, the MD sim u la tions re veal the nu cle ation of an ϵ phase that prop a gates along the sam ple in the shock di rec tion, whereas for the nanocrys talline sam ple, the α to ϵ phase tran si tion takes place within each grain, pre serv ing the ini tial grain bound ary struc ture. At later time, as the shock pro file re flects from the rear sur face of the sam ple, the re verse tran si tion (ϵ to α ) takes place. Sim i lar con clu sions were obtained in pre vi ous MD stud ies of uni ax ial com pres sion and re lease of poly crys talline iron [33]. These processes take place be fore any spal lation event.
Stress pro files in iron sam ples were cal cu lated from non-equi lib rium MD sim u la tions along the shock di rec tion for dif fer ent times around the be gin ning of spal la tion (Fig. 11). Pos i tive values of σ in di cate ten sion. As sum ing that the spall strength is equal to the peak ten sile stress along the shock di rec tion, the sin gle crys tal strength is 18 GPa, while the nanocrys tal strength is 14.5 GPa. There was no sig nif i cant strength vari a tion with grain size for the nanocrystalline sys tems mod eled here. The peak stress for the sin gle crys tal occurs in a re gion be tween 75 and 100 nm at about 56 ps, with a marked drop in stress 2 ps later. This drop sig nals the in cep tion of the spal lation event, with the for ma tion of one or more closely spaced voids in that re gion. The stress peak is broader for the nanocrys talline sam ple, re sem bling a plateau that spans around 50 -100 nm. In this in stance, as time evolves (≥ 54 ps) stress drops oc cur in sev eral po si tions along the plateau, sig nal ing a dis tri b u tion of the spal la tion event as voids open at sev eral lo ca tions in this re gion. The de lo cal ized spal la tion is due to the dis trib uted na ture of grain bound aries along the sam ple, two-di men sional de fects that pos sess less strength and more void nu cleation sites than in the sin gle crys tal.
In the sin gle crys tal sam ples, as voids nu cle ate due to twin-twin inter ac tions and grow, they start co a lesc ing, form ing larger flaws, weaken ing the sam ple's sec tion, and ul ti mately lead ing to the for ma tion of a spal la tion plane (Fig. 12a). For the nanocrys talline sim u la tions, however, the high frac tion of grain bound aries of fers en er get i cally fa vorable nu cle ation sites for the for ma tion of voids (Fig. 12b). The ori en tation of a grain bound ary with re spect to the wave prop a ga tion di rection is more im por tant than spe cific grain bound ary ori en ta tion. This pref er en tial fail ure along grain bound aries that are per pen dic u lar to the load ing di rec tion is in agree ment with pre vi ous stud ies on BCC met als [58]. Con se quently, twin ning is less pro nounced, but can still be found in rel a tively large grains. In ad di tion to ex per i men tal (Fig. 5d-e) and com pu ta tional ev i dence (Fig. S2) of in tra gran u lar plas tic ity un der spall con di tions, sim i lar ob ser va tions can be found in the atom istic sim u lation lit er a ture: Gunkel mann and co-work ers re port in tra gran u lar plastic ity in α -Fe for their nanocrys talline stud ies both un der ho mo ge neous com pres sion [36] and un der non-equi lib rium shock com pres sion [29]. Ex per i men tal ev i dence of in tra gran u lar plas tic ity is also found in both BCC and FCC met als [59][60][61]. Grain-bound ary plas tic ity is an other factor that can play an im por tant role, par tic u larly for grain sizes as small as the ones used in this work and un der the high stresses in duced by shock load ing. To add com plex ity, grain bound ary plas tic ity can also be rate de pen dent, as shown by grain-bound ary dis con nec tion mo tion stud ies [62]. Grain-bound ary slid ing and in tra gran u lar plas tic ity were iden ti fied in our sim u la tions and can be seen in Fig. S2.
The time-re solved evo lu tion of void nu cle ation and growth dur ing spall in the [001]-ori ented sin gle crys tal sam ple can be seen in Fig. 13. The poly he dral tem plate match ing al go rithm was used to clas sify the lo cal struc tural en vi ron ment of the atoms. Twins (or ange do mains) nucle ated dur ing spall form elon gated struc tures that, upon in ter sect ing, fa vor the nu cle ation and growth of voids (Fig. 13). Void for ma tion by twin-twin in ter ac tion was also re ported by Gunkel mann et al [14] in Fe spall sim u la tions and by Hahn et al. [27] in Ta spall sim u la tions.
Tem per a ture analy sis around a void was per formed fol low ing the dis cov ery of re gions that had the ap pear ance of molten and reso lid i fied ma te r ial on the spalled sur face of sin gle crys tal iron in ex per i ments. Tem per a ture was de ter mined by:  [55], (c) twin bound aries with mis orien ta tion in the lat tice, (d) twin in ter ac tions be tween pri mary and sec ondary twins, (e) grain bound aries [51] and triple points [49], (g) in ter nal cracks and in ter face debond ing at second-phase par ti cles [49].
where the atomic mass of iron m is equal to 55.85 u, k is Boltz man n's con stant, N is Avo gadro's num ber, v , v , and v are the com ponents of the atom ve loc ity vec tor, and v is the lo cal cen ter of mass trans la tional ve loc ity. Tem per a ture pro files cal cu lated from Eq. 3 for the sin gle crys tal and nanocrys talline sam ple dur ing the time of the spal la tion process show that tem per a ture is no tably higher in the vicinity of the spal la tion plane (Fig. 11). The global tem per a ture of the sam ple dur ing spall was cal cu lated to be 1000 K at most. How ever, due to the geo met ri cally-nec es sary plas tic de for ma tion around voids, the tem per a ture is in creased lo cally (Fig. 11), ap proach ing the de creased melt ing point caused by the ten sile state dur ing spall that re laxes the high pres sure con di tions [63]. This is in di cated by the iso lated data points (with er ror bars) in the re gion of the spall at 90 nm (and 60 ps) for the sin gle crys tal and by the three points be tween 60 and 100 nm for the nanocrys talline ma te r ial. The spall re gion is more lo cal ized for the sin gle crys tal; the nanocrys talline sam ple pro vides am ple re gions for void ini ti a tion and there fore the spall re gion is more dif fuse. As explained above, ten sile stresses in duce the for ma tion of voids and temper a ture cal cu la tions around the voids re veal even higher tem per atures, close to or above melt ing point.

Analytical model for dislocation generation around void surface
The MD pre dic tions of high dis lo ca tion den si ties and tem per a tures around voids and the ex per i men tal ob ser va tion of the ap par ent melt ing be hav ior stim u lated the de vel op ment of an an a lyt i cal model for voidgen er at ing dis lo ca tions that in crease the tem per a ture. A nanome tersized void is con sid ered to act as a nu cle ation site for GNDs that, upon prop a ga tion away from the void, lead to suc ces sive nu cle ation and prop a ga tion events (Fig. 14). This yields a quan tifi able dis lo ca tion den sity. The mo tion of dis lo ca tions, in turn, heats the sur round ings of the void due to plas tic ac tiv ity, pro duc ing a sig nif i cant in crease in lo cal tem per a ture. GNDs can be used to es ti mate the to tal dis lo ca tion length around a grow ing void by as sum ing that a cer tain num ber of dis lo cation shear loops are ini tially nu cle ated on the void sur face and that they trans port mat ter away from it [64][65][66]. This mech a nism was proposed and an a lyt i cally demon strated by Lubarda et al. [67]; later MD sim u la tions quan ti fied the emis sion of dis lo ca tion shear loops and their even tual trans for ma tion into pris matic loops by a " lasso" mech a nism for BCC crys tals [68,69]. In the analy sis pre sented be low only the shear loop emis sion is eval u ated be cause it is as sumed that pris matic loops do not form un der uni ax ial com pres sion [70], un til later when plas tic flow re duces the back ground shear stress and the voids grow un der greater stress tri ax i al ity [71]. For the BCC structure, eight loops that prop a gate along BCC slip planes ( ) are ini tially as sumed, and other sets of loops are also cre ated as the void growth pro ceeds. Fig ure 14 de picts the growth of a void by the emis sion of var i ous dis lo ca tions.
For every emis sion i of loops, the to tal length is, where L and L are the lengths of the screw and edge com po nents, respec tively. It is as sumed that each loop ex pands to a dis tance of about 5-10 times the ini tial void ra dius. This is a rea son able as sump tion consis tent with con tin uum plas tic ity treat ments of void growth un der hydro sta tic ten sile stresses. Once the loop reaches such dis tance, the applied shear stresses are suf fi ciently re laxed and the ap plied strain is accom mo dated by the emis sion of a new loop set. The to tal dis lo ca tion length around the void is: The for ma tion of the dis lo ca tion loops can be as sumed to ex pand the void by an in cre men tal vol ume dV [70] such that for every emis sion i, the in cre men tal and new void vol umes are: As sum ing spher i cal voids, the ra dius at the i'th emis sion event is: The strain is com puted as sum ing the in crease in vol ume due to void growth is ac com mo dated by an in crease in the ver ti cal di men sion, a , in the unit cell: (9) where a and a re main con stant dur ing uni ax ial ten sion and a increases as: (10) The cor re spond ing strain in cre ment dϵ and to tal strain are: The re sults for each gen er ated emis sion event can be found in Table 2 for an ex am ple case. They show that af ter ten dis lo ca tion emis sions, the void ra dius is dou bled, for an ini tial void ra dius of 1.5 nm. The work-hard ened vol ume is con sid ered as the vol ume lim ited by the spher i cal vol umes cor re spond ing to the work hard ened ra dius R = 10r and void ra dius r : (13) Dis lo ca tion den sity, , can then be cal cu lated as the ra tio between to tal dis lo ca tion length (Eq. 5) and work hard ened vol ume (Eq. 13): It is im por tant to note that to ac com mo date for the in creas ing strain, the an a lyt i cal model uses suc ces sive loop gen er a tion events, and that be yond 10% strain the dis lo ca tion con fig u ra tion be comes too complex for the sim ple as sump tions. The dis lo ca tion ve loc ity (in m/ s) as a func tion of shear stress, , can be com puted as sum ing a power law fit to the ex per i men tal re sults of Urabe and Weert man [72]: The shear stresses were com puted from our MD sim u la tions of shock com pres sion as [73]: (16) where σ , σ , and σ are the hy dro sta tic stresses in the x, y, and z geo met ri cal di rec tions. The shear stresses ob tained dur ing the void nucle ation and growth process and within that re gion are, on av er age, 600 MPa (0.6 GPa), giv ing an av er age dis lo ca tion ve loc ity of ap prox imately 500 m/ s. Sim i lar dis lo ca tion ve loc i ties were ob served in our MD sim u la tions of a sin gle void un der high strain rate uni ax ial ten sion and are also con sis tent with previ ous stud ies on BCC Ta un der high strain rate uni ax ial com pres sion [69]. The plas tic shear strain rate can be re lated to the mo bile dis lo cation den sity us ing Orowan's equa tion [74]: (17) where b is the Burg ers vec tor, ap prox i mately 0.27 nm for α iron. The mo bile dis lo ca tion den sity is a frac tion of the to tal dis lo ca tion density, : . is a pa ra me ter that is as sumed to grad u ally de crease with strain due to shear loops nu cle at ing, prop a gat ing, and pop u lat ing the work-hard ened vol ume. This frac tion takes the unity value as the first gen er a tion of loops leave the sur face, and lin early decreases to 0.1 as the last emis sion of dis lo ca tions takes place due to two main fac tors: (1) dis lo ca tion loops con sist of edge dis lo ca tion com ponents that slip with high mo bil ity, while screw com po nents have limited mo bil ity and (2) the rapid for ma tion of junc tions leads to a decrease in mo bile dis lo ca tions with re spect to the to tal num ber of dis loca tions [70].
As sum ing adi a batic ity, the tem per a ture in crease as so ci ated with plas tic de for ma tion is ex pressed as: (18) where is the ma te r ial den sity, is the spe cific heat ca pac ity, is the time-de pen dent shear stress, and is the Quin ney-Tay lor pa ra me ter that rep re sents the frac tion of rate of plas tic work dis si pated as heat [75], taken as equal to 1 for the sake of sim plic ity. In re al ity, the evo lu tion of dis lo ca tion den sity with plas tic de for mation is the re sult of the com bined ef fects of dis lo ca tion gen er a tion and an ni hi la tion, with char ac ter is tic rates and , re spec tively [76]: At the ini tial stages of plas tic ity, the ma te r ial can be con sid ered as pris tine, and dis lo ca tions can ex pand and pop u late the sur round ing volume freely. As the suc ces sive emis sion of loops takes place, the newly nu cle ated dis lo ca tions now prop a gate within a vol ume that con tains dis lo ca tions. As more and more loops are emit ted from the void surface, the plas tic vol ume in the vicin ity of the void has an in creas ing dis lo ca tion den sity and the dis lo ca tion for est re quires a re duc tion of the mean free path. In con se quence, an ni hi la tion events be come more prob a ble, to the limit that the an ni hi la tion rate be comes equal to the gen er a tion rate once the dis lo ca tion den sity reaches a crit i cal value. Here, the sat u ra tion is around 10 m , con sis tent with con di tions of shock com pres sion in BCC met als [45,77]. As a first ap prox i ma tion, the dis lo ca tion an ni hi la tion rate is con sid ered to be pro por tional to the dis lo ca tion den sity and in versely pro por tional to the mean free path before they en counter dis lo ca tions of op po site sign [78]. In deed, the Kocks-Meck ing the ory has dif fer ent power de pen den cies for the two terms, and [79] and a de tailed treat ment Figure 11. MD cal cu la tions of pres sure in the z-di rec tion (par al lel to load ing) and tem per a ture in the sin gle crys tal (left) and nanocrys tal (right) with 10 nm av er age grain size. Void T sym bols cor re spond to the peak tem per a ture in voids at 60 ns. Ini tial shock di rec tion goes from left to right. of the ex act func tional form for these high strain-rate con di tions is beyond the scope of this work. Es ti ma tions of the an a lyt i cal model are pre sented in Fig. 15. It is rea son able that, when dis lo ca tion ac tiv ity starts, the to tal and mo bile dis lo ca tion den si ties are sim i lar, but the rapid for ma tion of junc tions leads to a de crease in mo bile dis lo ca tions with re spect to the to tal num ber of dis lo ca tions. As the mo bile dis lo cation den sity reaches the sat u ra tion value (~10 m ), the and can be con sid ered to be equal, such that be comes zero. Thus, the to tal dis lo ca tion den sity,

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Proof PDF Figure 13. Slices of the sin gle crys tal sam ple along the shock di rec tion at (a) 56 ps, (b) 58 ps, and (c) 62 ps. Color ing cor re sponds to lo cal ori en ta tion of the atoms. Red cor re sponds to [100]-ori ented BCC struc ture and or ange are {110} ori ented do mains, which cor re spond to twins. Their in ter ac tion leads to the for ma tion of one (top row) or more voids (mid row) of spher oidal shape. The bot tom row shows that, at a later stage, the voids have grown and are closer to each other, lead ing to co a les cence. The strong changes in color in the sur round ing of the voids point to mas sive dis or der, typ i cally as so ci ated with melt ing.
con sid er ing both gen er a tion and an ni hi la tion events, re mains con stant at the sat u ra tion point. The den sity of mo bile dis lo ca tion de creases beyond a strain of 0.09. The com par i son of an a lyt i cal and MD pre dic tions is re veal ing. The MD-pre dicted dis lo ca tion den sity rises rapidly with plas tic strain up to 5 × 10 m and then in creases at a much lower rate. Al though there are dif fer ences be tween the two pre dic tions, the re sults are fairly consis tent. Ad di tion ally, the GND den sity es ti mated from re cov ered samples (10 m ) is within one or der of mag ni tude from MD sim u la tions and an a lyt i cal pre dic tions a rea son able dif fer ence given the as sumptions of the model and the un der es ti ma tion of the EBSD GND technique.
The emis sion and prop a ga tion of dis lo ca tions for ten suc ces sive gener a tion events in creases the tem per a ture to 590 K, an in crease of over 250 K above the ini tial value. This cor re sponds to an in crease in void ra dius from 1.5 to 3 nm. GND-pre dic tions can also be com pared with the tem per a ture pro files of the spall sim u la tions (Fig. 11). The tem pera ture rise at void ini ti a tion at 58 ps and 54-56 ps in sin gle and nanocrys tal iron, re spec tively, is sim i larly ~250 K. The void nu cle ation in the spall sim u la tions ex hibits im por tant dif fer ences not in cluded in the model; e.g. un der spall con di tions, voids nu cle ate at twintwin in ter sec tions in a heav ily stressed en vi ron ment, un der high strain rate con di tions, whereas the model pre sented in this sec tion con tains a pre-ex ist ing void un der equi lib rium con di tions. Sev eral other fac tors may also play a role un der shock load ing con di tions, such as phase trans for ma tions, mi crostruc tural evo lu tion, ther mal soft en ing, heat conduc tion or even elec tron-phonon cou pling [80,81]. In par tic u lar, heat con duc tion can play an im por tant role in void growth as pointed out by Wu et al. [81] since, de pend ing on the ef fi ciency of the heat con duction, voids grow ei ther adi a bat i cally or isother mally, in the lim it ing cases. One pos si bil ity or the other de pends mainly on the ini tial void size, void growth rate, and ma te r ial prop er ties such as ther mal con ductiv ity, mass den sity, spe cific heat and yield strain. Ac cord ing to Wu et al. [81], the cri te rion to de ter mine when the adi a batic void growth or the isother mal void growth ide al iza tions should be con sid ered is: where is the ini tial void ra dius, is the void growth rate, and , where is the ther mal con duc tiv ity, is the mass den sity, the spe cific heat ca pac ity and the yield strain. This opens the pos si bil ity for im prove ments of the cur rent model by in tro duc ing mod i fi ca tions to Eq. 18, that would re sult from in cor po rat ing a heat con duc tion term into the en ergy bal ance equa tion. At later stages of void growth dur ing spall, how ever, voids grow much more rapidly, justi fy ing a treat ment us ing an adi a batic ap prox i ma tion [81].
Al though the sim pli fied an a lyt i cal model pre sented here may not cap ture other ef fects stated above and that may play a role in void nucle ation and growth, the 250 K tem per a ture in crease is sim i lar to what is seen in MD. The in cor po ra tion of more com plex dis lo ca tion phe nomena and this promis ing re sult may pave the ground for fu ture de vel opments of such mod els.

Conclusions
The ef fect of ini tial nano/ mi crostruc ture on the spall be hav ior of iron at 10 s -10 s strain rate was ex per i men tally in ves ti gated in thin foils shocked by a 100 J 2ω laser pro duc ing a tri an gu lar shock wave (P ~60 GPa) in the ma te r ial and spal la tion at the rear free surface. This study of spall in iron pro vides in sight into the com plex dynamic fail ure process. The fol low ing are the main find ings of these system atic ex per i ments cou pled with an a lyt i cal cal cu la tions and MD sim ula tions: • A novel soft re cov ery tech nique was used in con junc tion with VISAR to ob tain high qual ity mi crostruc tural and ve loc ity data for every shot.
• Sin gle crys tal sam ples ex hi bit a char ac ter is tic dim pled spall sur face that is in dica tive of duc tile fail ure known to oc cur af ter the reversible α -ϵ phase tran si tion [18]. This is the re sult of nano-sized voids that nu cle ate, grow, and co a lesce, caus ing fail ure. Smaller grained poly-and nanocrys talline iron have a blis tered or streaked spall sur face due to fail ure along grain bound aries, in ad di tion to minor dim pling. Grain bound aries are known to be pref er en tial void nu cle ation sites, and are ul ti mately the cause of fail ure. There fore, the spall strength of sin gle crys tal iron is higher than that of polyand nanocrys talline iron at strain rates tested ow ing to the larger num ber of grain bound aries that weaken the ma te r ial in the lat ter cases. This " re verse" Hall-Petch re la tion ship is also seen in many other spalled ma te ri als, both BCC and FCC [7,8,63,[82][83][84], and is due to the greater strain-rate sen si tiv ity of plas tic flow in com par i son with grain-bound ary co he sion. Ex per i men tal re sults from this study

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Proof PDF Figure 15. Com par i son of model pre dic tions of dis lo ca tion den sity with a MD sim u la tion of iron sin gle crys tal with a void. When con sid er ing only gen er a tion events, the dis lo cation den sity in creases with strain (red). If an ni hi la tion events are in cluded (blue), the satu ra tion den sity of 10 m must be taken into ac count. The to tal dis lo ca tion den sity (black), in cludes den sity of mo bile dis lo ca tions (green).
pro vide the nec es sary ev i dence that the re verse phase trans for ma tion has oc curred: a duc tile spall sur face is ob served, and peak pres sures reached well above the α to ϵ phase tran si tion pres sure.
• MD sim u la tions show that in sin gle crys tal iron, the in ter ac tion of twin bound aries be comes a pref er en tial site for void nu cle ation. Further void growth re sults in a lo cal highly dis or dered struc ture that in creases the tem per a ture to near-melt ing con di tions around voids. As more voids nu cle ate and grow, they co a lesce, lead ing to the forma tion of a spal la tion plane. The large vol ume den sity of grain bound aries in nanocrys talline sim u la tions fa cil i tates void nu cle ation sites and leads to pref er en tial fail ure along bound aries that are perpen dic u lar to the shock di rec tion. Fail ure in sin gle crys tal iron appears to be more lo cal ized, while in nanocrys tal iron it is more distrib uted and leads to a lower spall strength. De spite dif fer ences in strain rates, the sim u lated spall strengths fol low the same trend as ex per i men tal re sults.
• An an a lyt i cal model was de vel oped which pre dicts a dis lo ca tion density that is con sis tent with MD sim u la tions and sat u rates at ap prox imately 10 m . This sim pli fied model uses nanome ter sized voids as nu cle ation sites for geo met ri cally nec es sary dis lo ca tions. As void growth pro ceeds, more dis lo ca tion loops are cre ated and heat the sur round ings as they prop a gate. It is as sumed that the evo lu tion of dis lo ca tion den sity is a com bi na tion of dis lo ca tion gen er a tion and an ni hi la tion rates.
• Both MD sim u la tions and an a lyt i cal cal cu la tions pre dict sig nif i cant tem per a ture rises which can, at peak, lead to melt ing of the void surfaces. They ex plain the ap par ent melt ing be hav ior ob served in postshock char ac ter i za tion.
• Ex per i men tal re sults, MD sim u la tions, and an a lyt i cal mod el ling predict den sity of geo met ri cally nec es sary dis lo ca tions to be 10 -10 m . This is a rea son able range of val ues con sid er ing the as sumptions made in the an a lyt i cal model and the un der es ti ma tion of GND den sity from the EBSD maps. Ad di tion ally, EBSD maps of dis lo ca tion den sity sup port sim u la tion find ings of cir cu lar void growth in sin gle crys tal iron and in tra gran u lar plas tic ity in poly crys talline iron.

Declaration of Competing Interest
The au thors de clare that they have no known com pet ing fi nan cial in ter ests or per sonal re la tion ships that could have ap peared to in fluence the work re ported in this pa per.