5 This paper provides new insights into the role of porous microstructure on adiabatic shear localization. For 6 that purpose, we have performed 3D nite element calculations of electro-magnetically collapsing thick-walled 7 cylinders. The geometry and dimensions of the cylindrical specimens are taken from the experiments of Lovinger 8 et al. (2015), and the loading and boundary conditions from the 2D simulations performed by Lovinger et al. 9 (2018). The mechanical behavior of the material is modeled as elastic-plastic, with yielding described by the 10 von Mises criterion, an associated ow rule and isotropic hardening/softening, being the ow stress dependent on 11 strain, strain rate and temperature. Moreover, plastic deformation is considered to be the only source of heat, 12 and the analysis accounts for the thermal conductivity of the material. The distinctive feature of this work is 13 that we have followed the methodology developed by Marvi-Mashhadi et al. (2021) to incorporate into the nite 14 element calculations the actual porous microstructure of 4 di erent additively manufactured materials {aluminium 15 alloy AlSi10Mg, stainless steel 316L, titanium alloy Ti6Al4V and Inconel 718{ for which the initial void volume 16 fraction varies between 0.001% and 2%, and the pores size ranges from 6m to110m. The numerical 17 simulations have been performed using the Coupled Eulerian-Lagrangian approach available in ABAQUS/Explicit 18 (2016), which allows to capture the shape evolution, coalescence and collapse of the voids at large strains. To 19 the authors' knowledge, this paper contains the rst nite element simulations with explicit representation of the 20 material porosity which demonstrate that voids promote dynamic shear localization, acting as preferential sites for 21 the nucleation of the shear bands, speeding up their development, and tailoring their direction of propagation. In 22 addition, the numerical calculations bring out that for a given void volume fraction more shear bands are nucleated 23 as the number of voids increases, while the shear bands are incepted earlier and develop faster as the size of the 24 pores increases. 25 Keywords: 26 Adiabatic shear bands, Porous microstructure, Thermal softening, Microstructural softening, Printed metals 27 Corresponding author. E-mail address. jarmarti@ing.uc3m.es Preprint submitted to { July 16, 2021 2 1. Introduction 28 The problem of shear band localization in metallic materials subjected to high strain rates has drawn the 29 attention of the Solid Mechanics community for almost a century (Zener and Hollomon, 1944). The concern on the 30 shear banding phenomena comes from the fact that shear localization frequently precedes fracture of structural 31 metals under dynamic loading conditions, e.g. during high-speed machining operations (Molinari et al., 2002; 32 Burns and Davies, 2002; Ye et al., 2013), blast and explosions (Wiehahn et al., 2000), ballistic impact (B rvik 33 et al., 2001; Zou et al., 2011) and high-velocity metalworking processes (Gaudilli ere, C. et al., 2010; Schmitz et al., 34 2020). 35 36 Many di erent experimental techniques have been developed over the last decades to study the mechanics of 37 dynamic shear banding in a controlled laboratory environment. For instance, Hartley et al. (1987), Marchand 38 and Du y (1988) and Du y and Chi (1992) used a Torsional Split Hopkinson Bar to twist thin-walled tubes at 39 nominal strain rates ranging from 103s1to 5103s1. The gauge section of the cylindrical specimens had a small 40 thickness of 0.4 mm to assist in locating and localizing the shear deformation. The results obtained for various steel 41 grades served to distinguish di erent stages in the process of nucleation and propagation of a shear band. At the 42 beginning of loading, the strain eld in the specimen was uniform; secondly, with the continuation of the loading 43 process, after the maximum stress was attained, the strain started to localize with a continuous reduction in the 44 width of the region over which the shear localization was occurring; and in the third stage the strain localized 45 further, leading to the nal fracture of the specimen. The advantage of the torsion experiments is that the thin- 46 walled tube is subjected to simple shearing during the whole loading process, so that the strain and the strain rate 47 are easily controlled (and determined), which facilitates the interpretation of results. Moreover, Ferguson et al. 48 (1967) and Campbell and Ferguson (1970) devised a double shear specimen to be tested using a Split Hopkinson 49 Bar apparatus which consists of two identical rectangular shear zones which connect a central portion loaded in 50 compression and two supporting ends. Preventing lateral displacements of the supporting ends during loading 51 {di erent designs of specimens and xtures have been developed for that purpose over the years{ it is possible to 52 minimize the specimen bending and to obtain near simple shear deformations in the gauge sections (Klepaczko, 53 1994; Rusinek and Klepaczko, 2001; Guo and Li, 2012; Shi et al., 2014). The latest advances in this technique 54 published by Xu et al. (2017, 2018, 2019a,b) enabled to study the shear deformation and fracture behaviors of 55 603 steel and pure Copper at strain rates up to 45000 s1. The stress and strain distributions were shown to 56 be rather uniform in the central part of the shear regions of the sample during the loading process. However, 57 3 the corners of the gauge sections acted as stress concentrators, promoting shear localization, so that the samples 58 generally fractured along the diagonal directions of the shear zones. On the other hand, Meyer and Manwaring 59 (1985) used a Split Hopkinson Pressure Bar to test axisymmetric hat-shaped specimens consisting of an upper hat 60 part and a lower brim part, so that when subjected to compression the material between the hat and the brim 61 parts shears. This technique enables to reach strain rates above 104s1, and has the advantage of producing 62 shear bands even in very ductile materials, making it possible to test metals and alloys with large di erences in 63 ductility using the same specimen geometry. The corners of the hat-shape specimen act as nucleation sites for 64 the shear bands, so that even materials that do not show strain localization spontaneously can be forced up to 65 shearing failure. This feature of the hat-shape sample has been exploited to study the mechanics of shear fracture 66 in a variety of metals and alloys (Meyers et al., 1995; Nemat-Nasser et al., 1998; Andrade et al., 1994; Bronkhorst 67 et al., 2006; Lins et al., 2007; Edwards et al., 2018; Lieou et al., 2019). However, this specimen design has the 68 disadvantage that a high hydrostatic pressure evolves in the shear region during loading, so that the interpretation 69 of the experimental results is not straightforward (see Peirs et al. (2010)). As an alternative to the hat-shaped 70 sample, Rittel et al. (2002) developed the shear-compression specimen, which consists of a cylinder with two slots 71 which are inclined with respect to the longitudinal axis, so that when loaded in compression, the gauge region of 72 the specimen undergoes a dominant shear deformation, leading to the formation of a shear band at large strains. 73 The shear-compression specimen is particularly attractive as it is easy to use in the standard Split Hopkinson 74 Pressure Bar setup, and it has been successfully employed to study dynamic shear localization and fracture in 75 di erent metallic materials (see Rittel et al. (2008a,b); Vural et al. (2003); Alkhader and Bodelot (2011); Dorogoy 76 et al. (2015); Zhang et al. (2021)). However, the numerical analysis of the specimen showed that the stress state 77 in the gauge section is three-dimensional, depends on the orientation of the slot, and evolves during loading (Vural 78 et al., 2011; Dorogoy et al., 2015). In addition, the traction free boundaries at the edges of gauge section act as 79 stress concentrators, favouring the shear fracture to start at these locations. 80 81 Note that the common basis for the experimental techniques described above is that they all refer to a state of 82 forced shear localization (Lovinger et al., 2015), so that both the nucleation and the trajectory of the shear band {a 83 single shear band is formed in these tests{ are controlled by the specimen geometry and the loading conditions. This 84 has advantages when it comes to obtain time and spatial resolved information of the shear localization process, yet 85 the initiation of the shear band is dictated geometrically, and therefore externally forced, so that these experiments 86 do not provide the true sensitivity of materials to shear localization. 87 4 88 In contrast, spontaneous shear localization can occur in radially symmetrical structures {like cylindrical 89 specimens{ when they are subjected to radially symmetrical loadings {e.g. internal or external pressures{, so 90 that multiple shear bands can be formed at locations that generally cannot be predicted in advance. For instance, 91 Nesterenko et al. (1989) developed the collapsing thick-walled cylinder technique in which a circular cross-section 92 cylindrical specimen is subjected to rapid radial collapse under nearly plane strain conditions, so that the state of 93 stress in the material approaches pure shear. The high pressures required for the collapse of the specimen are gen- 94 erated with the controlled detonation of an explosive charge located on the external wall of the cylinder. Multiple 95 shear bands form spontaneously {and almost simultaneously{ at large strains, being nucleated in the inner radius 96 of the cylinder where the strain is maximum, and propagate towards the external radius, with the tips following a 97 trajectory of maximum shear stress that makes a 450angle to the radial direction (Gu and Ravichandran, 2005). 98 The collapsing thick-walled cylinder technique was shown to produce controlled and repeatable helical shear band 99 patterns, which allowed to investigate {for the rst time{ the collective behavior and spacing of shear bands at 100 strain rates of104s1. Experiments performed with Copper, AISI 304L, pure Titanium, Ti6Al4V and Tantalum 101 specimens provided indications of the actual sensitivity of these materials to shear banding, and revealed that the 102 critical strain leading to shear bands formation and the average separation between shear bands varies from mate- 103 rial to material (Bondar and Nesterenko, 1991; Nesterenko and Bondar, 1994; Nesterenko et al., 1995, 1997, 1998; 104 Xue et al., 2002, 2004). Similar conclusions were obtained from the recent experiments performed by Yang et al. 105 (2011), Yang and Jiang (2016) and Yang et al. (2017) with di erent tempers of Ti-1300 titanium alloy and ZK60 106 magnesium alloy, which showed that the distribution, width, and spacing of the adiabatic shear bands varies with 107 the heat treatment applied to the specimen before testing. Collapsing thick-walled cylinder experiments using an 108 electromagnetic eld as the driving collapsing force were performed by Stokes et al. (2001). When compared to the 109 explosively driven experiments, the electromagnetic forces were shown to provide a more uniform collapse of the 110 specimen. Recently, Lovinger et al. (2011, 2015) developed a electromagnetic set-up for the collapse of thick-walled 111 cylinders at strain rates of 105s1, and tested seven metallic alloys with considerable di erences in strength 112 and ductility {AISI 304L, pure Titanium, wrought Ti6Al4V, additively manufactured Ti6Al4V, cast Al-A356, cast 113 Mg-AM50 and OFHC Copper{ in what probably is the most comprehensive experimental study on the formation 114 of spontaneous shear bands published to date. Large di erences in the number and length of the shear bands were 115 obtained for the materials tested, being steel 304L, magnesium alloy AM50 and pure Titanium the materials with 116 the greater failure strain and the smaller shear band spacing, while for OFHC Copper and aluminum alloy A356 117 5 no shear bands were observed. The experimental results, however, did not provide de nite information to explain 118 these di erences, and the mechanisms which control shear localization are still are a source of active debate in the 119 literature. 120 121 The classical explanation of Zener and Hollomon (1944) states that a shear band is formed when the ther- 122 mal softening, caused by the local increase of temperature generated by plastic dissipation at high strain rates, 123 overcomes the strain hardening of the material, leading to a rapid loss of the material capability to carry shear 124 stress. The mechanisms involved in the development of the shear band are assumed to be cumulative, i.e., the high 125 temperature band becomes progressively narrower, and ever higher strain rates and temperatures are attained in 126 this narrowing band, which accelerates the localization process (e.g., see Wang et al. (1988), Zhou et al. (1996a,b) 127 and Bonnet-Lebouvier et al. (2002)). This autocatalytic behavior, as it was named by Molinari and Clifton (1987), 128 has motivated that the dynamic shear bands were accepted as a thermomechanical (material) instability, so that 129 they have been customarily referred to as adiabatic shear bands. As stated by Molinari (1991), adiabatic shear 130 bands usually form in materials with low thermal conductivity, in which the heat conduction does not refrain local 131 heating. An alternative explanation to the formation of shear bands, which has been popularized following the 132 work of Rittel et al. (2008a), states that the dynamic stored energy of cold work {the portion of the plastic work 133 which is not dissipated into heat{ causes microstructural softening evolutions leading to shear localization (e.g., 134 see Osovski et al. (2012), Landau et al. (2016), Mourad et al. (2017), Lieou and Bronkhorst (2018), Lieou et al. 135 (2019) and Magagnosc et al. (2021)). However, the idea that microstructural evolutions promote shear localization 136 was introduced much earlier. For instance, Mathur and Backofen (1973) and Gil Sevillano et al. (1977) observed 137 that texture evolution induced softening causes shear bands to form at large strains, as further substantiated by 138 the nite element calculations reported by Dao and Li (2001), Kuroda and Tvergaard (2007) and Kuroda and 139 Needleman (2019), among others. The shear bands in which thermal softening plays negligible role are commonly 140 referred to as isothermal shear bands. Moreover, Dodd and Atkins (1983) theorized that for materials which 141 contain voids {generally nucleated from inclusions, second phase particles and micro-cracks formed during mate- 142 rials manufacturing and processing{ shear localization is possible due to the geometric softening induced by void 143 distortion and inter-void interaction. The presence of elongated and coalesced voids inside the shear bands formed 144 in the dynamic experiments of Bai et al. (1994), Xue et al. (2002), Odeshi et al. (2006), Yang et al. (2011) and 145 Lovinger et al. (2015) substantiated the hypothesis of Dodd and Atkins (1983), and made apparent that in porous 146 ductile materials dynamic shear localization is triggered by both thermal and geometric softening e ects. 147 6 148 However, few e orts have been dedicated so far to incorporate the e ect of porous softening on modeling 149 dynamic shear bands. Kubair et al. (2015) performed 1D nite di erence simulations of multiple shear band 150 formation in a metallic specimen with preexisting voids and subjected to high strain rate. The mechanical behavior 151 of the material was described with the constitutive model developed by Nahshon and Hutchinson (2008), who 152 developed a phenomenological extension of the ow potential of Gurson (1977) to account for void evolution 153 under shear deformation. It was shown that the porosity promotes early localization and reduces the shear bands 154 spacing. The constitutive model of Nahshon and Hutchinson (2008) was also employed in the recent paper of Pascon 155 and Waisman (2021) to perform 2D nite element calculations of two benchmark problems, a rectangular plate 156 subjected to axial extension and a notched bar under tensile loading. Consistent with previous results of Kubair 157 et al. (2015), the void shearing mechanism was found to accelerate material damage, leading to early shear band 158 formation. A di erent constitutive framework to account for void induced softening on dynamic shear localization 159 was employed by Dorothy and Long ere (2019). These authors used the continuum damage mechanics approach 160 proposed by Long ere and Dragon (2016) to perform 3D nite element simulations of shear band formation in 161 hat-shape specimens impacted at 30 m/s. The void softening e ect was derived from a phenomenological potential 162 which degrades the material properties as deformation evolves, so that the calculations showed that voids softening 163 accelerates the propagation of shear bands. However, while the qualitative and quantitative results reported by 164 Kubair et al. (2015), Pascon and Waisman (2021) and Dorothy and Long ere (2019) are physically sound and seem to 165 be consistent with the experimental evidence, it has to be noted that they were obtained using ad-hoc de nitions of 166 the e ect of void softening in the formation and development of shear bands, i.e., the role of porosity on shear band 167 localization was determined beforehand by the functional form of the constitutive relations employed. Note also 168 that in the constitutive models of Nahshon and Hutchinson (2008) and Long ere and Dragon (2016) the porosity is 169 represented by a damage function which does not account for the e ect of the spatial and size distributions of voids 170 in the mechanical behavior of the material, disregarding the inherent heterogeneity of the porous microstructure of 171 metals and alloys. These limitations of the approaches that are currently used to model the e ect of void induced 172 softening on dynamic shear banding have prevented from obtaining a complete understanding of the role of porosity 173 on shear localization, showing the need for developing alternative methodologies which consider additional features 174 of the materials' porous microstructure in the simulation of shear localization problems. 175 176 This is precisely the goal of this paper, in which we perform 3D nite element simulations of the formation of 177 7 spontaneous multiple shear bands in collapsing thick-walled cylinders which contain populations of voids represen- 178 tative of the porous microstructure of four di erent additively manufactured materials. The voids are randomly 179 distributed in the specimens, varying the initial void volume fraction between 0.001% and2%, and the 180 pores size from6m to110m. The mechanical behavior of the material is modeled as elastic-plastic, 181 with yielding described by the von Mises criterion, being the ow stress dependent on strain, strain rate and 182 temperature. The numerical simulations have been performed using the Coupled Eulerian-Lagrangian approach 183 available in ABAQUS/Explicit (2016), which allows to capture the shape evolution, coalescence and collapse of the 184 voids at large strains. The geometry and dimensions of the cylindrical specimens, and the loading and boundary 185 conditions, are taken from the experiments of Lovinger et al. (2015), and from the 2D simulations performed by 186 Lovinger et al. (2018), respectively. To the authors' knowledge, this paper contains the rst nite element simu- 187 lations ever performed of the rapid collapse of thick-walled cylinders which include an explicit representation of 188 the porous microstructure of the material, so that the e ect of voids evolution on dynamic shear localization is 189 unequivocally determined. Speci cally, the calculations have shown that the pores favor early plastic localization, 190 act as preferential sites for the nucleation of shear bands, speed up their development, and tailor their orientation. 191 2. Constitutive framework 192 The constitutive model used to describe the mechanical behavior of the collapsing cylinders follows the standard 193 principles of von Mises (1928) plasticity. The rate of deformation tensor dis decomposed as the sum of an elastic 194 partdeand a plastic part dp: 195 d=de+dp(1) The relation between the elastic part of the rate of deformation tensor and the stress rate is given by the 196 following hypo-elastic law: 197 _=C:de(2) where _is an objective derivative of the Cauchy stress tensor {which corresponds to the Green-Naghdi rate in 198 ABAQUS/Explicit (2016){ and Cis the tensor of isotropic elastic moduli given by: 199 8 C= 2G~I+K1 1 (3) whereGis the elastic shear modulus, Kis the bulk modulus, 1is the second-order unit tensor and ~Iis the 200 fourth-order deviatoric unit tensor. 201 202 Moreover, the plastic part of the rate of deformation tensor is related to the Cauchy stress tensor by an 203 associated ow rule: 204 dp=_@f @(4) where _is the plastic ow proportionality factor and f= Y0 is the yield function, with  =q 3 2s:s 205 being the von Mises e ective stress, where s=h:1andh=1 3:1. Moreover, Yis the yield stress of the 206 material de ned as: 207 Y=0("p+"0)n_"p _"0mT T0 (5) where "p=Rt 0_"p( )d is the e ective plastic strain, _"p=q 2 3dp:dpis the e ective plastic strain rate and Tis the 208 material temperature. The parameter 0is the initial yield stress, and n,mandare the strain hardening, strain 209 rate sensitivity and temperature sensitivity exponents, respectively. Moreover, "0, _"0andT0are the reference 210 strain, strain rate and temperature. 211 212 The energy equation is taken to be of the form: 213 Cp_T= _"p+kr2T (6) where plastic dissipation is the only source of heat. Moreover, is the current material density, Cpthe speci c 214 heat, the Taylor-Quinney coecient and kthe thermal conductivity. 215 216 9 The baseline constitutive parameters used in the calculations presented in Sections 5 and 6 correspond to 217 Titanium and CRS 1018 Steel, and they are taken from Molinari (1997), see Table 1. 218 Symbol Property and units Titanium CRS 1018 Steel 0 Initial density (kg =m3) 4510 7800 G Elastic shear modulus (GPa), Eq. (3) 43.28 78.85 K Bulk modulus (GPa), Eq. (3) 120.83 170.83 Cp Specic heat (J/Kg K), Eq. (6) 528 500 k Thermal conductivity (W/m K), Eq. (6) 19 50 Taylor-Quinney coecient, Eq. (6) 0.9 0.9 0 Initial yield stress parameter (MPa), Eq. (5) 1171.45 639.77 n Strain hardening exponent, Eq. (5) 0.15 0.015 m Strain rate sensitivity exponent, Eq. (5) 0.033 0.019  Temperature sensitivity exponent, Eq. (5) -1.7 -0.38 "0 Reference strain, Eq. (5) 5 :771035:77103 _"0 Reference strain rate (s1), Eq. (5) 0.001 0.001 T0 Reference temperature (K) , Eq. (5) 293 293 Table 1: Numerical values of initial density, elastic constants, physical constants and parameters of the yield stress corresponding to Titanium and CRS 1018 Steel. Data after Molinari (1997). 3. Porosity distribution 219 In this section, we brie y recall the methodology developed by Marvi-Mashhadi et al. (2021) to determine the 220 spatial and size distribution of pores present in four additively manufactured metals: aluminium alloy AlSi10Mg, 221 stainless steel 316L, titanium alloy Ti6Al4V and Inconel 718. 222 223 Cylindrical specimens with diameter and height of 6 mm were printed by Selective Laser Melting by the 224 company Materialise. In this paper, we only consider 4 out of the 12 samples investigated by Marvi-Mashhadi 225 et al. (2021), namely: Al3XY, SS5XY, Ti0.5XY and INC1XY, where Al, SS, Ti and INC stand for the material, 226 the following numbers indicate the nominal void volume fraction (%) provided by the manufacturer and XY tells 227 that the samples were printed in parallel to the longitudinal axis of the cylinder. These four specimens are selected 228 because they contain microstructures with large di erences in void volume fraction, number of voids, average void 229 size, and maximum void size, see Table 2, so that it is possible to obtain a clear depiction of the e ect of porosity on 230 the nucleation and collective behavior of shear bands at high strain rates. Moreover, analyzing 4 microstructures, 231 instead of 12, facilitates the presentation of results. Note that in this paper the additively manufactured materials 232 are solely employed to obtain representative porosity distributions to be used in the simulations of Sections 5 and 6; 233 10 the mechanical behavior of the collapsing cylinders being described by the constitutive model presented in Section 234 2 and the parameters of Table 1. 235 236 The porous microstructure of the printed materials was characterized by X-ray Computed Tomography (XCT). 237 The reconstruction of the tomograms was performed using an algorithm based on the ltered back-projection 238 procedure for Feldkamp cone beam geometry (e.g. see Stef et al. (2018)). Further analysis of 3D reconstructed 239 images was performed using the open source software tool ImageJ (Schneider et al., 2012), so that for each sample 240 we selected three cylindrical sub-volumes with diameter and height of 1.5 mm, with no overlapping between them, 241 to obtain the spatial distribution, shape and size of the voids in the samples investigated. For all the specimens, 242 the voids were roughly spherical and were rather homogeneously/uniformly distributed in the bulk of the material. 243 Note that voids with diameter smaller than 6 m could not be detected with the resolution of 3 33m3of voxel 244 size used in the X-ray tomography analysis. The experimental measurements for the void size distribution were 245 tted to a Log-normal statistical function with parameters given in Table 2. 246 Al3XY SS5XY Ti0.5XY INC1XY VVF 0(%) 2.17 0.0025 0.0013 0.0203 Voids (num:=mm3)5985 17 5 390 dmax(m) 110.53 41.40 26.93 45.52 dmin(m) 8.00 7.40 8.27 6.36 (m) 15.98 11.21 17.30 6.41 dev(m) 4.57 5.13 5.03 4.56 Table 2: Summary of the measurements obtained from the reconstructed 3D tomograms: initial void volume fraction ( VVF 0), number of voids per mm3(Voids ), maximum diameter of voids ( dmax), minimum diameter of voids ( dmin), and mean ( ) and standard deviation (dev) values of tted Log-normal distribution. Results taken from Marvi-Mashhadi et al. (2021). For a complete description of the X-ray tomography characterization of the porous microstructures considered 247 in this work, including 3D reconstructions of the distribution of voids in the specimens, the reader is referred to 248 Section 2 of Marvi-Mashhadi et al. (2021). 249 4. Finite element model 250 This section presents the 3D nite element model developed in ABAQUS/Explicit (2016) to investigate multiple 251 shear banding in thick-walled cylinders subjected to rapid radial collapse. While this problem is customarily 252 simulated using a 2D approach (Areias and Belytschko, 2007; Lovinger et al., 2011; Liu et al., 2016; Wang et al., 253 2020), in this paper we have developed a 3D version of the numerical model employed by Lovinger et al. (2018) 254 11 {which is based on the experimental setup of Lovinger et al. (2015){ in order to consider explicitly the porous 255 microstructure of the specimen. The nite element model consists of three sandwiched cylinders: an inner copper 256 cylinder (inner diameter 3.25 mm, outer diameter 3.5 mm), the specimen of the tested material (inner diameter 257 3.5 mm, outer diameter 5.0 mm) and an outer copper cylinder (inner diameter 5.0 mm, outer diameter 5.5 mm). 258 Material points are referred to using a cylindrical coordinate system with positions in the reference con guration 259 denoted as ( R;;Z). The origin of coordinates is located in the center of mass of the three coaxial cylinders, 260 see Fig. 1. The outer boundary of the external copper cylinder is subjected to the same pressure load used in 261 the simulations of Lovinger et al. (2018), which consists of half sinusoidal wave with an amplitude of 25 kbars 262 and a duration of 2.2 s. The inner copper cylinder controls the extend of the collapse and acts as a barrier to 263 the formation of opening cracks. We have only analyzed one eighth of the coaxial cylinders in order to reduce 264 the computational time, since very ne mesh is needed to capture the geometry of the porous microstructure 265 of the specimen (see the discussion below). The lateral surfaces of the modeled portions of the three coaxial 266 cylinders cannot move along the circumferential direction to maintain the axial symmetry of the problem, see 267 Fig. 1. Similarly, the axial length of the coaxial cylinders is taken to be 0.25 mm (while in the experiments of 268 Lovinger et al. (2011) varied between 15 mm and 30 mm), and their ends cannot move along the axial direction, 269 so that the kinematics of the simulations is consistent with the nearly plane strain conditions obtained in the mid 270 section of the cylinders in the experiments (Lovinger et al., 2011, 2018). Moreover, we assume that the dimensions 271 of the numerical model enable to include sucient number of voids in the specimen to obtain results which are 272 representative of the porosity distributions investigated. 273 274 The novelty of the numerical shear banding analysis carried out in this paper is that we have included in the 275 specimen the actual porous microstruture of four di erent additively manufactured materials, see Section 3. For 276 that purpose, we have followed the methodology developed by Marvi-Mashhadi et al. (2021), assuming that the 277 voids are spherical, and that they are randomly distributed in the material, consistently with the observations 278 derived from the X-ray tomography analysis, see Section 3. The steps taken to generate the porous microstruc- 279 ture are as follows. Firstly, a Force Biased Algorithm (Bargie l and Mo sci nski, 1991) is used to obtain a spatial 280 distribution of Mspheres, S(Ri;i;Zi;di), in a cubic volume element of 1 11 mm3,Ri, iandZibeing the 281 coordinates of the centers of the spheres and dithe diameters of the spheres, where the sub-index idenotes the 282 sphere number. The number of spheres and the distribution of spheres diameters are taken from the X-ray tomog- 283 raphy measurements satisfying the Log-normal distribution with parameters given in Table 2. The intersection 284 12 Al3XY SS5XY Ti0.5XY INC1XY Voids 171729 52 33 10415 Table 3: Average number of voids and corresponding standard deviation ( Voids ) in the thick-walled cylindrical specimens modeled in ABAQUS/Explicit (2016) for the 4 microstructures investigated. between spheres is not allowed, so that any overlapping ocurring during the generation of the microstructure is 285 eliminated removing one of the overlapped spheres (the maximum number of spheres removed in this process is 286 less than 2% of the total). This operation is repeated to obtain 5 random packed con gurations of 1 11 mm3287 which are placed next to each other {such that the whole specimen is included inside the arrangement of packed 288 con gurations{ to generate the porous microstructure of the specimen. The spheres that lie outside the specimen 289 boundaries, as well as the spheres overlapping with the boundaries of the specimen are dismissed, see Fig. 1. 290 Next, the remaining spheres which lie inside the specimen boundaries are cut out to obtain actual voids in the 291 nite element model. For each of the 4 porous microstructures considered (see Section 3) we have created three 292 realizations of void size and position distribution which ful ll the same Log-normal statistical function, in order 293 to take into account the scatter in the nite element results caused by the random spatial distribution of pores. 294 These three realizations will be referred to as R1, R2 and R3. The average number of voids and the corresponding 295 standard deviation included in the portion of the collapsing cylinders modeled, for the three realizations that we 296 have generated per sample, and for the 4 microstructures investigated, are shown in Table 3. Note that the inner 297 and outer copper rings are taken to be fully dense. 298 299 The numerical simulations have been performed using the Coupled Eulerian-Lagrangian approach available in 300 ABAQUS/Explicit (2016) which allows to combine Lagrangian and Eulerian elements in the same analysis, so 301 that while the inner and outer copper cylinders are modeled with Lagrangian elements, the specimen is considered 302 to be Eulerian, see Fig. 2. Modeling the specimen as Eulerian material allows to capture the shape evolution, 303 coalescence and collapse of the voids at large strains without the use of any failure criterion (Becker, 2017; Becker 304 and Callaghan, 2020), and prevents from element distortion problems that appear in Lagrangian analysis involving 305 extreme deformations {recall that in a Lagrangian analysis nodes are xed within the material, so that the elements 306 deform as the material deforms{. In an Eulerian analysis nodes are xed in space, and material ows through 307 elements that do not deform, so that the Eulerian mesh extends beyond the initial specimen boundaries, giving the 308 material space in which to move and deform, see Fig. 2. Eulerian and Lagrangian parts of the nite element model 309 interact each other through the Eulerian-Lagrangian contact, which is based on an enhanced immersed boundary 310 13 Figure 1: 3D nite element model developed in (ABAQUS/Explicit, 2016) to investigate multiple shear banding in thick-walled cylinders subjected to rapid radial collapse. The model consists of three sandwiched cylinders: an inner copper cylinder, the specimen and an outer copper cylinder. The pressure load is applied on the outer boundary of the outer copper cylinder. The specimens contains a spatial and size distribution of spherical voids representative of the microstructure Al3XY. 14 method (ABAQUS/Explicit, 2016). In this method, the Lagrangian mesh occupies void regions inside the Eulerian 311 mesh. The contact algorithm automatically computes and tracks the interface between the Lagrangian and the 312 Eulerian materials, and the contact constraints are enforced using a penalty method. The frictionless and hard 313 pressure-overclosure contact constraint available in ABAQUS/Explicit (2016) has been used to de ne the contact 314 between the Eulerian specimen and the Lagrangian copper cylinders. On the other hand, the Eulerian-to-Eulerian 315 contact during the pores collapse occurs by default in ABAQUS/Explicit (2016) with a sticky behavior because of 316 the kinematic assumption that a single strain eld is applied to all materials within an element. 317 318 Figure 2: Top view of the 3D nite element model developed in ABAQUS/Explicit (2016) to investigate multiple shear banding in thick-walled cylinders subjected to rapid radial collapse. Identi cation of the Eulerian domain and the Lagrangian elements. The Eulerian domain (specimen domain) is discretized with eight-node thermally-coupled solid Eulerian ele- 319 ments with reduced integration (EC3D8RT in ABAQUS notation), with initial dimensions 444m3. The 320 volume fraction of material associated with each Eulerian element at the beginning of the computation has to be 321 initialized {note that at a particular time step, a single Eulerian element can contain material, void or both{ using 322 the Volume Fraction Tool available in ABAQUS. Moreover, the inner and outer copper cylinders are discretized 323 15 with eight-node thermally-coupled solid elements, with reduced integration and hourglass control (C3D8RT in 324 ABAQUS notation), with dimensions 101010m3, so that the whole nite element model contains 16 325 million elements. The calculations were performed using two workstations with 12 and 48 cores, respectively, with 326 the following features Intel Xeon Gold 6128 @ 3.7 GHz and Intel Xeon Gold 5220R @ 4.0 GHz. The computational 327 cost of each simulation shown in this paper ranged between 3 and 7 days, depending on the the loading velocity, 328 the material behavior and the microstructure considered, using simultaneously all the cores of the workstations 329 indicated before. 330 331 A key point for the analysis is to obtain the evolution of the void volume fraction in the specimen during the 332 simulations. For that purpose, we have developed the following two-steps methodology. 333 1. The matrix material volume, MMV , at each time step, is computed as. 334 MMV =nelemX n=1MVFnEVOLn (7) where,nelemis the number of Eulerian elements, and MVF andEVOL are the material volume fraction and 335 the volume of the corresponding Eulerian element, respectively. 336 337 2. Top-view and bottom-view screenshots of the sample are exported to MATLAB®as greyscale images with 338 contrasting background. Images obtained for all time steps are processed using the Image Processing module 339 of MATLAB®to nd the area of the specimen contained in the R-  plane in both top-view and bottom- 340 view. We assume a linear variation of the area in the axial direction (since there is a small di erence between 341 the areas of both the surfaces after shear bands are formed). The total volume, V, at each time step, is then 342 approximated by multiplying the calculated average area of the top and bottom surfaces by the axial length 343 of the specimen (0.25 mm). 344 The void volume fraction, VVF , at each time step, is then computed as. 345 VVF =VMMV V(8) 346 The constitutive behavior of the specimen presented in Section 2 has been implemented into ABAQUS/Explicit 347 16 (2016) through a user subroutine VUMAT. Moreover, the constitutive model used for the inner and outer copper 348 cylinders is preimplemented in ABAQUS/Explicit (2016) and consists of linear elasticity and von Mises plasticity 349 with hardening described by the Johnson-Cook model (Johnson and Cook, 1983). The material parameters for 350 copper are taken from Table 1 of Lovinger et al. (2018). 351 5. Salient results 352 The nite element simulations reported in this section correspond to Titanium, see Table 1. In Section 5.1, we 353 compare calculations performed with fully dense material and with porous microstructure Al3XY for realization 354 R1. The results illustrate the e ect of actual voids on dynamic shear localization. Note that the microstructure 355 Al3XY contains the larger amount of voids, and the greater voids, among the porous microstructures considered, 356 see Table 3. Section 5.2 compares results obtained with realizations R2 and R3 of the microstructure Al3XY. The 357 topology of the porosity distribution is shown to a ect the localization pattern and the speci c locations where the 358 shear bands nucleate. In Section 5.3, we perform calculations with realization R2 of porous microstructure Al3XY 359 for di erent values of the temperature sensitivity exponent , see Table 1. The results bring to light the in uence 360 of thermal softening on dynamic shear localization in porous materials. 361 5.1. The comparison between porous and fully dense materials 362 Fig. 3 shows contours of e ective plastic strain  "pfor microstructure Al3XY and realization R1. Four di erent 363 loading times are considered. t= 0.5s, 1s, 1.5s and 2s. The color coding of the isocontours is such that 364 e ective plastic strains ranging from 0 to 1.5 correlate with a color scale that goes from blue to red. E ective 365 plastic strains above 1.5 remain red. Note that the inner and outer copper cylinders are not shown. Subplots 366 (a)-(b)-(c)-(d) correspond to solid representations of the specimen which illustrate the formation and development 367 of multiple shear bands, while subplots (d)-(f)-(g)-(h) correspond to empty representations of the specimen which 368 show the evolution of the porous microstructure. For t= 0.5s the distribution of plastic strains in the specimen 369 is largely homogeneous, the deformation gradients being imperceptible for the colour scale used to represent the 370 isocontours. The microstructure remains virtually undeformed with the pores being almost spherical. For t= 1s 371 we observe the onset of plastic localization in speci c locations of the specimen which correspond to the position 372 of large voids and clusters. The zoomed areas of the porous microstructure display deformed voids leading to 373 geometric softening and promoting localization (e.g. see Nahshon and Hutchinson (2008) and the discussion in the 374 following paragraphs). The nucleation of multiple shear bands in the inner radius of the specimen becomes apparent 375 fort= 1.5s. Consistent with experiments reported elsewhere (Nesterenko et al., 1995; Lovinger et al., 2011), 376 17 the shear bands propagate towards the external radius of the sample, along trajectories that make a 450angle 377 to the radial direction. The zoomed area of the microstructure shows that the voids are elongated and sheared, 378 nding close resemblance with the shape of the pores observed inside shear bands in the collapsing thick-walled 379 experiments of Xue et al. (2002) and Lovinger et al. (2015) {see Figs. 8 and 9 therein{. For a loading time of 2 s, 380 the specimen contains multiple shear bands with di erent degrees of development. The shear bands intersect each 381 other, displaying irregular trajectories modulated by the porous microstructure (intervoid interaction, as stated by 382 Dodd and Atkins (1983)), leading to a topological shear bands distribution originated from the spatial distribution 383 of voids (shear bands are linking the voids, as in the experiments with voided materials of Weck et al. (2006)). 384 The intersection of shear bands occurs near large pores and clusters that triggered early plastic localization (as 385 mentioned before) and at the boundaries of the specimen (edge e ects, e.g. see Zaera et al. (2015)). For this 386 loading time, the void volume fraction has been signi cantly reduced (see Fig. 8 and the discussion therein), and 387 many pores have collapsed, specially near the outer radius of the specimen. The closure of pores is caused by 388 the hydrostatic pressure that develops in the specimen during loading (which is most likely attributed to dynamic 389 e ects). 390 391 Fig. 4 shows contours of e ective plastic strain for the fully dense material. No voids are included in the 392 nite element model, and the small round-o errors present in the nodal coordinates, and the errors associated 393 to the numerical integration of the balance laws and the constitutive equations, are responsible for breaking the 394 symmetry of the problem and triggering localization (see the nite element calculations performed by Guduru 395 and Freund (2002) and Rusinek and Zaera (2007), among others, for ring expansion problems in which numerical 396 perturbations trigger the formation of multiple necks along the circumference of the specimen). Similar strategy 397 was followed by Lovinger et al. (2018), who performed 2D calculations of collapsing thick-walled cylinders with a 398 Lagrangian hydrocode in which the unstructured mesh used to discretize the specimen was responsible for breaking 399 the symmetry of the problem. Lovinger et al. (2018) showed that, while the initial perturbations acted as a trigger 400 for the localization to occur, the shear bands distribution was eventually determined by the waves travelling along 401 the circumference of the cylinder and by the mechanical behavior of the material. The colour coding is the same 402 used in Fig. 3, however, the contours correspond to greater loading times, namely t= 2s, 2.5s, 3s and 403 3.5s, showing that the localization process is delayed for the fully dense material (i.e., porosity favours shear 404 localization). For t= 2s, the plastic ow has not localized yet, so that the plastic strains increase smoothly 405 towards the inner radius of the specimen. Note that for the same loading time the shear bands pattern was fully 406 18 Figure 3: Results corresponding to Titanium with microstructure Al3XY and realization R1. Contours of e ective plastic strain  "pfor di erent loading times: (a)-(e) t= 0.5s, (b)-(f)t= 1s, (c)-(g)t= 1.5s and (d)-(h) t= 2s. Subplots (a)-(b)-(c)-(d) correspond to solid representations of the specimen which show the formation and development of multiple shear bands. Subplots (d)-(f)-(g)-(h) correspond to empty representations of the specimen which show the evolution of the porous microstructure. 19 formed for the microstructure Al3XY{R1, see Fig. 3(d). With the continuation of the loading process, a series of 407 shear bands become visible for t= 2.5s. These bands further develop to form at t= 3s an array of spiral shear 408 bands which progress towards the outer radius of the specimen. At t= 3.5s the e ective plastic strain inside the 409 shear bands is above 4. Note that the shear bands for the fully dense material are more regularly spaced than in 410 the case of the microstructure Al3XY{R1, and their degree of development is also more uniform {compare Figs. 411 3(d) and 4(d){. 412 Quantitative results of the development of the shear bands patterns are provided in Fig. 5, which shows the 413 evolution of the e ective plastic strain  "pin the inner radius of the specimen with the normalized inner perimeter 414 (also referred to as normalized radial angle in the gures) of the ring P=4 for the porous microstructure Al3XY{ 415 R1 and the fully dense material, Figs. 5(a) and 5(b), respectively. The loading times are the same considered 416 in Figs. 3 and 4. The y-axis scale is half in Fig. 5 (a) because plastic localization occurs earlier in the porous 417 material. Note that for the porous microstructure the e ective plastic strain attains a maximum value of 2 at 418 2s, while for the fully dense material this level of strain is reached at 3 s (see the dashed yellow lines). The 419 "p-Pcurves, which are roughly horizontal at the beginning of loading (see the dashed blue curves), eventually 420 display a succession of peaks and valleys for later loading times, where the peaks denote the formation of shear 421 bands. Consistent with the observations derived from Figs. 3 and 4, while the number of shear bands (peaks) is 422 larger in the case of the porous microstructure, their relative degree of development (relative height of the peaks) 423 is more heterogeneous. Assuming that each peak represents a shear band, the average shear bands spacing for 424 the porosity distribution Al3XY{R1 (14 peaks at t= 2s), referred to the undeformed con guration, is 0.2 425 mm. This value is in reasonable agreement with the experimental measurements of Xue et al. (2002) and Lovinger 426 et al. (2015), who reported values for the spacing of 0.18 mm and0.21 mm, respectively, for commercial pure 427 Titanium, and0.53 mm and0.46 mm for Ti6Al4V. The most developed shear bands, 6 in the case of the 428 porous microstructure and 7 in the case of the fully dense material, are indicated in the graphs, and in the contour 429 plots of Figs. 3(d) and 4(d). 430 5.2. The comparison between di erent microstructural realizations 431 Fig. 6 compares contour plots of e ective plastic strain for calculations performed with the porous microstruc- 432 ture Al3XY and realizations R2 and R3. The results correspond to two di erent loading times, 1 s and 2s. 433 Qualitatively, the number, distribution and degree of development of the shear bands is similar for both real- 434 izations. The localization patterns are irregular, so that the shear bands intersect and branch/bifurcate as they 435 propagate towards the outer radius of the specimen, showing a de nite resemblance with the experimental ob- 436 20 Figure 4: Results corresponding to fully dense Titanium. Contours of e ective plastic strain  "pfor di erent loading times: (a) t= 2s, (b)t= 2.5s, (c)t= 3s and (d)t= 3.5s. 21 Figure 5: Results corresponding to Titanium. E ective plastic strain  "pversus normalized inner perimeter of the ring P=4 . (a) Microstructure Al3XY and realization R1. Four di erent loading times: t= 0.5s, 1s, 1.5s and 2s. (b) Fully dense material. Four di erent loading times: t= 2s, 2.5s, 3s and 3.5s. The horizontal yellow dashed lines correspond to  "p= 2. For interpretation of the