7 This paper investigates the e ect of porous microstructure on the necking formability of ductile sheets subjected 8 to dynamic in-plane stretching. We have developed an original approach in which nite element calculations which 9 include actual void distributions obtained from additively manufactured materials are compared with simulations in 10 which the specimen is modeled with the Gurson-Tvergaard continuum plasticity theory (Gurson, 1977; Tvergaard, 11 1982) which considers porosity as an internal state variable. A key point of this work is that in the calculations 12 performed with the continuum model, the initial void volume fraction is spatially varied in the specimen according 13 to the void distributions included in the simulations with the actual porous microstructure. The nite element 14 computations have been carried out for di erent loading conditions, with biaxial strain ratios ranging from 0 (plane 15 strain) to 0 :75 (biaxial tension) and loading rates varying between 10000 s1and 60000 s1. We have shown that 16 for the speci c porous microstructures considered, the necking forming limits obtained with the Gurson-Tvergaard 17 continuum model are in qualitative agreement with the results obtained with the calculations which include the 18 actual void distributions, the quantitative di erences for the necking strains being generally less than 25% (the 19 calculations with actual voids systematically predict greater necking strains). In addition, the spatial distribution 20 of necks formed in the sheets at large strains is very similar for the actual porosity and the homogenized porosity 21 models. The obtained results demonstrate that the voids promote plastic localization, acting as preferential sites 22 for the nucleation of fast growing necks. Moreover, the simulations have provided individualized correlations 23 between void volume fraction, maximum void size and necking formability, and highlighted the in uence of the 24 heterogeneity of the spatial distribution of porosity on plastic localization. 25 Keywords: 26 Formability, Necking, Inertia, Porous microstructure, Finite element simulations 27 Corresponding author. E-mail address: jarmarti@ing.uc3m.es Preprint submitted to International Journal of Plasticity September 27, 2022 2 1. Introduction 28 Ductile fracture of metals and alloys commonly occurs by plasticity-mediated growth and coalescence of voids, 29 which are generally nucleated from inclusions, second phase particles and micro-cracks formed during materials 30 manufacturing and processing (Tipper, 1949; Goods and Brown, 1979; Marino et al., 1985; Pineau et al., 2016). 31 For instance, solidi cation defects, such as porosity and hot cracking, commonly observed in metal-based additive 32 manufacturing processes (e.g. Aboulkhair et al. (2014), Gong et al. (2015), Samei et al. (2019) and Marvi-Mashhadi 33 et al. (2021)), are customarily assumed to be responsible for the inferior ductility and fatigue life of 3D-printed 34 metals as compared to their counterparts manufactured by traditional processes (casting, forging, extrusion, etc.), 35 e.g., Zhao et al. (2018), Voisin et al. (2018), Li et al. (2019) and Laursen et al. (2020). In fact, inconsistent quality 36 and densi cation problems characteristic of additively-manufactured metallic materials are considered limiting 37 factors for the application of 3D-printed parts and components in high loading environments (Slotwinski et al., 38 2014; Ngo et al., 2018; Yuan, 2019). 39 40 Nevertheless, papers published so far assessing the performance of printed structures subjected to severe me- 41 chanical solicitations are quite scarce. It has to be highlighted the recent work of Kristo ersen et al. (2020), who 42 carried out the tensile characterization and ballistic performance evaluation of additively-manufactured AlSi10Mg. 43 The printed samples contained a substantial number of voids, with the largest pores having diameter of 200m, 44 and displayed severe plastic anisotropy. Tensile and high-velocity perforation tests were repeated using tradi- 45 tionally die-cast specimens having the same chemical composition as the powder used in the 3D printing. The 46 die-cast specimens were very dense, with only a few pores of diameter smaller than 2 m, and displayed less 47 pronounced plastic anisotropy than the printed samples. In addition, the ow stress was found to be lower and 48 the fracture strain higher for the cast material. However, despite the di erences in the microstructure and in the 49 macromechanical properties of printed and cast specimens, the perforation experiments showed that the ballistic 50 perforation resistance was almost identical. Moreover, the ballistic experiments performed on the printed speci- 51 mens were simulated using ABAQUS/Explicit, modeling the target material with the isotropic Hershey-Hosford 52 yield criterion (Hershey, 1954; Hosford, 1972) with an exponent equal to 8, associated ow rule, a relationship 53 for the yield stress evolution which included strain hardening, strain rate hardening and thermal softening e ects, 54 and the Cockcroft and Latham fracture criterion. The discrepancy between numerical and experimental results for 55 the ballistic limit varied between 15% and35%, depending on the set of parameters used for the yield stress 56 evolution, with the calculations repeatedly underestimating the experimental measurements. While it is dicult 57 3 to determine the speci c reason for this disagreement {which may not be considered particularly large within the 58 context of ballistic simulations{, the authors pointed out the in uence of the constitutive model parameters in the 59 numerical predictions. In addition, no special measures were undertaken to account for the fact that the target 60 was produced by additive manufacturing, so that no information was provided regarding the e ect that porosity 61 has on the impact behavior of printed AlSi10Mg. 62 63 A physically sound constitutive framework to model the macromechanical behavior of porous metals subjected 64 to high strain rate impacts was developed by Molinari and Mercier (2001). This approach, which followed the 65 works of Carroll and Holt (1972) and Wang (1997), among others, was derived using a dynamic homogenization 66 procedure which assumes that the total stress is additively decomposed into a static part and a dynamic part. 67 The static contribution is generally computed using a Gurson-type ow potential (Gurson, 1977) and the dynamic 68 term {which assumes that the pores are initially spherical and remain spherical during loading{ accounts for the 69 e ect that the local acceleration elds that develop around the voids at high loading rates have on the macroscopic 70 material response. The dynamic stress, customarily referred to as microinertia, brings out that the initial void size 71 is a natural length scale that modulates the overall dynamic response of the material (Molinari and Mercier, 2001; 72 Czarnota et al., 2006, 2008; Jacques et al., 2015). This key feature is not captured by standard damage models in 73 which the porosity stands for the single damage parameter with no contribution of the void size. Czarnota et al. 74 (2008) and Jacques et al. (2010) implemented the constitutive framework of Molinari and Mercier (2001), coupled 75 with a speci c void nucleation model, into the nite element code ABAQUS/Explicit, and simulated the spall 76 fracture observed in the plate impact experiments performed by Roy (2003) and Lin et al. (2004). The models of 77 Czarnota et al. (2008) and Jacques et al. (2010) provided accurate predictions for the velocity in the rear side of the 78 impacted plate, and for the spatial and size distribution of the voids in the spall plane. The nite element analysis 79 also brought out the stabilizing e ect of microinertia, which controls the level of the maximum stress and the growth 80 of porosity in the spall plane. Moreover, Jacques et al. (2012) investigated the role of microinertia in dynamic 81 ductile crack growth. Finite element calculations of notched bars and edge cracked specimens subjected to dynamic 82 tension showed that microinertia e ects refrain void growth, delaying the strain localization process that precedes 83 fracture, lowering the crack speed and increasing the dynamic fracture toughness. Recently, Czarnota et al. (2017, 84 2020) employed the constitutive framework of Molinari and Mercier (2001) to investigate the e ect of microinertia 85 in the propagation of shock waves in porous ductile metals. For low shock pressures, microinertia e ects were 86 shown to be negligible and the shock structure is mostly determined by the material strain rate sensitivity and the 87 4 initial void volume fraction, with no e ect of the distribution of void sizes. The increase of the shock amplitude 88 was shown to boost the role of microinertia, so that the rate of collapse of the voids and the structure of the 89 shock front become determined by the acceleration elds that develop around the void boundaries. Speci cally, 90 microinertia e ects increased the shock width {which was scaled by the initial void radius{ and refrained the voids 91 compaction. However, to this day, the structural problems that have been addressed using the microinertia-based 92 approach are limited to the three applications discussed above: spall fracture, dynamic ductile crack growth and 93 shock wave propagation. It is therefore evident that much research still needs to be done to determine the speci c 94 loading conditions and material behaviors for which microinertia has an important e ect on the plastic ow and 95 fracture of porous metals. In fact, most of the papers available in the literature modeling localization and fracture 96 of porous metals under dynamic loading do not consider microinertia e ects (e.g., Guduru and Freund (2002), 97 Becker (2002), Osovski et al. (2015) and N'souglo et al. (2018)), which raises the question of whether the standard 98 Gurson-type porous plasticity theories yield sound results for dynamic problems, provided that the acceleration of 99 the material particles in the vicinity of the voids is not critically large. 100 101 The present paper addresses the question of the in uence of the voids distribution on the dynamic formability 102 of porous materials. For this purpose, the necking of metal sheets subjected to biaxial stretching is modeled with 103 (i) nite element simulations in which the porous microtructure is explicitly represented and (ii) the Gurson- 104 Tvergaard continuum plasticity theory (Gurson, 1977; Tvergaard, 1982). Di erent porous microstructures for 105 which the void volume fraction varies from 0:003% to0:03%, and the size of the pores between 6 m 106 and 100m, have been considered (Marvi-Mashhadi et al., 2021). The calculations have been carried out in 107 ABAQUS/Explicit (2019) for loading rates varying between 10000 s1and 60000 s1. The forming limit diagrams 108 obtained with the continuum plasticity theory nd qualitative resemblance with the calculations which include 109 the actual voids, the quantitative di erences for the limit necking strains being generally less than 25%, so that 110 the calculations containing explicitly resolved pores systematically predict greater formability limits. In addition, 111 selected calculations have been performed with the constitutive model proposed by Jacques et al. (2012), which 112 extends the Gurson-Tvergaard theory to consider microinertia e ects using the dynamic homogenization procedure 113 developed by Molinari and Mercier (2001). The stabilizing e ect of microinertia has been shown to be relevant, 114 only, after necking localization, so that the necking strains obtained with the Gurson-Tvergaard model and with the 115 formulation of Jacques et al. (2012) are virtually the same. Moreover, the calculations with explicitely resolved voids 116 have provided individualized correlations between initial porosity, maximum voids size and specimen formability, 117 5 bringing to light that the void volume fraction distribution is the main factor controlling the limit strains and the 118 spatial distribution of the necks in the localization pattern that emerges in the sheets at large strains. Note that 119 numerical simulations of specimens containing a large population of voids was performed by Becker and Callaghan 120 (2018, 2020) to assess the load path and mean stress dependence of void growth, by Marvi-Mashhadi et al. (2021) 121 to identify the role of material defects in multiple necking and fragmentation of metal rings subjected to dynamic 122 expansion, by Vishnu et al. (2022a) to investigate the in uence of porosity on the formation of multiple shear 123 bands in thick-walled cylinders subjected to rapid radial collapse, and by Vishnu et al. (2022b) to study the e ect 124 of porous microstructure on the formation of shear localization bands in thin-walled tubes subjected to dynamic 125 torsion. However, to the authors' knowledge, this is the rst paper ever that (i) computes dynamic forming limit 126 diagrams for specimens with actual distributions of porosity and (ii) provides a systematic comparison with the 127 results obtained with the Gurson-Tvergaard continuum plasticity theory. 128 2. Constitutive modeling 129 The mechanical behavior of the material is modeled as elastic-plastic, with yielding de ned by the von Mises 130 criterion (Mises, 1928) in the simulations with explicitly resolved porosity, see Section 2.2, and by the Gurson- 131 Tvergaard criterion (Gurson, 1977; Tvergaard, 1982) in the calculations with homogenized porosity, see Section 2.3. 132 The Gurson-Tvergaard criterion considers the material to display a periodic porous microstructure approximated 133 by an array of representative volume elements idealized as hollow spheres with a central hole and with matrix 134 described by the von Mises criterion. 135 2.1. General equations for elastic/plastic materials 136 We assume the additive decomposition of the total rate of deformation tensor dinto an elastic part deand a 137 plastic part dp: 138 d=de+dp(1) where the elastic part of the rate of deformation tensor is related to the rate of the stress by the following linear 139 elastic law: 140 O=L:de(2) 6 whereO=_+  is the Green-Naghdi objective derivative of the Cauchy stress tensor used by ABAQUS/Explicit 141 (in order to achieve incremental objectivity of the constitutive equations), where =_RRTwithRbeing the polar 142 rotation tensor. Note that_() denotes di erentiation with respect to time. Moreover, Lis the tensor of isotropic 143 elastic moduli given by: 144 L= 2GI0+K1 1 (3) with1andI0being the unit second-order tensor and the unit deviatoric fourth-order tensor, respectively. Moreover, 145 GandKare the shear modulus and the bulk modulus. 146 147 Assuming an associated plastic ow rule, the plastic part of the rate of deformation tensor is: 148 dp=_@ @(4) where _is the rate of plastic multiplier and  is the ow potential. 149 150 The formulation of the constitutive model is completed with the Kuhn-Tucker loading-unloading conditions: 151 _>0; 0; _ = 0 (5) and the consistency condition during plastic loading: 152 _ = 0 (6) 153 2.2. Actual porosity: von Mises yield criterion 154 Specialization of the ow potential for the von Mises (1928) yield criterion leads to: 155 7  = Y1 (7) where =q 3 2s:sis the e ective von Mises stress, with s=h1being the deviatoric part of the Cauchy 156 stress tensor, and h=1 3:1is the hydrostatic stress. Moreover, Yis the ow strength of the material: 157 Y=0 Y+K("p)n_"p _"refmT Tref (8) where0 Yrepresents the initial ow strength, Kis the plastic modulus, nis the strain hardening exponent, m 158 andare the strain rate and temperature sensitivity parameters, and _ "refandTrefare the reference strain rate 159 and temperature, respectively. The e ective plastic strain is  "p=Rt 0_"pd , with _"pbeing the e ective plastic strain 160 rate. Moreover, Tis the current temperature. 161 162 The e ective plastic strain rate is obtained from the work conjugacy relation: 163 _"p=:dp Y(9) 164 Assuming adiabatic conditions of deformation (no heat ux) and considering that plastic work is the only 165 source of heat, the evolution of the temperature is given by: 166 _T= Y_"p 0Cp(10) where0is the initial material density, Cpthe speci c heat and the Taylor-Quinney coecient. 167 2.3. Homogenized porosity: Gurson-Tvergaard yield criterion 168 Specialization of the ow potential for the Gurson-Tvergaard yield criterion (Gurson, 1977; Tvergaard, 1982) 169 leads to: 170 8  = Y2 + 2q1fcosh3q2h 2Y 1(q1f)2(11) whereq1andq2are material parameters, and fis the porosity (void volume fraction). Recall from Section 2.2 171 that andhare the e ective von Mises stress and the hydrostatic stress, respectively. The ow strength of the 172 matrix material Yis de ned by equation (8). 173 174 The e ective plastic strain rate in the matrix material is obtained assuming that the rate of macroscopic plastic 175 work is equal to the rate of e ective plastic work in the matrix material: 176 _"p=:dp (1f)Y(12) 177 Assuming the incompressibility of the matrix material, the evolution of the void volume fraction is de ned as: 178 _f= (1f)dp:1 (13) We consider that the evolution of the porosity is due only to the growth of preexisting voids (void nucleation is 179 neglected). If the initial material porosity f0is set to zero, the Gurson-Tvergaard model reduces to the von Mises 180 yield criterion. Moreover, the evolution of temperature is computed using equation (10). 181 2.4. Material parameters 182 The values of the material parameters used in the simulations reported in Section 4 are given in Table 1, 183 and they correspond to AISI 430 Steel (Vaz-Romero et al., 2015), except the parameters ,q1andq2which 184 take standard values used in the literature. Additional calculations with constitutive parameters corresponding to 185 Copper and Tantalum are included in Appendix A. The idea is to substantiate the conclusions of this paper with 186 di erent material behaviors. 187 9 Symbol Property and units Value oInitial density (kg =m3), Eq. (10) 7740 Cp Speci c heat (J =kg K), Eq. (10) 460 K Bulk modulus (GPa), Eq. (3) 196 G Shear modulus (GPa), Eq. (3) 75 0 Y Initial yield strength (MPa), Eq. (8) 175.67 K Strain hardening modulus (MPa), Eq. (8) 530.13 n Strain hardening exponent, Eq. (8) 0.167 _"ref Reference strain rate (s1), Eq. (8) 0.01 m Strain rate sensitivity exponent, Eq. (8) 0.0118 Tref Reference temperature, Eq. (8) 300  Temperature sensitivity exponent, Eq. (8) -0.51 q1 Material parameter, Eq. (11) 1.25 q2 Material parameter, Eq. (11) 1 Taylor-Quinney coecient, Eq. (10) 0.9 Table 1: Material parameters corresponding to AISI 430 Steel used in the nite element simulations reported in Section 4. Data after Vaz-Romero et al. (2015), except the parameters ,q1andq2which take standard values used in the literature. 3. Finite element modeling 188 The problem addressed is that of a plate subjected to in-plane biaxial stretching. In order to reduce the 189 computational time of the calculations, we have only modeled a representative volume element consisting of a 190 strip of initial length L0 x= 8 mm, and square cross-section of initial thickness L0 y=L0 z= 0:5 mm, with symmetry 191 boundary conditions (corresponding to a plate thickness of 1 mm), see Fig. 1. The Lagrangian Cartesian coordinate 192 system associated to the applied loading is denoted by ( X;Y;Z ). The origin of coordinates is located at the rear 193 bottom left corner of the nite element model (i.e., at the center of mass of the specimen if no symmetry boundary 194 conditions are applied). The nite element calculations are performed with ABAQUS/Explicit (2019) under the 195 following imposed initial and boundary conditions: 196 Vx(X;Y;Z; 0) = _"0 xxX;Vy(X;Y;Z; 0) = _"0 yyY;Vz(X;Y;Z; 0) = _"0 xx+ _"0 yy Z T(X;Y;Z; 0) =T0=Tref 10 Vx L0 x;Y;Z;t = _"0 xxL0 x;Vy X;L0 y;Z;t = _"0 yyL0 y Ux(0;Y;Z;t ) = 0;Uy(X;0;Z;t) = 0;Uz(X;Y; 0;t) = 0 with _"0 xxand _"0 yybeing the imposed initial strain rates, see Fig. 1. The loading condition is determined by 197 the ratio = _"0 yy=_"0 xx, which is varied between 0 (plane strain tension) and 0 :75 (near equibiaxial stretching) 198 in the calculations presented in this paper (Z-direction is parallel to the thickness of the plate). Hereinafter, 199 _"0 xxand _"0 yywill be referred to as the imposed initial major and minor strain rate, respectively, and as the 200 loading path ( _ "0 xxwill be also called loading rate). We have investigated imposed initial major strain rates varying 201 from 10000 s1to 60000 s1. These strain rates are the highest values generally attained in dynamic forming 202 processes (e.g. see Dariani et al. (2009), Golovashchenko et al. (2013) and Li et al. (2017), among others). For 203 these strain rates, the dimensionless parameter customarily used to represent the inertial resistance to motion 204 ^H1= 2r 0( _"0xxL0z)2 0 Y, see Zhou et al. (2006) and N'souglo et al. (2021), varies from 0:07 to0:4. Note that 205 nite element simulations reported by N'souglo et al. (2020) and Zheng et al. (2020) indicated that dynamic e ects 206 start to play an important role in the inception of necks for ^H1in the range 0 :060:1. Moreover, the imposed 207 initial conditions allow to minimize the propagation of stress waves within the specimen caused by the abrupt 208 motion of the strip at t= 0. Otherwise, the waves generated due to the application of the velocity boundary 209 conditions could lead to instantaneous plastic localization (Needleman, 1991; Xue et al., 2008). Moreover, note 210 that considering a thin plate subjected to dynamic loading with these initial conditions leads to the formation 211 of multiple necks at large strains (Zaera et al., 2015). Note also that for isotropic materials subjected to biaxial 212 stretching, the preferential orientation of the necks is perpendicular to the major loading direction (Zaera et al., 213 2015; Rodr guez-Mart nez et al., 2017), leading to local thinning of the thickness of the strip. Notice that the initial 214 and boundary conditions of the nite element model are consistent with a thin plate subjected to in-plane loading, 215 so that modeling a wider plate does not a ect the results. Moreover, since for the considered loading conditions 216 and isotropic material behavior the orientation of the necks is perpendicular to the major loading direction (St oren 217 and Rice, 1975), the specimen width does not a ect the necking pattern. 218 3.1. Actual porosity 219 The specimen includes the actual porous microstructure of three additively manufactured metallic materials: 220 stainless steel 316L, titanium alloy Ti6Al4V and Inconel 718, see Fig. 2. The mechanical behavior of the matrix 221 11 Figure 1: Schematic of the geometry and boundary conditions of the problem addressed: a strip of initial length L0 x= 8 mm and square cross-section of initial thickness L0 y=L0 z= 0:5 mm, subjected to biaxial stretching. The Lagrangian Cartesian coordinate system associated to the applied velocity eld is denoted by ( X;Y;Z ). The origin of coordinates is located at the rear bottom left corner of the nite element model. material is modeled with the constitutive framework described in Section 2.2, which has been implemented in 222 ABAQUS/Explicit (2019) through a user subroutine VUMAT. As in Marvi-Mashhadi et al. (2021) and Vishnu 223 et al. (2022a,b), the additively manufactured materials are solely employed to obtain representative porosity 224 distributions to be used in the nite element models with actual porosity (no attention is paid to the speci c 225 mechanical response of the printed materials from which the microstructures are taken). 226 The voids size distributions obtained from the X-ray tomography analysis carried out by Marvi-Mashhadi et al. 227 (2021) are imported to the nite element model following the methodology developed by Marvi-Mashhadi et al. 228 (2021) and Vishnu et al. (2022a,b). The voids are taken to be spherical, and they are randomly distributed in the 229 strip. The voids size distribution of the three porous microstructures {referred to as SS5Z, Ti0.5Z and INC1XY 230 in Table 1 of Marvi-Mashhadi et al. (2021){ is approximated by a Log-normal statistical function with parameter 231 values given in Table 2. For each of the three porous microstructures, we have generated three realizations of voids 232 size and positions distribution (R1, R2, R3) which ful ll the Log-normal probability function. The goal is to assess 233 the scatter in the nite element results induced by the random spatial distribution of pores and the distribution 234 of void sizes. Moreover, Table 3 shows the number of voids, the initial void volume fraction and the number of 235 voids per mm3included in the strips, for the 3 microstructures investigated, and for the 3 realizations that we have 236 generated per specimen. The di erences between the void volume fractions and the void densities obtained from 237 the experimental measurements, Table 2, and included in the nite element models, Table 3, are attributed to 238 12 the random nature of the position of the voids and to the statistical distribution of void sizes in the experimental 239 specimen (see Marvi-Mashhadi et al. (2021)), which are carried over to the computational model, leading to 240 deviations from the number of voids, the maximum size of the voids and the initial void volume fraction measured 241 experimentally (note that the X-ray tomography analysis is performed over a greater volume, see Marvi-Mashhadi 242 et al. (2021)). 243 Figure 2: Finite element model with actual porosity. Strip of initial length L0 x= 8 mm and square cross-section of initial thickness L0 y=L0 z= 0:5 mm with porous microstructure INC1XY and realization R1. The spherical voids are randomly distributed in the specimen. The distribution of void sizes ful lls the Log-normal statistical function with parameter values given in Table 2. SS5Z Ti0.5Z INC1XY Initial void volume fraction, f0(%) 0.0290 0.0033 0.0203 Voids density, N v(num.=mm3) 18 18 390 Maximum diameter of voids, d max(m) 100.00 31.44 45.52 Minimum diameter of voids, d min(m) 7.44 7.44 6.36 Mean of Log-normal distribution, (m) 18.45 14.30 6.41 Standard deviation of Log-normal distribution, dev ( m) 9.85 9.31 4.56 Table 2: Summary of the measurements obtained from the X-ray tomography analysis: initial void volume fraction f0(%), voids density N v(num.=mm3), maximum diameter of voids d max(m), minimum diameter of voids d min(m), and mean (m) and standard deviation dev ( m) values of tted Log-normal distribution. Results taken from Marvi-Mashhadi et al. (2021). The nite element models were discretized using C3D4 linear tetrahedral elements (ABAQUS/Explicit, 2019), 244 13 SS5Z Ti0.5Z INC1XY R1 R2 R3 R1 R2 R3 R1 R2 R3 Number of voids, N t(num.) 30 31 32 38 34 37 719 748 760 Initial void volume fraction, f0(%) 0.0378 0.0423 0.0117 0.0018 0.0033 0.0029 0.0193 0.0267 0.0190 Voids density, N v(num.=mm3) 15 15.5 16 19 17 18.5 359.5 374 380 Table 3: Number of voids N t(num.), initial void volume fraction f0(%) and voids density N v(num.=mm3) included in the nite element models corresponding to the 3 microstructures investigated, and the three realizations (R1, R2, R3) that we have generated per microstructure. see Fig. 3. For the microstructures SS5Z, Ti0.5Z and INC1XY the average number of elements and the standard 245 deviation are 710428 213135, 13034159903 and 2248399 225540, respectively. Following Marvi-Mashhadi et al. 246 (2021), the meridian of the smallest pore was meshed with 5 elements, such that the number of elements surrounding 247 the voids increases with the size of the pores. The calculations were performed using 48 cores of a workstation 248 with processor Dual Intel Xeon Gold 5220R @ 2 :24:0 GHz. The computational time of each simulation 249 varied between 4 hours and 6 days, depending on the microstructure considered, such that the computational cost 250 increases with the number of elements in the mesh. We have performed a mesh sensitivity analysis, increasing 251 the number of elements, and checked that the actual porosity results presented in this paper are little dependent 252 on the discretization (see Appendix B). For instance, increasing the number of elements by a factor of 7 for the 253 microstructure SS5Z and realization R1, the loading rate 60000 s1, and the loading path = 0, does not modify 254 the position and the number of necks, while the necking strain only decreases 9%. Moreover, we have also 255 carried out the same calculation using C3D10 elements, obtaining very similar necking pattern, while the necking 256 strain drops14%. 257 3.2. Homogenized porosity 258 The strip does not contain explicitly resolved voids. The mechanical behavior of the material is described with 259 the constitutive framework presented in Section 2.3, which has been implemented in ABAQUS/Explicit (2019) 260 through a user subroutine VUMAT. The porous microstructure is represented by the void volume fraction, a scalar 261 parameter resulting from the porosity homogenization, which acts as an internal variable of the constitutive model, 262 see equations (11) and (13). 263 The spatial and size distribution of the voids is captured using an original methodology which consists of 264 dividing the strip into several cubic cells of size Lc, see Fig. 4, with initial void volume fraction computed by 265 equation (14) from the positions and the sizes of the voids included in the nite element models with actual 266 porosity: 267 14 Figure 3: Finite element model with actual porosity. Cut-view of the mesh. Strip of initial length L0 x= 8 mm and square cross-section of initial thickness L0 y=L0 z= 0:5 mm with porous microstructure INC1XY and realization R1. f0 i=PNi j=01 6d3 j Vc(14) wheref0 iandNiare the initial void volume fraction and the number of voids in the cell i, respectively, Vcdenotes 268 the volume of the cells, and djis the diameter of a given void. The cell size is taken to be a sub-multiple of the 269 dimensions of the strip, and it has to be larger than the mean spacing between voids, so that the void volume 270 fraction in the cell is di erent from zero (the whole specimen domain is considered porous). For the microstructures 271 SS5Z and Ti0.5Z, the average spacing between voids is 0:4 mm, which is close to the thickness of the strip, 272 such that the cell size matches the cross-section of the specimen ( Lc= 0:5 mm). For the microstructure INC1XY, 273 the average void spacing is smaller ( 0:14 mm), so that the cell size can be taken to be either Lc= 0:5 mm or 274 0:25 mm. We have carried out simulations for both values of the cell size and have checked that the results are 275 similar (see Appendix C). All the calculations reported in Section 4 correspond to Lc= 0:5 mm (for the three 276 microstructures investigated). 277 Figures 5(a) and 5(b) compare the evolution of the initial void volume fraction along the normalized axial 278 coordinate of the strip with the initial diameter of the voids of the actual porous microstructure for SS5Z-R1 279 and INC718XY-R1, respectively. The cells with large void volume fraction correspond to big voids of the resolved 280 porosity microstructure, showing the connection between the void volume fraction heterogeneity in the homogenized 281 15 Figure 4: Illustration of the methodology used to de ne the initial void volume fraction distribution in the homogenized porosity nite element model. The strip domain is divided into several cells of size Lcwith initial void volume fraction computed from the positions and sizes of the voids included in the nite element models with actual porosity. porosity model and the void size distribution in the actual porosity model. 282 283 The nite element model was discretized using solid elements with tri-linear interpolation function, reduced 284 integration and hourglass control (C3D8R in ABAQUS/Explicit (2019) notation). The dimensions of the elements 285 are 255050m3(the smallest value corresponding to the axial direction of the strip), so that the total number 286 of elements is 32000. The calculations were performed using a single core of a laptop computer with a processor 287 Intel(R) Core(TM) i5-8350U CPU @ 1.70 GHz. The computational cost of a simulation varied between 15 and 288 60 minutes, depending on the loading path and the strain rate, i.e., these calculations are signi cantly faster than 289 the simulations with the actual porous microstructure. We have performed a mesh sensitivity analysis, increasing 290 the number of elements, and checked that the homogenized porosity results presented in this paper are hardly 291 dependent on the discretization (see Appendix B). Note that previous papers of Marvi-Mashhadi et al. (2021) 292 and Vishnu et al. (2022a,b) did not provide a comparison between calculations performed with actual porous 293 microstructures and simulations with homogenized porosity. 294 16 020406080100120 00.10.20.30.40.5 0 0.25 0.5 0.75 1B DInitial void diameter (actual porosity) Initial void volume fraction (homogenized porosity) LcMicrostructure: SS5Z - R1Initial void diameter, d ( μm)Initial void volume fraction, f 0 (%) Normalized axial coordinate, X12 34 (a) 0204060 00.020.040.060.080.1 0 0.25 0.5 0.75 1B D LcMicrostructure: INC718XY - R1 Normalized axial coordinate, XInitial void diameter (actual porosity) Initial void volume fraction (homogenized porosity)Initial void diameter, d ( μm)Initial void volume fraction, f 0 (%) (b) Figure 5: Examples of initial microstructures: (a) SS5Z and realization R1, and (b) INC718XY and realization R1. Initial diameter of the voids dalong the normalized axial coordinate X=X L0xin the actual porosity model. Comparison with the initial void volume fractionf0(%) in the homogenized porosity model. The cell size in the homogenized porosity model is Lc= 0:5 mm. Note that Lc=Lc L0xis the normalized cell size. 4. Results 295 The presentation of results is split up into four sections: the procedure for comparison of the localization 296 patterns obtained from the calculations with actual porosity and homogenized porosity is described in Section 4.1, 297 Section 4.2 investigates the in uence of the porous microstructure on the necking formability of the strip, Section 298 4.3 shows the stabilizing e ect of the loading rate on the necking strain, and Section 4.4 presents forming limit 299 diagrams for the three microstructures considered and di erent loading rates. 300 4.1. Salient features 301 The results presented in this section correspond to the three realizations generated for SS5Z. This microstructure 302 displays the greater void volume fraction and the larger voids, see Table 2, which helps to enlighten the role of 303 porosity in necking formation. The calculations correspond to = 0 and _"0 xx= 60000 s1, this combination of 304 loading path and loading rate being chosen because plane strain is the loading path that most favors the nucleation 305 of necks (see Section 4.4), and increasing the loading rate increases the number of necks (see Section 4.3). 306 307 Fig. 6 shows contour plots of e ective plastic strain  "pcorresponding to realization R1 for both the actual 308 porosity model and the homogenized porosity model. The color coding of the isocontours for the calculation with 309 explicitly resolved pores is such that e ective plastic strains ranging from 0 :55 to 0:8 correlate with a color scale 310 17 that goes from blue to red. E ective plastic strains above 0 :8 remain red and below 0 :55 remain blue. In the 311 case of the calculation with homogenized porosity, the threshold values for the e ective plastic strain are 0 :5 and 312 0:7, respectively. The snapshots are taken at the necking time tc, i.e., at the loading time leading to necking 313 formation. The necking pattern is assumed to form when a material point of the path indicated in Fig. 1 rst 314 meets the conditiond"xx dt<103s1, so that outside the necks the plastic strain rate becomes negligible. To 315 apply the necking criterion, we track the evolution of the major strain "xxfor all material points in the path along 316 time and identify the rst one reaching the necking condition. The necking criterion corresponds to the specimen 317 unloading, indicating that plastic strain has localized, giving rise to the formation of necking instabilities. As an 318 illustration, Figure D.27 in Appendix D shows that the major strain outside the necks saturates, so that the 319 material is unloaded (see also Sections 2 and 3 in Xue et al. (2008)). Moreover, note that, while 103s1is an 320 arbitrary value for the necking condition, we have checked that the necking strain is largely independent of this 321 threshold value, provided that it is small enough so that the sections outside the necks are unloaded (we have 322 checked that the results obtained withd"xx dt<104s1, 102s1and 101s1, are virtually the same). Similar 323 criterion and methodology have been used by N'souglo et al. (2021) and Jacques and Rodr guez-Mart nez (2021) 324 to determine the formation of necks in plates and rings subjected to dynamic stretching. The saturation value of 325 the major strain outside the necks is referred to as the major necking strain "c xx, and the minor strain measured at 326 the same material point, and at the same loading time, is called the minor necking strain "c yy(note that"c yy= 0 for 327 = 0, and that major and minor necking strains will be used in Section 4.4 to construct forming limit diagrams). 328 For this realization, the major necking strain and the necking time for the actual porosity model, "c xx= 0:45 329 andtc= 11:3s, respectively, are greater than for the calculation with homogenized porosity, "c xx= 0:34 and 330 tc= 8:9s, which predicts earlier plastic localization. On the other hand, the spatial distribution of the necks 331 in the strip is very similar for both modeling approaches (4 necks indicated in Fig. 6 at the locations numbered 332 in Fig. 5(a)). The correspondence between the position of the necks, Fig. 5(a), the location of large voids when 333 the strip is modeled with explicitly resolved porosity, Fig. 6(a), and the cells with greater void volume fraction 334 when the strip is modeled with the Gurson-Tvergaard plasticity theory, Fig. 6(b), makes apparent that the porous 335 microstructure plays a key role in the necks spacing. Note that we have not included contours of e ective plastic 336 strain rate in the paper as they do not provide additional information to the contours of e ective plastic strain 337 shown in Figs. 6 and 10 (the strain rate in the necked sections is large, and outside the necks, after the necking 338 criterion is met, it becomes negligible). 339 340 18 Figure 6: Microstructure SS5Z and realization R1. Contours of e ective plastic strain  "pcorresponding to calculations performed for loading path = 0 (plane strain stretching) and loading rate _ "0 xx= 60000 s1: (a) actual porosity model and (b) homogenized porosity model. The loading time corresponds to the necking condition. For interpretation of the