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2 Mechanics of Materials xxx (2016) xxx xxx Contents lists available at ScienceDirect Mechanics of Materials journal homepage: Determination of the constitutive relation an critical conition for the shock compression of cellular solis Q1 Yongle Sun a, Q.M. Li a, c,, S.A. McDonal b, P.J. Withers b a School of Mechanical, Aerospace an Civil Engineering, The University of Manchester, Sackville Street, Manchester M13 9PL, UK b Henry Moseley X-ray Imaging Facility, School of Materials, The University of Manchester, Upper Brook Street, Manchester, M13 9PY, UK c State Key Laboratory of Explosion Science an Technology, Beijing Institute of Technology, Beijing , China a r t i c l e i n f o a b s t r a c t Article history: Receive 19 August 2015 Revise 18 March 2016 Available online xxx Keywors: Foam material Compressive behaviour Shock Hugoniot Compute tomography This stuy aims at unerstaning the constitutive relation an critical conition for the shock compression of cellular solis. A 2D virtual foam is constructe from the cross-section of a close-cell aluminium foam image by micro X-ray compute tomography, which enables the realistic consieration of mesoscale structural effect in numerical moelling. Quasi-static an shock compressions of the 2D foam are simulate. A series of Hugoniot relations between shock spee (an other mechanical quantities) an impact spee are etermine from the FE simulations. It is foun that the shock spee increases approximately linearly with impact spee, similar to that observe for conense solis, but the relate material constants for cellular solis have ifferent physical implications, whereas the shock strain, stress an energy increase with impact spee nonlinearly, ue to shock-enhance cell compaction an cell-wall eformation. Base on conservation laws in continuum mechanics, other Hugoniot relations are erive from the basic linear one, which agree well with those obtaine from FE simulations. It is thus emonstrate that the unique linear Hugoniot relation can be use to characterise the shock constitutive behaviour which is istinct from the quasi-static one. Furthermore, a new analytical metho base on the linear Hugoniot relation is propose to estimate the critical impact spee for shock initiation, which has reasonable agreement with the present FE simulation an previous experimental an numerical results, an outperforms the existing methos Elsevier Lt. All rights reserve Introuction Cellular solis are characterise by high porosity (usually exceeing 70%), leaing to their istinctive mechanical, thermal, electromagnetic an other properties attractive for various engineering applications ( Gibson an Ashby, 1997 ). Uner high spee impact or intensive blast, shock compression can be initiate in cellular solis an the loa transmitte can be significantly increase ( Elnasri et al., 2007; Li an Meng, 2002; Rei an Peng, 1997; Tan et al., 2012; Tan et al., 2005a ), which may enhance the energy absorption but not benefit structural protection. Therefore, it is important to unerstan the shock behaviour of cellular solis. Extensive experiments have ientifie two prominent features of shock compression in cellular solis ( Barnes et al., 2014; Rafor et al., 2005; Rei an Peng, 1997; Tan et al., 2012; Tan et al., 2005a ): (1) a significant enhancement of stress measure at the impact en; an (2) a localisation of cell crushing ajacent to the Corresponing author. aress: qingming.li@manchester.ac.uk (Q.M. Li) / 2016 Elsevier Lt. All rights reserve. impact en. Meanwhile, numerical stuies on the ynamic crush- 17 ing of iealise 2D cellular solis have been reporte to uner- 18 stan the effects of cell irregularity ( Zheng et al., 2005 ), non- 19 uniform cell-wall thickness ( Li et al., 2007 ), cell micro-topology 20 ( Liu an Zhang, 2009 ) an structural efects ( Zhang et al., 2010 ) 21 on the shock behaviour. However, Sun et al. (2015; 2016b ) have 22 emonstrate the importance to use realistic cell geometry in the 23 simulation of ynamic crushing in orer to capture the istinctive 24 shock behaviour. Therefore, it is necessary to apply more realistic 25 cellular soli in moelling. 26 Shock, as one of the eformation moes of cellular solis, 27 is use here as a term for the propagation of the planar inter- 28 face (i.e. shock front) separating the crushe an uncrushe cells 29 in ynamic compression, which has similar feature to shock wave 30 propagation in a conense soli ( Davison, 2008; Meyers, 1994; 31 Wang, 2007 ). To emonstrate the shock feature of the shock 32 front, Zou et al. (2009) performe finite element simulations an 33 numerically observe a jump or macroscopic iscontinuity in the 34 key mechanical quantities (i.e. velocity, stress an strain) across 35 the shock front (about one cell size in thickness) of hexagonal-cell 36

3 2 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx honeycombs. Liao et al. (2013) numerically confirme the existence of shock front for iealise irregular 2D cellular structures. The measurement of the spee of shock front (i.e. shock spee for brevity) has been also attempte in experiments an simulations. For instance, Barnes et al. (2014) experimentally observe that the variation of shock spee with impact spee generally follows a linear tren for open-cell aluminium Duocel foam an a linear equation was use to fit the experimental ata. Liao et al. (2013) erive the relation between shock spee an impact spee for 2D Voronoi foam base on various iealisations of the quasi-static stress-strain relation; but their analytical preictions showe marke ifference from the numerical result, especially at high impact spees (see Fig. 13 in Ref. ( Liao et al., 2013 )). Similarly, Pattofatto et al. (2007) compare the analytical preictions of shock spee erive from the quasistatic stress-strain relation for close-cell aluminium Alporas foam an they state that their nonlinear equation (see Eq. 9 in Ref. ( Pattofatto et al., 2007 )) gave a satisfactory preiction. More recently, Zheng et al. (2014) measure the ynamic stress-strain relation for 3D Voronoi foam numerically an recommene a ynamic material moel, i.e. Eq. (14) in Ref. ( Zheng et al. (2014)), base on which, we foun that a linear relation between shock spee an impact spee, i.e. = v + D/, can be erive (see Appenix) where is the shock spee, v is the impact spee, is the ensity an D is a material constant efine therein. The simple linear relation between shock spee an impact spee seems more funamental for characterising an moelling the shock behaviour of cellular solis. However, it requires more experimental an numerical supports an a better unerstaning of its physical implications. On the other han, increasing effort s have been mae to establish the constitutive relation uner shock compression. In the early stuy, Rei an Peng (1997) propose a shock moel base on a rigi, perfectly-plastic, locking (r-p-p-l) material moel to analyse the impact response of woo, which was subsequently aopte by other researchers to analyse the shock behaviour of various cellular solis ( Li an Rei, 2006; Ma et al., 2009; Ruan et al., 2003; Tan et al., 2005b; Zou et al., 2009 ). The continuous improvement of this shock moel is mainly from the aoption of more realistic constitutive equations such as linear harening ( Zheng et al., 2012 ), power-law harening ( Zheng et al., 2013 ) an complicate nonlinear harening ( Karagiozova et al., 2012 ) equations. However, the basic assumption that the shock state of the material is inepenent of the local crushing velocity has not been change. In other wors, the quasi-static compression tests were consiere to be able to provie complete input parameters for the shock moel. Recently this assumption has been proven questionable. For instance, Zheng et al. (2014) employe a 3D Voronoi finite element moel to establish a ynamic stress-strain relation which is shown ifferent from the quasi-static one. Barnes et al. (2014) experimentally measure the mechanical variables of the Duocel foam at ifferent impact spees an emonstrate that a complete escription of shock behaviour requires the irect measurement of Hugoniot relations (i.e. the loci of all shocke states) an the material states uner shock cannot be etermine from the quasi-static stress-strain relation. These recent finings o not support the conclusion mae by Pattofatto et al. (2007) that shock enhancement effect shoul not be taken into account at the level of the constitutive law itself. Equally importantly, previous experimental an numerical stuies have shown that there exists a critical impact spee, above which shock occurs ( Barnes et al., 2014; Tan et al., 2005a; Zou et al., 2009 ). For compression at subcritical impact spees, localise cell crushing occurs in presumably weak cells or sites, an the crush bans are ranomly istribute an the bounary between crushe an uncrushe cells is not necessarily flat ( Barnes 102 et al., 2014; Liu et al., 2009; Sun et al., 2014; Tan et al., 2005a; 103 Zheng et al., 2014 ), in contrast to the shock eformation moe. The 104 critical conition for shock initiation is of funamental importance 105 an practical interest. However, it still lacks a recognise analytical 106 metho to etermine this critical impact spee, an there is con- 107 fusion about the factors that influence the critical impact spee 108 ( Wang et al., 2013 ). 109 The objective of this stuy is to clarify above outstaning is- 110 sues. A combine image-base moelling an continuum-base 111 theory were applie. A 2D virtual foam was create from the com- 112 pute tomography (CT) image of a cross-section of a close-cell 113 aluminium Alporas foam sample an a finite element (FE) moel 114 with the same meso-scale complexity of the real foam was evel- 115 ope for compression simulations. First, the quasi-static compres- 116 sion of the 2D foam was simulate for comparison purpose. Then, 117 the shock compression was simulate at ifferent impact spees 118 to obtain the complete Hugoniot relations (e.g. the epenences of 119 shock spee an material states on impact spee, an the stress- 120 strain relation) an to efine the shock constitutive relation. Fur- 121 thermore, a new analytical metho base on the basic Hugoniot 122 relation (shock spee vs. impact spee) is propose to estimate the 123 critical impact spee for shock initiation Image-base moelling Two-imensional foam 126 A iametral X-ray compute tomography (CT) slice image of a 127 cylinrical sample (Ø30 30 mm) of close-cell aluminium Alpo- 128 ras foam was use to construct a 2D virtual foam (30 30 mm) 129 with realistic cell geometry, as shown in Fig. 1. The CT scanning 130 of the real foam sample was performe in a Nikon Metris CT sys- 131 tem house in a customise bay at the Henry Moseley X-ray Imag- 132 ing Facility (HMXIF, The University of Manchester). An acceleration 133 voltage of 70 kv, a current of 280 μa, an effective voxel size of μm, an an exposure time of 500 ms for each of 2000 pro- 135 jections over 360 egrees were use. The X-ray raiographs were 136 reconstructe using Nikon Metris CT-Pro software into CT images. 137 A greyscale-base segmentation metho was then use to extract 138 the cell structure of the foam, which aopte a threshol of grey 139 values to ensure the separation of all the soli parts from the sur- 140 rouning air. 141 It is clearly seen from Fig. 1 that the cell morphology an topol- 142 ogy of the 2D virtual foam are much more complex than those of 143 the iealise 2D cellular solis such as hexagonal-cell, circular-cell 144 an Voronoi ones ( Liu et al., 2009; Sun an Li, 2015; Zheng et al., ). Some structural imperfections, e.g. corrugation, bowe or 146 amage walls, an non-uniform cell-wall thickness, are evient, 147 which are relate to the liqui-state foaming process of Alporas 148 foam ( Simone an Gibson, 1998a ). These kins of structural e- 149 fects may play important roles in the etermination of macroscopic 150 material properties ( Chen et al., 1999; McDonal et al., 2006; Si- 151 mone an Gibson, 1998b, 1998c ). However, ue to the ifficulty to 152 control the variation of the structural imperfection in the selecte 153 sample, the influence of structural imperfection will not be the fo- 154 cus of this stuy. 155 To take account of realistic geometrical features, continuum el- 156 ements have to be use in a finite element moel (FEM), which 157 ramatically increases the computational expense an may also 158 cause numerical problems ue to element istortion. The 3D FE 159 moelling of the compressive behaviour of Alporas foam has been 160 attempte by the same authors ( Sun et al., 2016b ). However, 161 the 3D FE simulation is still restricte to relatively small sam- 162 ples an less intensive eformations for limite loaing cases 163

4 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx 3 Fig. 1. (a) CT slices in three orthogonal irections of an Alporas foam sample (Ø30 30 mm) an the 3D renering of the central part (Ø20 20 mm) of the sample; (b) a binarise iametral slice corresponing to an actual cross-section of the sample Table 1 Structural parameters of the 2D foam. Average Stanar Maximum Minimum eviation Cell size (mm) Cell-wall thickness (mm) ( Jeon et al., 2010; Maire an Withers, 2014; McDonal et al., 2011; Sun et al., 2016a; Sun et al., 2016b; Sun et al., 2014 ), which hiners the analysis of shock compression relate to extreme cell crushing an complex cell-wall contact at high spee impact. Therefore, the present stuy focuses on the shock compression of the 2D virtual foam constructe from the CT slice image. As the 2D foam is base on an actual cross-section taken from its 3D counterpart, it may be regare as a 2D replica of closecell aluminium foams having similar cross-sectional geometry to Alporas foam. However, it shoul be note that 3D foams have more complex geometrical elements (e.g. cell faces) an eformation moes (e.g. membrane stretch). This fact must be borne in min as the analysis using the 2D foam moel is essentially qualitative to provie general insights. But it has avantages over the iealise foam moels (e.g. Hönig an Stronge, 2002; Liu et al., 2009; Ma et al., 2009 ) in the sense of realistically incorporating the existing structural efects. The 2D foam has a relative ensity (RD, i.e. area ratio between cell walls an foam sample) of 12.5%. The size for each cell is obtaine as the iameter of a circle having the same area as the cell an the average cell size is 2.05 mm. There are 148 cells in total, of which 111 are insie the sample an 37 are locate on the bounaries. The local cell-wall thickness at a given material point (pixel in the igital image) is efine as the iameter of the largest circle which contains the point an fits completely insie the wall ( Hilebran an Rüegsegger, 1997 ), an the 2D foam has an average cell-wall thickness of 0.36 mm. The structural parameters are summarise in Table 1. Besies the large scatter of cell size an cell-wall thickness, the cell shape also exhibits a wie variety (see Fig. 1 b), inicating the heterogeneous nature of the cell structure at the meso-scale Finite element moel 195 The meshing of the 2D foam structure obtaine from the CT 196 slice image (see Fig. 1 b) was accomplishe by using ScanIP (Sim- 197 pleware Lt, UK), which is esigne to irectly convert CT image 198 into high-quality FE mesh ( Young et al., 2008 ). A total number of ,703 linear elements with reuce integration were use, as 200 shown in Fig. 2. The FE solver Abaqus/Explicit was employe for 201 the numerical simulations. The large eformation effect was in- 202 corporate an the plane strain conition was assume. Herein, 203 the cell-wall yiel strength reporte for Alporas foam, i.e. 172 MPa 204 ( Simone an Gibson, 1998a ), was aopte in a rate-inepenent 205 perfect plasticity material moel. The elastic moulus, Poisson s ra- 206 tio an ensity were taken to be 68 GPa, 0.33 an 2710 kg/m 3, re- 207 spectively. 208 The loaing was applie across the two ens of the 2D foam 209 sample by two rigi platens (not shown in Fig. 2 ). During loa- 210 ing, the top platen move ownwar with a constant spee (var- 211 ie in ifferent loaing cases an enote as V i ), while the bot- 212 tom platen was fixe to support the sample. Frictionless contact 213 was consiere between the platens an cell walls, as well as be- 214 tween the cell walls themselves. No irect constraints were applie 215 to the foam. Uner such loaing, the foam was subjecte to uniax- 216 ial compression macroscopically. In orer to visualise an quantify 217 the heterogeneity of the eformation, we use a particle array 218 approach as illustrate in Fig. 2 where a vertical line of 79 par- 219 ticles was use to represent the average motion of 79 equally 220 space cross-sections through the foam (excluing the two en 221 surfaces), which is similar to the metho propose by Zou et al. 222 (2009). 223 The nominal (engineering) compressive stresses measure at 224 the two sample ens are obtaine as the reaction forces at the cor- 225 responing rigi platens ivie by the original with of the sam- 226 ple. The nominal (engineering) compressive strain is efine as the 227 ratio of the vertical isplacement of the loaing platen to the orig- 228 inal sample height. The shock strain for the crushe zone behin 229 shock front is efine over the original height of the shocke cells, 230 of which the etermination metho will be elaborate later when 231 presenting the relevant numerical results. 232

5 4 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx Fig. 2. The 2D foam after meshing an the representation of 1D particle array for the uniaxial compression. Each particle i is locate at the same Y coorinate as that of the corresponing transverse cross-section i of the uneforme 2D foam an the particle motion represents the average of the FE noal motion of the cross-section. Fig. 3. Quasi-static compressive eformation of the 2D foam at a nominal strain of In (a) the Mises stress istribution in the cell walls is shown by the colour scale to ientify the locations of stress concentration; whereas in (b) the cell-wall strain contours are plotte in the original reference configuration an the enlarge views show the etails of the strain istribution in one typical cell wall with its eforme configuration inicate by the arrow. (For interpretation of the references to colour in this figure legen, the reaer is referre to the web version of this article.) Results 3.1. Quasi-static compression The compression at a quasi-static strain-rate ( s 1 ) was simulate first, wherein inertia effect is negligible. The nominal stress-strain curve obtaine from the 2D compression simulation, as shown in Fig. 4, captures the key characteristics of the quasistatic stress-strain relation of cellular solis, i.e. the stress increases to a local maximum an then rops ue to cell collapse after which the stress remains more or less constant leaing to a plateau regime followe by a rapi stress rise ue to foam ensification. Fig. 3 shows the quasi-static compressive eformation of the 2D foam at a nominal strain of A crush ban (somewhat incline) forms near the top en. The cell-wall stress is mainly concentrate within the crush ban, as seen from the Mises stress istributions shown in Fig. 3 a. The plastic buckling an bening of cell walls, as a primary collapse moe, contribute most to the nominal eformation. Plastic hinges are prevalent in the cell walls, as shown in Fig. 3 b Shock compressive stress an eformation 251 With the increasing of the loaing spee the inertia effect be- 252 comes important, which can lea to a significant imbalance be- 253 tween the reaction forces at the impact an support ens. Fig shows the comparison between the quasi-static an ynamic com- 255 pressive stress-strain curves. At a moerate impact spee (e.g. V i = m / s ), the nominal stresses at the impact an support ens are 257 close an the ynamic stress-strain curve is similar to the quasi- 258 static one, suggesting that the inertia effect is insignificant in this 259 case. By contrast, when the impact spee further increases (e.g. 260 V i = 90 m / s ), the stress magnitue at the impact en becomes sig- 261 nificantly larger than both the supporting stress an the quasi- 262 static stress, inicating that in such a loaing case the global nomi- 263 nal stress-strain relation cannot escribe the constitutive compres- 264 sive behaviour. It shoul be also note that a severe stress fluc- 265 tuation is observe at the high spee impact, which partly results 266 from the contact interaction an the numerical solution algorithm 267 (Abaqus/Explicit) use in the FE simulation, as iscusse in Refs. 268 ( Zheng et al., 2014; Zou et al., 2009 ). The stress enhancement at 269

6 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx 5 velocity rop of approximately a half of the impact velocity, which 316 is able to have the resolution limite to one average cell size, since 317 the shock front usually has a thickness of at least one average cell 318 size Shock spee an material states 320 To obtain the shock spee V s, the variations of the position 321 of shock front with the time are measure for ifferent impact 322 spees, as shown in Fig. 7 a. It is clearly seen that the position 323 of the shock front is approximately linearly epenent on loa- 324 ing time, an thus the shock spee can be obtaine as the slope 325 of the linear fitting line, as suggeste in Refs. ( Elnasri et al., 2007; 326 Liao et al., 2013 ). The variation of shock spee with impact spee 327 is shown in Fig. 7 b, from which a linear relation is evient, an a 328 linear fitting leas to Fig. 4. Comparison of the quasi-static an ynamic stress-strain curves. the impact en has been generally recognise as a result of shock effect (see Section 1 ). The cell eformation uner shock compression, as shown in Fig. 5, is istinct from that uner quasi-static compression (see Fig. 3 ). Note that the cell eformation at moerate impact spees (e.g. V i = 18 m / s ) is not shown here since it is similar to the quasi-static one. Uner shock compression, the cell eformation is concentrate at the impact en, from which the cell crushing propagates towars the support en in a layer-by-layer manner, an the shock eformation is intensifie when impact spee increases, as shown in Fig. 5. This is consistent with previous numerical an experimental observations, as introuce in Section 1. It is also noteworthy that the lateral eformation uner shock is negligible, espite the cell compaction in the crushe zone. To examine the cell-wall eformation at ifferent impact spees, the plastic strain istribution in cell walls is also shown in Fig. 5. It is clearly seen that plastic hinges are wiely sprea in the cell walls, resulting from the plastic bening an buckling of the cell walls. In general, the junction portions are less eforme an the plastic eformation is mainly istribute in thin walls. Furthermore, when impact spee increases more plastic hinges are prouce in the cell walls, which has significant implication for energy absorption of foam materials uner shock compression an will be further iscusse later. Fig. 6 shows the istributions of uniaxial cross-sectional velocity. Although the original cross-section becomes uneven when cells are crushe, as shown in Fig. 5, it is expecte that the cell compaction reuces the velocity variation between ifferent material points in the crushe zone. Therefore, the velocity of the original cross-section is use to inicate the average of the vertical velocities of the material points locate within each cross-section. It is evient that the cross-sectional velocities in the crushe zone (ajacent to the top en) are close to the impact spee, while those in the uncrushe zone (ajacent to the bottom en) are almost zero. Consequently there is a jump in the velocity istribution, which efines a shock front as shown in Fig. 5. However, the shock front, which represents a macroscopic iscontinuity, actually has a finite thickness within which the material velocity changes rapily but continuously at the meso-scale. This thickness is at least one cell size for the 2D foam with heterogeneous cell structure. A similar observation has also been reporte by Zou et al. (2009) for regular hexagonal-cell honeycombs. Base on the cross-sectional velocity profile, the etermination of the position of the shock front can be implemente. The position of the shock front is etermine at the cross-section having a V s = V r + S V i (1) where V r is a reference spee ( V s intercept), V i is the impact spee 330 an S is an material parameter. Here we have V r = m / s an 331 S = accoring to Fig. 7 b. 332 As the cell crushing of the 2D foam uner shock compression 333 propagates in a similar manner to the 1D shock wave in a contin- 334 uum meium, 1D conservation laws for a continuum soli can be 335 use to establish the governing equations for the shock compres- 336 sion of the 2D foam. The conservation equations of mass, momen- 337 tum an energy in a continuum soli wherein a 1D shock wave 338 propagates are ( Wang, 2007 ) 339 V b V a = V s ( ε b ε a ) (2) σ b σ a = V s ( V b V a ) (3) U b U a = 1 2 ( σ b + σ a )( ε b ε a ) (4) where V, ε, σ an U are material velocity, engineering strain, en- 342 gineering stress an internal energy ensity, respectively; is the 343 initial ensity; the subscript s, a an b enotes the shock front, 344 the material ahea of the shock front an the material behin the 345 shock front, respectively. Note that in the above equations it is as- 346 sume that the 2D foam has been homogenise. In aition, as 347 the cell-wall material is assume to be rate-inepenent an have 348 no other irreversible processes (e.g. heating an friction) involve 349 except plastic issipation in the moelling, the internal energy is 350 equal to strain energy in this case. 351 Fig. 8 illustrates the shock propagation an the material states 352 across the shock front. Since the material ahea of the shock 353 front is uncrushe (almost uneforme, see Fig. 5 ), their e- 354 formation state can be approximate as ε a 0 (i.e. h a H a ) an 355 V a 0 m / s. From this approximation, we further have U a 0 J / m 3, 356 but the stress σ a still has to be etermine since negligible efor- 357 mation oes not necessarily lea to negligible stress (see Fig. 4 ). 358 For the material behin the shock front, the only approximation 359 can be mae is that V b V i (see Fig. 6 ) ue to the compaction 360 of the crushe cells behin the shock front. The above approxi- 361 mations together with Eqs. (1) (4) lea to the erivations of sub- 362 sequent Eqs. (5) (8) following the similar way presente in Ref. 363 ( Barnes et al., 2014 ). 364 The estimate strain behin the shock front base on 365 Eqs. (1) an (2) is 366 ε b = ε a + V b V a V s V i V r + S V i (5) The material state behin the shock front also can be irectly 367 measure from the FE results. For instance, the instantaneous 368 shock nominal strain ( ε i ) over the height of the crushe zone, b

7 6 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx Fig. 5. Shock eformation of the 2D foam (top) an the equivalent plastic strain istribution in cell walls (bottom) at a nominal strain of 0.30: (a) V i = 60 m / s ; (b) V i = 120 m / s. The cell-wall strain contours are plotte in the original reference configuration an the enlarge views show the etails of the strain istribution in one typical cell wall with its eforme configuration inicate by the arrow (m/s) velocity Cross-sectional V i = 30 m/s V i = 60 m/s V i = 90 m/s V i = 120 m/s Normalise istance from sample bottom Fig. 6. Distributions of the material velocity ( ε = ) average over the crosssections efine in the original reference configuration. The istance is normalise by the original sample height. is etermine by ( H b h b )/ H b (H h a h b )/(H h a ) when H a h a ue to the negligible eformation ahea of the shock front, where H is the initial sample height, h b an h a are the heights of the crushe an uncrushe zones measure in the current configuration, respectively. To eliminate the ata scatter cause by either the heterogeneity of the cell structure or the numerical algorithm, the shock nominal strain ( ε b, or simply shock strain) in the crushe zone is obtaine as the average over a series of mea- 377 surements up to a global nominal strain of 0.40 (equivalent to a 378 certain perio of time uring the impact). The eterminations of 379 other state variables escribe later follow the same metho. Ac- 380 coringly, the error bars of the ata presente in the following fig- 381 ures inicate the stanar eviation of the measurement at iffer- 382 ent global nominal strains. 383 Fig. 9 a shows the comparison between the ε b estimate by 384 Eq. (5) an that etermine from the FE results in a way escribe 385 above. A goo agreement is achieve an the errors are mainly 386 cause by the limite accuracy to etermine the position of the 387 shock front. Also, in Fig. 9 a the shock strain ε b is compare with 388 the quasi-static ensification strain ε qs = which is etermine 389 using an energy absorption efficiency metho ( Li et al., 2006; Tan 390 et al., 2005a ). It reveals that ε b at high spee impact is signifi- 391 cantly larger than ε qs. This issue has been also aresse in recent 392 experimental tests for open-cell aluminium foam ( Barnes et al., ) an numerical simulations for 3D Voronoi foam ( Zheng et al., ). 395 The shock stress (i.e. stress behin the shock front) can be ap- 396 proximate as 397 σ b = σ a + V s ( V b V a ) σ qs c + V i ( V r + S V i ) (6) where the approximations that V a 0 m / s, V b V i an σ a 398 σ qs c ( σ qs c = 0. 5 MPa, the quasi-static collapse stress) are aopte. 399 The approximation of σ a σ qs c is acceptable as verifie by the 400 shock supporting stress an quasi-static stress shown in Fig. 4, al- 401 though the average σ a appears slightly smaller than σ qs c, as also 402 foun in previous simulations for 2D Voronoi cellular structures 403 ( Liao et al., 2013 ). It shoul be note that in the FE measurement 404

8 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx 7 (a) Distance from sample bottom (mm) 30 V i = 30 m/s V i = 60 m/s V i = 120 m/s Shock front Linear fits (b) S hock spee V s (m/s ) FEM Linear fit Time (ms) Impact spee V i (m/s) Fig. 7. (a) The variation of the position of shock front with loaing time at ifferent impact spees; (b) the variation of shock spee with impact spee. Fig. 8. Schematic of the shock propagation in the 2D foam: (a) the position of the shock front in the original reference configuration (Lagrangian configuration); (b) shock position an material states in the current configuration (Eulerian configuration) the stresses σ a an σ b are obtaine by the average reaction forces at rigi platens ivie by the original sample with, since both material velocities in the crushe an uncrushe zones are almost constant, as shown in Fig. 6. It shoul be also note that in a recent FE analysis, Zheng et al. (2014) use a similar metho to obtain σ b, while they i not irectly measure σ a but instea they erive σ a from Eq. (3), which le to a conclusion that σ a is larger than σ qc c. In fact, the value of σ a epens on the location at which the material is consiere as immeiately ahea of shock front since there is a meso-scale continuous rop of stress from the impact en to the support en. As it still lacks a strict efinition of the shock front bounary for heterogeneous cell structure, the σ a an σ b obtaine here are base on the measurements on the support an impact ens of the sample, respectively. In Fig. 9 b, the σ b estimate by Eq. (6) an that obtaine from the FE result is also compare, which shows a reasonable agreement. Accoring to the σ b / σ qs c shown in Fig. 9 b, the shock stress is significantly larger than the quasi-static collapse stress an increases with impact spee. Using Eqs. (5) an (6), the strain energy ensity of the material 424 behin the shock front can be expresse in terms of impact spee, 425 i.e. 426 U b = U a ( σ b + σ a )( ε b ε a ) σ qs c V i + 1 V r + S V i 2 V 2 i (7) The preiction from Eq. (7) also compares well with the FE re- 427 sult, as shown in Fig. 10. Furthermore, it is seen that the energy 428 absorption is significantly enhance at high spee impact. How- 429 ever, such enhancement is not ue to the intrinsic rate epen- 430 ence of foams as suggeste by Rafor et al. (2005), since in 431 the FE moel a rate-inepenent perfect plasticity material moel 432 was aopte. A check on the internal energy calculate by Abaqus 433 confirms that the plastic issipation is ominant an the artificial 434 viscosity in explicit FE simulations ( Fleck an Deshpane, 2005 ) 435 is negligible. This inicates that the enhance energy absorption 436 capacity results from the increase cell compaction uner shock 437 compression which prouces more plastic hinges in the cell walls 438 with increasing impact spee, as shown in Fig

9 8 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx Fig. 11. Comparison between the σ b ε b Hugoniot relation an the quasi-static stress-strain relation. Note that the material states across the shock front are connecte by the Rayleigh line Fig. 9. Variations of material states with impact spee: (a) local nominal strain behin the shock front; (b) nominal stresses ahea of an behin the shock front. Fig. 10. Strain energy ensity of the material behin the shock front at ifferent impact spees. 4. Discussion 4.1. Shock constitutive relation The Hugoniot relations between the material states an impact spee have been establishe in Section 3.3, wherein the linear Hugoniot relation, i.e. Eq. (2), has been chosen as the constitutive relation to erive other Hugoniot relations in conjunction with conservation laws. It is also of interest to know the Hugoniot relations between ifferent state variables, especially that between stress an strain which is often use as a constitutive relation. It 448 is straightforwar to erive the σ b ε b Hugoniot relation by elim- 449 inating V i in Eqs. (5) an (6), i.e. 450 ( ) σ b σ qs V 2 c + ε r b 1 S ε b (8) The preiction of the above equation captures the FE stress- 451 strain states at ifferent impact spees, as shown in Fig. 11. Uner 452 shock compression the material states across the shock front are 453 connecte by the Rayleigh line. In other wors, the stress jumps 454 when the shock front passes an it oes not follow the path e- 455 fine by the quasi-static stress-strain curve. Moreover, it is evi- 456 ent that the σ b ε b Hugoniot curve is markely ifferent from 457 the quasi-static stress-strain curve, which confirms the recent ob- 458 servations for 2D Voronoi foam ( Liao et al., 2013 ), 3D Voronoi foam 459 ( Zheng et al., 2014 ) an open-cell aluminium foam ( Barnes et al., ). This funamental ifference suggests that it is not really 461 appropriate to use a quasi-static stress-strain relation to eter- 462 mine the shock state quantities as one in many previous stu- 463 ies. This nees to be realise an aresse as the quasi-static 464 stress-strain relation is still wiely use in shock analysis, e.g. 465 Refs. ( Karagiozova an Alves, 2014; Zheng et al., 2016 ). On the 466 other han, it shoul be note that the shock stress-strain rela- 467 tion applies to ifferent impact spees as long as shock is ini- 468 tiate, in contrast to the strain-rate effects at intermeiate loa- 469 ing spees which change the stress-strain curve at ifferent strain- 470 rates ( Zheng et al., 2014 ). 471 One may suggest that the σ b ε b Hugoniot relation can serve 472 as a unique ynamic constitutive stress-strain relation, as one 473 by Zheng et al. (2014). In such a treatment, other Hugoniot re- 474 lations, incluing the V s V i one, can be erive from this y- 475 namic stress-strain relation in conjunction with the conservation 476 laws, see for example the erivation presente in Appenix. How- 477 ever, we notice that to escribe this ynamic relation an empiri- 478 cal nonlinear equation is usually neee, the form of which may 479 not be unique to achieve a satisfactory ata fitting. In the recent 480 FE stuy by Zheng et al. (2014) an empirical equation ( Eq. 14 in 481 ( Zheng et al., 2014 )) having similar strain terms to Eq. (8) is pro- 482 pose to escribe the stress-strain states for 3D Voronoi foam un- 483 er shock compression. It appears that the function selection in 484 Zheng et al. (2014) for stress-strain relation is not unique. In con- 485 trast, as long as a unique linear V s V i relation exists, the form 486 of the stress-strain relation is naturally etermine, which seems 487 more logic an reuces uncertainties. Therefore, we recommen 488 the establishment of the constitutive relation through measuring 489

10 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx V s V i, as one in present numerical stuy an previous experimental stuy ( Barnes et al., 2014 ). In aition, from Eq. (8) the stress wave spee can be erive as follows C s = σ b / ε b = 1 + S ε b 1 S ε b V r (9) Accoringly, the erivative of the stress wave spee is obtaine as C s ε b = S(2 + S ε b ) (1 S ε b ) 2 1 S 2 ε 2 b V r (10) which is constantly greater than zero. This implies that the stress wave spee increases with the increase of strain, which satisfies the requirement of the formation of shock wave in a material exhibiting nonlinear stress-strain relation ( Wang, 2007 ). Therefore, the linear V s V i relation can naturally meet the necessary conition for the formation of shock wave an thus seems more funamental to characterise shock behaviour in cellular solis. Similar linear V s V i Hugoniot relations have also been observe for conense solis ( Davison, 2008; Meyers, 1994 ), porous materials ( Morris, 1991 ), polystyrene foams ( Morris, 1991 ) an open-cell aluminium foams ( Barnes et al., 2014 ). For conense solis the constant V r in Eq. (1) is normally close to the spee of soun ( Davison, 2008; Meyers, 1994 ), but it is not applicable to the 2D foam stuie here or open-cell aluminium foam ( Barnes et al., 2014 ). The soun spee C 0 for the 2D foam, which is measure as the ratio of the initial sample height to the time uration before the stress at the support en becomes non-zero, is about 4010 m/s, while the calculate soun spee of Alporas foam is about 1720 m/s base on its overall Young s moulus an ensity. Both are far larger than V r. This represents a funamental ifference in the shock physics between cellular solis an conense solis. As the coefficient S is close to one for cellular solis, the V r inicates the ifference between shock spee an impact spee (also material spee in the crushe zone accoring to Fig. 6 ). The consequence of a small V r is that the shock spee is close to material spee in cellular solis. This contrasts with most of conense solis (e.g. aluminium) wherein shock wave travels much faster than material points ( Meyers, 1994 ). Such a ifference arises from the cell structural response nature of the shock in cellular solis, i.e. the macroscopic iscontinuity across shock front is associate with the local ensification of the cells, rather than atomic-level mechanisms controlling the shock in conense solis ( Meyers, 1994 ). Nevertheless, shock response in cellular solis is still governe by macroscopic wave equations, as emonstrate in Section 3.3. It is also of interest to explore the physical implication of the coefficient S in the linear V s V i relation. It is evient that when V i the upper limit of ε b etermine by Eq. (5) is 1 /S = , which is almost ientical to the full ensification, i.e. 1 RD = when lateral eformation is negligible. It confirms that the shock occurring in the 2D foam is ominate by the structural ensification, which is intensifie when impact spee increases, as shown in Fig. 5. In other wors, increasing impact spee prouces more plastic hinges in cell walls rather than switching eformation moe to the uniaxial compression of the cell walls. Consequently, the shock strain, stress an energy all increase with impact spee Critical impact spee for shock initiation It is important to know the critical impact spee for shock initiation in cellular solis. The etermination of the critical impact spee often epens on the assume constitutive stress-strain relation. For instance, when a rigi, linear harening plastic, locking (R-LHP-L) constitutive moel is assume, the critical impact spee 548 is given by ( Wang et al., 2013; Zheng et al., 2012 ) 549 V c _ s = ε E h (11) where E h is the harening moulus, ε is the ensification strain 550 an σ 0 is the initial ensity of the cellular material. However, when 551 a rigi, perfectly plastic, locking (R-PP-L) constitutive moel is as- 552 sume, shock will be initiate at any impact spee larger than zero 553 ( Wang et al., 2013 ). This unreasonable conclusion is correcte by 554 consiering the elastic precursor wave ( Wang et al., 2013 ) an the 555 threshol of shock enhancement of stress ( Ashby et al., 2000; Tan 556 et al., 2005b ). For the former correction, the critical impact spee 557 becomes 558 E V c _ s = ε Y (12) where E is the elastic moulus an ε Y is the yiel strain (normally 559 aopte as the collapse strain). For the latter correction, the critical 560 impact spee epens on an empirical parameter α, i.e. 561 V c _ s = α σ pl ε (13) where σ pl is the plateau stress an α satisfies σ = ασ pl where 562 σ is the stress enhancement ue to shock. The plateau stress can 563 be obtaine by energy absorption equivalence at the plateau stage 564 ( Li et al., 2006; Tan et al., 2005a ). The value of α was aopte 565 as 0.1 by Ashby et al. (2000) an 2.0 by Tan et al. (2005b). Al- 566 though Tan et al. (2005b) claime that their aoption of α = is base on the so-calle kinematic existence conition for contin- 568 uing steay-shock an the preiction agrees with their experi- 569 mental results, Wang et al. (2013) prove that the theoretical basis 570 of this kinematic existence conition is incorrect an the mistake 571 is cause by the confusion in basic concept between the energy 572 conservation in an isolate system an the energy conservation 573 across a shock wave. 574 The further improvement of the preiction using a similar ap- 575 proach to that escribe above may be mae by consiering a 576 more realistic constitutive stress-strain relation, such as a non- 577 linear one. For instance, Zheng et al. (2013) aopte a power law, 578 i.e. σ = σ 0 + K ε n where σ 0 is the yiel stress, K is the strength 579 inex an n is the strain-harening inex, to escribe the quasi- 580 static stress-strain relation, an then gave the following equation 581 to preict the critical impact spee 582 V c _ s = ε (n+1) / 2 K (14) The common problem of above methos for the etermina- 583 tion of critical impact spee is that they are base on quasi-static 584 stress-strain relation, which is actually ifferent from that uner 585 shock compression. Experimental ( Barnes et al., 2014; Gibson an 586 Ashby, 1997; Sun et al., 2014; Tan et al., 2005a ) an numerical 587 ( Liu et al., 2009; Ruan et al., 2003; Zheng et al., 2005 ) observa- 588 tions have emonstrate that uner quasi-static (incluing low 589 ynamic compressions) the cell crush bans are istribute ran- 590 omly in terms of their location an orientation, which are sen- 591 sitive to the structural efects an the complicate transmission 592 of loa between neighbouring cells, in contrast to the shock e- 593 formation. Consequently, the shock stress-strain relation is istinct 594 from the quasi-static one, as shown in Refs. ( Barnes et al., 2014; 595 Liao et al., 2013; Zheng et al., 2014 ) as well as in Fig. 11. This sug- 596 gests that a constitutive relation without involving shock is not 597 applicable to shock compression, an the preiction of the criti- 598 cal impact spee for shock initiation base on continuum approach 599

11 10 Y. Sun et al. / Mechanics of Materials xxx (2016) xxx xxx Table 2 Preictions of the critical impact spee for shock initiation of the 2D foam using ifferent methos (unit: m/s). V r (n+1)/ 2 K FE result ( Eq. 15 ) ε 1 / ε sh S ( Eq. 14 ) 0. 1 σ pl ε ( Eq. 13 ) 2 σ pl ε ( Eq. 13 ) ε E Y ( Eq. 12 ) ε E h ( Eq. 11 ) Table 3 Preictions of the critical impact spee for shock initiation of open-cell aluminium foam ( Barnes et al., 2014 ) using ifferent methos (unit: m/s). Test result V r 1 / ε sh (n+1)/ 2 ( Eq. 15 ) ε S K ( Eq. 14 ) 0. 1 σ pl ε ( Eq. 13 ) 2 σ pl ε ( Eq. 13 ) ε Y E ( Eq. 12 ) ε E h ( Eq. 11 ) Table 4 Preictions of the critical impact spee for shock initiation of 3D Voronoi foam ( Zheng et al., 2014 ) using ifferent methos (unit: m/s). FE result V r 1 / ε sh (n+1)/ 2 ( Eq. 15 ) ε S K ( Eq. 14 ) 0. 1 σ pl ε ( Eq. 13 ) 2 σ pl ε ( Eq. 13 ) ε Y E ( Eq. 12 ) ε E h ( Eq. 11 ) shoul take account of the Hugoniot relations, which, however, has not been attempte before. Eq. (5) gives a relationship between the impact spee an the shock strain behin the shock front. This equation can be use to etermine the critical impact spee for the occurrence of the shock compression when the critical shock strain is etermine, i.e. V c _ s = V r 1 / ε c _ s S (15) 606 where ε c _ s is the critical shock strain, V r an S are parameters 607 in the linear V s V i Hugoniot relation, namely Eq. (1). A charac- 608 teristic strain, e.g. lock-up or ensification strain, base on quasi- 609 static stress-strain relation has been use previously to etermine 610 the critical impact spee, i.e. Eqs. (11), (13) an (14). Here, we 611 aopte ε c _ s ε sh where ε sh is the shock ensification strain ob- 612 energy absorption efficiency metho 613 which proves capable of capturing the onset of foam ensifica- 614 tion ( Li et al., 2006; Tan et al., 2005a ). Accoring to our simula- 615 tion results, i.e. ε sh = 0. 49, V r = m / s an S = , we ob- 616 taine that V c _ s = 22 m / s, which is close to the critical impact 617 spee ( 24 m/s) ientifie irectly from a series of FE simulations. 618 We also applie this equation to the experimental results for the 619 open-cell aluminium foam teste by Barnes et al. (2014), an ob- 620 taine that V c _ s = 56 m / s which falls in the spee range reporte 621 (40 60 m/s) wherein the foam compression transits into shock. For 622 3D Voronoi foam, the critical impact spee is preicte as 69 m / s 623 using the material parameters reporte in Ref. Zheng et al., 2014 ), 624 in comparison with the critical impact spee of 57 m / s reporte 625 therein. It is evient that V r /( 1 / ε sh S ) gives a quick an reason- 626 other methos base on 627 quasi-static material parameters, i.e. Eqs. (11) ((14), see the com- 628 parison in Tables 2 4. In aition, it is interesting to note that V c _ s 629 is very close to V r for the 2D foam stuie here. 630 It shoul be note that the ifference of cell eformations un- 631 er quasi-static an shock compressions has been reflecte in the 632 istinctive shock stress-strain relation use to estimate the critical 633 shock strain. For shock compression, the critical shock strain corre- 634 sponing to the onset of the collective cell ensification behin the 635 shock front. This is consistent with the shock eformation mecha- 636 nism, as shown in Fig Conclusions Base on a cross-sectional geometry obtaine from X-ray CT image of a close-cell aluminium Alporas foam sample, a 2D image-base FE moel is evelope, for the first time of this type, 640 to stuy the shock compression of cellular solis. This stuy leas 641 to the following conclusions: The shock compression occurs above a critical impact spee 643 which can be estimate by shock Hugoniot relations, an un- 644 er shock compression, the compaction in crushe cells an 645 the plastic eformation in cell walls are enhance when im- 646 pact spee increases, leaing to the nonlinear increase of shock 647 strain, stress an energy with impact spee; A linear shock Hugoniot relation between shock spee an im- 649 pact spee is obtaine, of which the constant term is much 650 lower than soun spee an implies the local structural en- 651 sification of cells, in contrast to the constant term close to 652 soun spee controlle by atomic-level mechanisms in con- 653 ense solis; Other shock state variables erive from the linear shock Hugo- 655 niot relation an conservation laws in continuum mechanics are 656 in goo agreement with the irect FE measurement, an the 657 measurement of shock spee provies sufficient information for 658 unerstaning the shock behaviour; The unique linear Hugoniot relation is preferential to charac- 660 terise the shock constitutive relation as it avois the uncer- 661 tainty about the selection of a nonlinear function for the shock 662 constitutive stress-strain relation which is istinct from the 663 quasi-static one; A new analytical metho base on the linear Hugoniot relation 665 is propose to estimate the critical impact spee for shock ini- 666 tiation, which is verifie by the present FE simulations an pre- 667 vious experimental an numerical results, an outperforms the 668 existing methos. 669 Acknowlegements 670 The authors woul like to acknowlege the use of Computa- 671 tional Share Facility at The University of Manchester (UoM). In a- 672 ition, the first author is grateful for the research scholarship from 673 UoM an for the imaging technical support from Dr T. Lowe an 674 Dr R. Braley. The first author also appreciates the iscussion with 675 Mr Y. Ding, Prof Z. Zheng, Prof L.-L. Wang an Prof S. Kyriakies 676 on the shock behaviour of foams. The Manchester X-ray Imaging 677 Facility was fune in part by the EPSRC (grants EP/F007906/1, 678 EP/F001452/1 an EP/I02249X/1 ). The secon author, as an a- 679 junct professor at Beijing Institute of Technology, acknowleges the 680