5 In this paper, we have investigated void growth in ductile materials with actual porous microstructures. For that 6 purpose, we have performed calculations of cubic unit-cells subjected to periodic boundary conditions and containing 7 porosity distributions representative of three additively manufactured materials, namely, aluminium alloy AlSi10Mg, 8 stainless steel 316L and Inconel 718. The initial void volume fraction in the calculations varies between 0 .00564% and 9 1.75%, the number of voids between 14 and 5715, and the pores size from 2 .3µm to 110 µm. Several realizations with 10 different void sizes and positions have been generated for each of the porous microstructures considered. The simulations 11 have been carried out with random spatial distributions of voids and with clusters of different sizes. The matrix material 12 is modeled using isotropic linear elasticity and von Mises plasticity with an associated flow rule and isotropic hardening, 13 being the flow stress dependent on strain and strain rate. The macroscopic stress state in the unit-cell is controlled by 14 prescribing constant triaxiality ( T) and Lode parameter ( L) throughout the loading. We have performed calculations 15 with stress states resulting from a combination of three different triaxiality and Lode parameter values, i.e., T= 1,2,3 16 andL=−1,0,1. To the authors’ knowledge, this is the first and the most comprehensive study that performs 3D 17 unit-cell calculations with actual representation of porous microstructures, and analyzes the effects of size and spatial 18 distribution of voids on the macroscopic response of the porous aggregate and on the collective behavior of individual 19 pores. The results obtained with the actual porous microstructures have been compared with unit-cell calculations 20 having an equivalent single central pore, and with calculations in which the material behaviour is modeled with Gurson 21 plasticity. It has been shown that both initial void volume fraction and distribution of void sizes affect the macroscopic 22 response of the porous aggregate and the void volume fraction evolution. Moreover, the calculations with random spatial 23 distribution of voids have brought out that different realizations of the same microstructure carry significant variations to 24 the effective behaviour of the porous aggregate, and that the interaction between neighboring pores dictates the volume 25 evolution of individual voids, especially at higher macroscopic triaxiality. The calculations with clusters have shown that 26 pores clustering promotes coalescence localization due to increased interaction between the voids, which results in an 27 increased growth rate of voids in clusters with large number of pores. 28 ∗Corresponding author. E-mail address. jarmarti@ing.uc3m.es Preprint submitted to Elsevier February 27, 2023 2 Keywords: 29 Porous microstructure, Unit-cell calculations, Void growth, Coalescence, Clustering 30 1. Introduction 31 Ductile fracture in metals and alloys has been the subject of many studies over the past decades and it is known to 32 (generally) occur by nucleation, growth (or closure) and coalescence of voids (Cox and Low, 1974; Benzerga and Leblond, 33 2010a; Benzerga et al., 2016; Pineau et al., 2016). Nucleation of voids usually starts at large inclusions and second-phase 34 particles, by particle cracking or by decohesion of the particle–matrix interface. Void growth largely depends on the 35 stress state, such that at low triaxiality voids tend to compress and collapse to ultimately form micro-cracks, and at high 36 triaxiality they grow rapidly by diffuse plastic deformation of the surrounding matrix. When void growth is substantial, 37 void coalescence occurs. The most common void coalescence mode is by internal necking of the intervoid ligament, 38 i.e., the collapse of the ligaments between adjacent voids along a localization band perpendicular to the main loading 39 direction that involves the formation of regions of elastic unloading separated from regions of highly localized plastic 40 flow (the ligaments). Void growth induced softening may also initiate failure without void coalescence per se (Teko˘ glu 41 et al., 2015; Reboul et al., 2020). 42 43 The development of many existing ductile fracture models was motivated by the analysis of void growth in a plastic 44 matrix. Notable are the pioneering studies of McClintock (1968), Rice and Tracey (1969), Hancock and Mackenzie (1976) 45 and Gurson (1975, 1977), who described the growth of an isolated cylindrical or spherical void in an infinite rigid plastic 46 solid. In particular, using micromechanical considerations, Gurson (1975, 1977) developed one of the most widely used 47 criteria for porous solids containing spherical (or cylindrical) voids. The derivation of the Gurson (1975, 1977) model was 48 based on a limit-analysis of a hollow sphere (or cylinder) of finite radius surrounded by the matrix material described 49 with von Mises (1928) criterion and subjected to homogeneous boundary strain rate. Due to its intrinsic limitations to 50 spherical or cylindrical voids and plastically isotropic materials, several extensions of the Gurson (1975, 1977) model 51 have been proposed in various directions during the last decades to account for void nucleation (Chu and Needleman, 52 1980), void coalescence (Tvergaard and Needleman, 1984), void shape effects (Thomason, 1985; Gologanu et al., 1997; 53 Jackiewicz, 2011), void size effects (Wen et al., 2005; Monchiet and Bonnet, 2013), void orientation (Danas and Ponte- 54 Casta˜ neda, 2009; Danas and Aravas, 2012) and distinct features of the constitutive model of the matrix material such as 55 strain hardening (Leblond et al., 1995), viscoplasticity (Duva, 1986; G˘ ar˘ ajeu et al., 2000), plastic anisotropy (Benzerga 56 and Besson, 2001; Benzerga et al., 2004; Chen and Dong, 2009) or pressure sensitivity (Cheng and Guo, 2007; Guo et al., 57 3 2008; Thor´ e et al., 2009). Few attempts have been also made to include the Lode angle in the Gurson (1975, 1977) 58 model to analyze the role of the third stress invariant in the ductile fracture of the porous material (Xue, 2008; Nahshon 59 and Hutchinson, 2008; Benallal et al., 2014; Leblond and Morin, 2014; Vadillo et al., 2016). All the studies cited in 60 this paragraph are based on the micromechanical analysis of a single void embedded in a plastic material. However, the 61 presence of a non-uniform distribution of voids breaks the symmetry within the surrounding matrix and suggests the 62 possibility of a void interaction effect. In a real material, the effect of void distribution observed from X-ray tomography 63 experiments either for random (Maire and Withers, 2014) or clustered voids (Hannard et al., 2017) has been recognized 64 to have an important contribution to ductile fracture. 65 66 Finite element unit-cell computations have been extensively used to provide fundamental understanding of the me- 67 chanical response of porous aggregates. Since the pioneering works of Needleman (1972) and Tvergaard (1981), a large 68 number of unit-cell analyses has been conducted taking into account different cell configurations and loading conditions 69 (Faleskog et al., 2000; Pardoen and Hutchinson, 2003; Kim et al., 2004a; Danas and Ponte-Casta˜ neda, 2012; Keralavarma 70 and Benzerga, 2010; Dæhli et al., 2018; Hosseini et al., 2022; Teko˘ glu and Ko¸ chan, 2022). In the original unit-cell frame- 71 work, a single void is explicitly modeled and embedded in a matrix material with prescribed periodic boundary conditions. 72 Nevertheless, this configuration does not provide information about microstructural features such as void evolution and 73 interaction in a porous aggregate with non-periodic distribution of voids. In order to overcome this limitation, different 74 works have extended the original unit-cell approach by modeling unit-cells embedding multiple voids. For instance, 75 Thomson et al. (1998, 2003) performed finite element simulations of 3D unit-cells where few (three, four or eight) spher- 76 ical pores were aligned with different orientations with respect to the loading directions. The orientation of the void 77 distribution was shown to be a very important factor for void evolution and fracture. McVeigh et al. (2007) modeled 78 the nucleation, growth and coalescence of several interacting pores in 2D and 3D unit-cells. Void spacing and relative 79 void position were found to play an important role in void-sheet localization for zero stress triaxiality (shear loading). 80 Tvergaard (2016, 2017) compared the behavior of several non-periodic distributions of pores embedded in representative 81 volume elements against the behavior of a single pore with equal void volume fraction, and showed that faster void 82 growth occurs for certain non-periodic void arrangements as compared to the single void case. Khan and Bhasin (2017) 83 carried out three-dimensional finite element studies modeling explicitly both primary and secondary voids (i.e., larger 84 and smaller voids). The ductile behavior of the porous aggregate was observed to significantly depend on the respective 85 position of primary and secondary voids. Trejo Navas et al. (2018) carried out 3D finite element simulations of a material 86 with a small population of voids and showed that for the same applied far-field stress, the growth of a void highly depends 87 4 on its relative position in the cell. Shakoor et al. (2018) assessed the competition between particle fragmentation and 88 particle debonding and analyzed their respective influence on void coalescence considering a 3D unit-cell with a realistic 89 population of particles (33 particles) obtained from X-ray tomography data. Recently, Hure (2021) and Cadet et al. 90 (2021, 2022) simulated void growth and coalescence of random distributions of voids (initially spherical and identical in 91 size) embedded in a cubic 3D cell. Specifically, Hure (2021) determined numerically the yield surfaces of porous isotropic 92 materials containing random spatial distributions of voids. The comparison of 3D unit-cells with different number of 93 voids (up to 64), showed that the maximum stress in the simulations –which was used to compute the numerical yield 94 surfaces– is less dependent on the porosity distribution as the number of voids increases. The numerical yield surfaces 95 were found to be consistent with a multi-surface yield criterion accounting for both homogeneous and inhomogeneous 96 material yielding. Moreover, the coalescence process was found to be substantially affected by the distribution of voids, 97 such that the coalescence strain was smaller for the material with random spatial distribution of voids as compared to 98 calculations performed with periodic porous microstructures. Cadet et al. (2021) investigated plastic strain localization 99 in 3D cubic cells made of an elastic-perfectly plastic matrix with random spatial distribution of identical non-overlapping 100 spherical voids (number of voids equal to 27). Various proportional loading conditions with controlled stress triaxiality 101 and Lode parameter were applied (up to the final fracture of the cell), with the microstructures with random spatial 102 distribution of voids showing earlier coalescence as compared to unit-cells with a single central void (and the same void 103 volume fraction). The random distribution of voids led to a large variability of failure strains due to inhomogeneous 104 plastic strain field induced by the porous microstructure. The work of Cadet et al. (2021) was extended shortly after by 105 Cadet et al. (2022) to consider loading cases in which the principal axes of the applied stress are systematically varied 106 with respect to the unit-cell axes. The minimum fracture strain of the porous aggregate was found to draw a U-shape 107 curve function of the Lode parameter ( L), with a minimum value near L= 0 (generalized shear stress). 108 109 Finite element simulations including distributions of voids obtained from X-ray tomography analysis of porous ma- 110 terials bring about opportunities to study ductile damage in more realistic situations. The idea is to elucidate the role 111 of real void sizes or real intervoid distances on the mechanisms of ductile fracture. However, to the authors’ knowledge, 112 an experimentally-based void configuration with a number of pores large enough to ensure a significant statistical rep- 113 resentation of the porous microstructure was never fully mapped within a 3D representative volume element. This is 114 precisely the gap we intend to fill in this paper. For that purpose, we have developed a microstructurally-informed finite 115 element unit-cell model to determine the role of actual porosity on the macroscopic response of the porous aggregate. 116 The cubic unit-cell is subjected to different loading conditions characterized by prescribed (constant) stress triaxiality 117 5 and Lode parameter. We have created porous microstructures with random spatial distributions of voids based on 118 the X-ray tomography measurements reported by Marvi-Mashhadi et al. (2021) for 3 different additively manufactured 119 materials, namely, aluminium alloy AlSi10Mg, stainless steel 316L and Inconel 718. The pores are taken to be initially 120 spherical, consistent with the X-ray tomography measurements and the SEM micrographs reported by Nieto-Fuentes 121 et al. (2022a). Several realizations have been generated for each of the porous microstructures considered and the results 122 have been compared with unit-cells with a single central pore, and with calculations in which the material is modeled 123 using Gurson plasticity. The main novelty of this research as compared to recently published papers is: (1) considering 124 larger (and real) population of spherical voids to ensure that the numerical results are statistically significant and (2) 125 including experimentally-measured distributions of void sizes. The calculations provide new insights into the effects of 126 size and spatial distribution of voids on the macroscopic response of the porous material and allow for characterization 127 of the collective behavior and interaction of individual pores. 128 129 The paper is organized as follows. Section 2 shows the elasto-plastic constitutive framework used to define the 130 mechanical behavior of the material. Section 3 describes the unit-cell finite element model and the main features of 131 the porous microstructures investigated. The effect of stress triaxiality and Lode parameter on the effective behavior of 132 the unit-cell, on the evolution of the void volume fraction and on the growth of individual voids is analyzed in Section 133 4. A parametric study on the influence of initial void volume fraction and distribution of void sizes on the mechanical 134 behavior of the porous aggregate is performed in Section 5, including comparisons with results obtained with unit-cells 135 containing a single central pore, and with unit-cells modeled using Gurson plasticity. Section 6 shows calculations with 136 clustering microstructures having the same initial void volume fraction and different number of clusters of different sizes. 137 A summary of the main findings of the paper is given in Section 7. 138 139 2. Constitutive framework 140 The mechanical behavior of the material is assumed to follow isotropic linear elasticity and von Mises (1928) plasticity. 141 The total rate of deformation tensor dis decomposed into elastic ( de) and plastic ( dp) components: 142 d=de+dp(1) The elastic deformation rate is related to the rate of the stress: 143 6 ˙σ=C:de(2) where ˙σis an objective derivative of the Cauchy stress tensor and C= 2G˜I+K1⊗1is the tensor of isotropic 144 elastic moduli, with Gbeing the elastic shear modulus, Kthe bulk modulus, 1the second-order unit tensor and ˜Ithe 145 fourth-order deviatoric unit tensor. 146 147 The plastic deformation rate follows an associated flow rule: 148 dp=˙λ∂f ∂σ(3) where ˙λis the plastic flow proportionality factor and f= ¯σ−σY≤0 is the yield condition, with ¯ σ=q 3 2s:sbeing the 149 von Mises effective stress, where s=σ−σh:1andσh=1 3σ:1. Moreover, σYis the yield stress of the material which 150 is assumed to be a function of the effective plastic strain (¯ ϵp) and the effective plastic strain rate ( ˙¯ϵp): 151 σY(¯ϵp,˙¯ϵp) =σ0(¯ϵp+ϵ0)n˙¯ϵp ˙ϵ0m (4) where ¯ ϵp=Rt 0˙¯ϵp(τ)dτand˙¯ϵp=q 2 3dp:dp. The parameter σ0is the initial yield stress, and nandmare the strain 152 hardening and strain rate sensitivity exponents, respectively. Moreover, ϵ0and ˙ϵ0are the reference strain and strain 153 rate. 154 155 The baseline numerical values used in the finite element simulations of Sections 4, 5 and 6 for the initial density, the 156 elastic constants and the parameters of the yield stress correspond to aluminum alloy 2090-T3 (Yoon et al., 2006; Cvitani´ c 157 et al., 2008). Moreover, additional calculations with parameters corresponding to aluminum alloys 6111-T4 and 6013 158 are included in Appendix A to illustrate the effect of material behavior on the evolution of the porous microstructure. 159 While these three materials display plastic anisotropy (Barlat et al., 2005; Kim et al., 2010; Ha et al., 2018), all the finite 160 element simulations in this paper are carried out using isotropic von Mises plasticity to facilitate the interpretation of 161 results. The effect of plastic anisotropy on the evolution of the porous microstructure will be studied in a future work. 162 7 163 Symbol Property and units Aluminium alloy 2090-T3 ρ0 Initial density (kg /m3) 2700 G Elastic shear modulus (GPa) 26.92 K Bulk modulus (GPa) 58.33 σ0 Initial yield stress parameter (MPa), Eq. (4) 646 n Strain hardening exponent, Eq. (4) 0.227 m Strain rate sensitivity exponent, Eq. (4) 0.01 ϵ0 Reference strain, Eq. (4) 0.025 ˙ϵ0 Reference strain rate (s−1), Eq. (4) 0.0001 Table 1: Numerical values of initial density, elastic constants and parameters of the yield stress corresponding to aluminum alloy 2090-T3 (Yoon et al., 2006; Cvitani´ c et al., 2008). 3. Finite element model 164 The finite element model is a cubic unit-cell containing spatially distributed spherical voids of different sizes, which 165 is considered to be a representative volume element of a porous material, see Fig. 1. The equations for the nodal 166 displacements reported in Appendix A of Dakshinamurthy et al. (2021) have been used to impose periodic boundary 167 conditions on the unit-cell, so that the displacement of opposed external nodes is coupled (i.e., the displacement of the 168 nodes in the outer faces of the unit-cell is coupled). Material points are referred to using a 3D Cartesian coordinate 169 system ( x,y,z) with origin located at the bottom right corner of the cell, see Fig. 1. The loading directions are determined 170 by the axes x,yandz(see below). Note that the contour plots in Figs. 3, 18, 23 and 27 are referred to this coordinate 171 system. The initial configuration of the unit-cell is defined by the domain 0 ≤x≤L0, 0≤y≤L0, and 0 ≤z≤L0, 172 with L0= 1 mm. 173 174 The macroscopic stress tensor is taken as the volumetric averaging of the microscopic (local) Cauchy stress tensor 175 (Vadillo et al., 2016; Hosseini et al., 2022): 176 Σ=1 VcellZ VcellσdVcell(5) where Vcellis the total volume of the unit-cell. 177 178 8 Figure 1: Unit-cell finite element model: (a) semi-transparent view displaying the porous microstructure, (b) loading conditions with Σ 1, Σ2 and Σ 3being the principal values of the macroscopic stress tensor and (c) cut-view showing the fine mesh around the voids. The macroscopic strain tensor is defined as the volumetric averaging of the microscopic (local) logarithmic strain 179 tensor (Dakshinamurthy et al., 2021; Hosseini et al., 2022): 180 ε=1 VmatrixZ VmatrixϵdVmatrix(6) where Vmatrix=Vcell−Vvoidsis the volume of the matrix material, with Vvoidsbeing the volume of all the voids 181 included in the unit-cell. 182 183 The multi-point constraint subroutine developed by Dakshinamurthy et al. (2021) has been used to enforce constant 184 and controlled values of the macroscopic stress triaxiality T=Σh Σand the macroscopic Lode parameter L=2Σ2−Σ1−Σ3 Σ1−Σ3185 during the calculations, where Σ h=Σ1+Σ2+Σ3 3andΣ =q 3 2Σ′:Σ′are the macroscopic hydrostatic stress and the 186 macroscopic effective stress, respectively, and Σ 1, Σ2and Σ 3(Σ1≥Σ2≥Σ3) are the principal values of the macroscopic 187 stress tensor, with Σ′=Σ−Σh1. The loading directions are aligned with the principal directions of the macroscopic 188 stress tensor, so that the major loading direction corresponds to the principal stress direction associated to Σ 1(parallel 189 tox), and the minor loading direction corresponds to the principal stress direction associated to Σ 3(parallel to z), see 190 Fig. 1. Moreover, the macroscopic effective strain is ε=q 2 3ε′:ε′, where ε′=ε−εh1andεh=ε1+ε2+ε3 3, with ε1,ε2 191 9 andε3(ε1≥ε2≥ε3) being the principal values of the macroscopic strain tensor. Note that the tensors Σandεdo not 192 determine the stress and strain states in a material point, but they are rather used for the definition of the macroscopic 193 stress and strain scalars required for the graphical representation and interpretation of results. 194 195 The novelty of the numerical simulations is that we have incorporated into the unit-cell calculations the porous 196 microstructure obtained from three additively manufactured metals –aluminium alloy AlSi10Mg (Al3XY), stainless steel 197 316L (SS5XY) and Inconel 718 (INC1Z)– following the methodology developed by Marvi-Mashhadi et al. (2021) and 198 later adopted by Vishnu et al. (2022a,b) and Nieto-Fuentes et al. (2022b) to study the effect of actual porosity on the 199 formation of necks and shear bands at high loading rates (i.e., so far, the methodology has been used to address problems 200 other than void growth in unit-cell calculations under controlled triaxiality and Lode parameter). Recall that the pores 201 are taken to be spherical, consistent with the X-ray tomography measurements and the SEM micrographs reported 202 by Nieto-Fuentes et al. (2022a) for AlSi10Mg specimens. Intersections between voids, and between voids and unit-cell 203 boundaries are not allowed. We have created porous microstructures with random spatial distribution of voids and with 204 void clusters. 205 The main features of the porous microstructures investigated, which are obtained from the X-ray tomography mea- 206 surements of Marvi-Mashhadi et al. (2021), are reported in Table 2: initial void volume fraction ( f0), number of voids 207 per mm3(Nv), maximum voids diameter ( dmax), minimum voids diameter ( dmin), and mean ( µ) and standard deviation 208 (dev) of fitted Log-normal distribution. 209 For each of the three porous microstructures considered, for the microstructures with random spatial distribution of 210 voids, we have generated up to five realizations of void size and position distribution which meet the same Log-normal 211 statistical function. The goal is to take into account the scatter in the finite element results caused by the random spatial 212 distribution of pores, see Section 4 for details. These realizations will be referred to as R1, R2, ... , R5. The initial void 213 volume fraction ( f0), the number of voids ( Nt), and the maximum void diameter ( dmax) in the unit-cell calculations of the 214 realizations created for each of the microtructures investigated are shown in Table 3. The difference in the void volume 215 fraction between the experimental measurements and the computations –compare Tables 2 and 3– is partially caused by 216 the random nature of the position of the voids and of the distribution of void sizes in the experimental specimen (see 217 Marvi-Mashhadi et al. (2021)), which is carried over to the finite element model, leading to deviations from the number 218 of pores measured in the tomograms (which are taken over a greater volume, see Marvi-Mashhadi et al. (2021)). Note 219 also that the void volume fraction in the finite element models is less than the experimental data reported in Table 220 2 because the intersections between voids, and between voids and specimen boundaries are not allowed (as mentioned 221 10 before), so that several pores are eventually removed from the finite element models. 222 Moreover, separate attention has been paid to the effect of pore clustering on the evolution of the macroscopic 223 response of the unit-cell and the void volume fraction. For the microstructures with clusters, the methodology developed 224 by Graham-Brady (2010) has been adapted to generate finite element models based on the microstructures of Table 2 225 with different number of void clusters, see Section 6 for details. 226 227 Al3XY SS5XY INC1Z f0(%) 2.17 0.0025 0.1363 Nv(num ./mm3)5985 17 147 dmax(µm) 110.53 41.40 78.93 dmin(µm) 8.00 7.40 7.45 µ(µm) 15.98 11.21 16.58 dev(µm) 4.57 5.13 7.71 Table 2: Summary of the porous microstructures investigated in this work: initial void volume fraction ( f0), number of voids per mm3(Nv), maximum diameter of voids ( dmax), minimum diameter of voids ( dmin), and mean ( µ) and standard deviation ( dev) values of fitted Log-normal distribution. Experimental measurements of Marvi-Mashhadi et al. (2021). Al3XY SS5XY INC1Z R1 R2 R3 R1 R2 R3 R1 R2 R3 R4 R5 f0(%) 1.75 1.62 1.60 0.00663 0.00626 0.00564 0.0735 0.0416 0.0418 0.0666 0.0679 Nt(num.) 5715 5642 5647 15 17 17 143 143 138 141 144 dmax(µm) 86.77 91.33 86.88 30.34 31.28 37.56 58.91 60.29 36.16 52.83 64.67 dmin(µm) 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 Table 3: Initial void volume fraction ( f0), number of voids ( Nt), maximum void diameter ( dmax) and minimum void diameter ( dmin) in the finite element models corresponding to the realizations generated for the microstructures with random spatial distribution of voids. 228 The finite element calculations have been performed using ABAQUS/Standard (2019). The unit-cell has been 229 discretized with ten-node quadratic tetrahedral elements (C3D10 in ABAQUS notation). A very fine mesh near the 230 pores is necessary to capture the geometry and the subsequent shape evolution of the voids during loading, see Fig. 1. 231 The number of elements increases with the number of pores, so that ≈5000000, ≈200000 and ≈350000 elements are 232 used to mesh the unit-cells corresponding to microstructures Al3XY, SS5XY and INC1Z, respectively. The calculations 233 have been performed using a workstation AMD Milan 7453 @ 2 .75 GHz with 56 cores. The computational cost of each 234 simulation ranged between 2 and 12 days, depending on the microstructure considered, using simultaneously all the cores 235 of the workstation. 236 11 237 The evolution of the void volume fraction ( f) in the unit-cell during loading is calculated as: 238 f=Vcell−Vmatrix Vcell(7) where the volume of the matrix material is computed as: 239 Vmatrix=nelemX n=1EV OL n (8) where EV OL is the elemental volume and nelemis the total number of elements in the model. We have also calculated 240 the volume evolution of individual voids using the Quickhull algorithm (Barber et al., 1996) available in MATLAB®to 241 compute the smallest convex set containing the nodal coordinates of the void surface at each time step. 242 243 Sections 4, 5 and 6 include calculations for three different values of macroscopic triaxiality T= 1, 2 and 3, and 244 macroscopic Lode parameter L=−1, 0 and 1. Note that for L=−1 we have that Σ 1>Σ2= Σ 3(axisymmetric 245 tension), for L= 0 the principal values of the macroscopic stress tensor are such that Σ 1>Σ2=Σ1+ Σ 3 2>Σ3 246 (generalized shear), and for L= 1 we have that Σ 1= Σ 2>Σ3(axisymmetric compression). 247 4. Salient results 248 The calculations correspond to porous microstructure INC1Z and realizations R1, R2, ... , R5. The effect of 249 microstructural realization, stress triaxiality and Lode parameter on the macroscopic effective behavior of the unit-cell, 250 on the evolution of the normalized void volume fraction and on the growth of individual voids is investigated in Sections 251 4.1, 4.2 and 4.3, respectively. 252 4.1. The effect of microstructural realization 253 Fig. 2 compares calculations performed with realizations R1, R2, ... , R5 for stress triaxiality T= 3 and Lode 254 parameter L=−1 (all the calculations in Section 4.1 are performed with T= 3 and L=−1). The evolution of the 255 normalized macroscopic effective stress ¯Σ/σ0with the macroscopic effective strain ¯ εis shown in Fig. 2a. The differences 256 between realizations increase with the effective strain, e.g., the maximum effective stress is 5% larger for R2 than for 257 12 R1. The influence of the realization on the ¯Σ/σ0−¯εcurves comes from the differences in the void volume fraction 258 between R1, R2, ... , R5, and also from the different spatial and size distribution of voids. Fig. 2b shows the evolution 259 of the normalized void volume fraction f/f0with the macroscopic effective strain ¯ ε. The f/f0−¯εcurves display a 260 concave-upward shape, so that the porosity growth rate increases as the loading progresses (due to the large value of 261 imposed triaxiality, e.g., see Hosseini et al. (2022)). Notice that the calculations which show greater porosity growth 262 rate, R2, R3 and R5, are the same displaying faster strain softening in Fig. 2a. The key outcome of these results is that, 263 despite all realizations correspond to the same porous microstructure, there are significant differences in the evolution 264 of macroscopic stress and porosity, consistent with the scatter generally observed in mechanical characterization tests of 265 additively manufactured materials (Kristoffersen et al., 2020). 266 267 Unit-cells showing contour plots of effective plastic strain in the matrix material ¯ ϵpfor realizations R1 and R3 are 268 included in Fig. 3. The results correspond to different values of macroscopic effective strain ¯ ε= 0,0.033,0.066 and 0 .1, 269 which are indicated in Fig. 2 with yellow markers. The color coding of the isocontours is such that effective plastic strains 270 ranging from 0 to 1 correlate with a color scale that goes from blue to red. Effective plastic strains above 1 remain red. The 271 semi-transparent view of the specimen in pictures (a)-(h) shows the evolution of the porous microstructure, and the solid 272 cut-views in pictures (d) and (h) illustrate the localization of plastic deformation and the pores coalescence localization. 273 There are no qualitative differences between the contours corresponding to R1 and R3. Semi-transparent pictures (a) and 274 (e) show the unit-cells before loading starts. The microstructures contain pores of different sizes randomly distributed in 275 the sample. For ¯ ε= 0.033, see pictures (b) and (f), the pores have already significantly grown, and the effective plastic 276 strain near the voids has reached values greater than 1. Notice the contrast between macroscopic strain and local plastic 277 deformation near the voids, the latter being much greater (the relationship between macroscopic strain and local plastic 278 deformation will be further discussed below). The contours in (c) and (g) for ¯ ε= 0.066 correspond approximately to the 279 maximum macroscopic effective stress, see Fig. 2a, and the void volume fraction is roughly ten times more than at the 280 beginning of loading, see Fig. 2b. Notice that the localization of plastic deformation at specific locations of the outer 281 faces of the unit-cell comes from the growth of nearby voids. Semi-transparent pictures (d) and (h) are taken for ¯ ε= 0.1, 282 during the strain softening process. The porosity for realizations R1 and R3 is nearly 30 and 40 times greater than 283 initially, respectively. The cut-view pictures show cross-sections of voids of different sizes and illustrate the interaction 284 between nearby pores that have grown, leading to large values of plastic deformation in the intervoid ligament. While 285 fracture is not accounted for in the finite element calculations, coalescence of voids is apparent (see pinkish arrows), as 286 they are just separated by very elongated necked sections. Teko˘ glu et al. (2015) determined coalescence localization to 287 13 occur when all the additional plastic deformation is confined within the ligaments between the voids, consistent with the 288 early modeling of coalescence of Thomason (1990) and subsequent researchers (Benzerga and Leblond, 2010b; Teko˘ glu 289 et al., 2012). Fig. 4 shows different contours of effective plastic strain in the surface of the two pores indicated in the 290 cut-view of image 3(d), from the onset of loading until ¯ ε= 0.116, for intervals of macroscopic strain of 0 .016. The pores 291 remain virtually spherical until ¯ ε= 0.05 (for this large value of triaxiality T= 3, the voids which do not interact with 292 nearby pores tend to grow maintaining the spherical shape). At this stage of the loading process, the entire surface of 293 the voids shows effective plastic strains greater than 1. Note that the plastic strain surrounding the pores increased from 294 0.5 to more than 1 within a narrow interval of macroscopic strain (0 .016≤¯ε≤0.05) which is fifteen times less, i.e., 295 the local plastic strain increases fifteen times faster than the macroscopic strain. For macroscopic strains greater than 296 0.066, the growth of the pores make them to interact and flatten, leading to the formation of a thin intervoid ligament 297 subjected to large plastic strain which shows coalescence localization (coalescence would be completed by considering a 298 fracture criterion in the simulations). 299 (a) (b) Figure 2: Results corresponding to porous microstructure INC1Z and realizations R1, R2, ... , R5 for stress triaxiality T= 3 and Lode parameter L=−1. (a) Normalized macroscopic effective stress ¯Σ/σ0versus macroscopic effective strain ¯ ε. (b) Normalized void volume fraction f/f0versus macroscopic effective strain ¯ ε. The yellow markers correspond to realizations R1 and R3, for different values of the macroscopic effective strain ¯ ε= 0,0.033,0.066 and 0 .1 shown in the contour plots of Fig. 3. (For interpretation of the