In this paper, we have performed a microstructurally-informed nite element analysis on the e ect of porosity on the formation of multiple necks and fragments in ductile thin rings subjected to dynamic expansion. For that purpose, we have characterized by X-ray tomography the porous microstructure of 4 di erent additively manufactured materials (aluminium alloy AlSi 10Mg, stainless steel 316L, titanium alloy Ti 6Al4V and Inconel 718L) with initial void volume fractions ranging from 0:0007% to 2%, and pore sizes varying between 6m and110m. Three-dimensional analysis of the tomograms has revealed that the voids generally have nearly spherical shape and quite homogeneous spatial distribution in the bulk of the four materials tested. The pore size distributions quanti ed from the tomograms have been characterized using a Log-normal statistical function, which has been used in conjunction with a Force Biased Algorithm that replicates the experimentally observed random spatial distribution of the voids, to generate ring expansion nite element models in ABAQUS/Explicit (2016) which include actual porous microstructures representative of the materials tested. We have modeled the materials behavior using von Mises plasticity, and we have carried out nite element calculations for both elastic perfectly-plastic materials, and materials which show strain hardening, strain rate hardening and temperature softening e ects. Moreover, we have assumed that fracture occurs when a critical value of e ective plastic strain is reached. The nite element calculations have been performed for expansion velocities ranging from 50 m =s to 500 m=s. A key point of this investigation is that we have established individualized correlations between the main features of the porous microstructure (i.e. initial void volume fraction, average void size and maximum void size) and the number of necks and fragments formed in the calculations. In addition, we have brought out the e ect of the porous microstrucure and inertia on the distributions of neck and fragment sizes. To the authors' knowledge, this is the rst paper ever considering actual porous microstructures to investigate the role of material defects in multiple localization and dynamic fragmentation of ductile metallic materials. Keywords: Corresponding author. Tel. +34 916249904; Fax: +34 916249430. E-mail address: jarmarti@ing.uc3m.es Preprint submitted to International Journal of Plasticity April 2, 2021 2 Additive manufacturing, Printed metals, Porosity, Multiple necking, Fragmentation, Finite elements 1. Introduction The pioneering statistical theory of Mott (1947) postulated that fragmentation of metals subjected to intense impulsive tensile loads occurs due to the activation of weak points of the material {e.g. material defects like pores, cavities and cracks formed during materials processing{ which are scattered throughout the specimen and determine the distribution of fractures sites. It also considered that material failure occurs instantaneously, disregarding the localization of plastic deformation that precedes fracture in ductile metals, as well as the fracture resistance and the fracture energy associated to the breakage process. The instantaneous fracture releases stress waves (or the Mott's waves) that propagate away from the fracture, unloading the specimen. The material cannot continue deforming within the regions of the specimen subjected to the action of the release waves, preventing additional fractures, so that the size of the fragments is determined by the distance traveled by the Mott's waves. Despite the relative simplicity of Mott's theory, and the number of assumptions made to make the model mathematically tractable and applicable to actual engineering problems, Grady (2002) highlighted that Mott's original publication contains seminal theoretical concepts from which numerous later modeling e orts emerged. For instance, Grady (1981) applied concepts of survival statistics to the theory of Mott to account for spatially random fracture nucleation, and derived fragments size distribution curves that found satisfactory agreement with the experimental data reported by Wesenberg and Sagartz (1977) for thin 6061-T6 aluminum cylinders subjected to rapid radial expansion at strain rates of 104s1. Moreover, Kipp and Grady (1985) expanded the analyses of Mott to explicitly account for energy dissipation during the fracture process, and obtained expressions for the nominal fragment size, the fracture time, and the dynamic fracture strain, which were in reasonable agreement with the experimental data obtained by Grady and Benson (1983) for the fragmentation of OFHC copper and 1100-O aluminum rings subjected to radial expansion at strain rates of 104s1. Grady and Olsen (2003) performed ring expansion fragmentation tests with U6N specimens, for expansion velocities ranging from 50 to 300 m =s, and compared the experimental results with the distribution of fragment masses obtained with Mott's statistical theory and the energy-based fragmentation theory of Kipp and Grady (1985). Both theoretical approaches were in reasonable accord with the experimental data. In addition, the predictions of the energy-based fragmentation theory for the evolution of the number of fragments with the expansion velocity were compared with the experimental evidence, and sensible agreement was found for the whole range of expansion velocities investigated. Moreover, Zhang and Ravi-Chandar (2006) performed ring expansion experiments using Al 6061-O specimens with rectangular cross-section of thickness 0 :5 mm and length 1 mm, at velocities ranging from 80 to 200 m =s, and 3 measured the distributions of neck spacings and fragment sizes. Following Mott (1947) postulates, Zhang and Ravi- Chandar (2006) suggested that the distribution of lengths between consecutive necks is governed by the spatial distribution of necking strains in the specimen and the unloading waves emanating from growing necks. Relying on Mott (1947) scaling analysis, and assuming that the variation in the necking strain is 2%, Zhang and Ravi- Chandar (2006) determined the necks spacing to be between 17 mm and 7 mm for the lower and higher expansion velocities tested, respectively, being these estimates in good agreement with the experimental results which showed a decrease in the average distance between consecutive necks as the loading rate increases. As to the distribution of fragment sizes, Zhang and Ravi-Chandar (2006) showed that if the length of the fragment is normalized by the characteristic unloading length, the normalized fragment lengths distribution, resulting from putting together the results for di erent expansion velocities, can be adequately represented by a Weibull probability density function. Zhang and Ravi-Chandar (2006) interpreted this result as an indicator that there is a random distribution of weak points scattered throughout the circumference of the ring, and that the release wave mechanism sets the scale for fragmentation. Zhang and Ravi-Chandar (2008) performed further ring expansion experiments with Al 6061-O, Al 1100-H14 and Cu 101 specimens with di erent length-to-thickness aspect ratios (di erent lengths and xed thickness of 0 :5 mm) and di erent cross-section sizes (varying both length and thickness), for expansion velocities ranging from 65 to 230 m =s. The tests using Al 1100-H14 and Cu 101 specimens with cross section of thickness 0:5 mm and length 1 mm yielded statistical distributions of neck spacings similar to that obtained in the experiments performed with Al 6061-O rings in Zhang and Ravi-Chandar (2006), such that with increasing strain rate, more necks were nucleated at shorter distances. Zhang and Ravi-Chandar (2008) suggested that this behavior occurs at high strain rates because the unloading engendered by any localization does not travel far enough quickly to inhibit further nucleation at neighboring locations. The tests using samples with di erent cross-section sizes showed that, as the specimen cross-sectional size increases, the strain at the onset of necking, which is roughly the Consid ere strain for the specimens with the smallest cross-section, increases. The size of the cross-section of the sample also seemed to a ect the spatial distribution of necks such that the scatter in the distribution of neck sizes decreases with the increase of the cross-section dimensions. Moreover, the tests using samples with di erent length-to-thickness ratios showed that the increase of the length of the specimen (thickness was xed) leads to a decrease of the average neck spacing and to a gradual removal of the long segments without necks that are frequently observed in rings with small length-to-thickness relation. In order to provide further insights into the experimental evidence, Zhang and Ravi-Chandar (2008) performed nite element calculations using ABAQUS/Explicit of specimens with di erent length-to-thickness ratios expanded at 250 m=s, modeled with von Mises plasticity and a power-law hardening model, and with material properties corresponding to Al 6061- O. In order to provide an idealized modeling of the material properties variation at the microstructural level, a 4 random distribution of punctual defects (at the element level), with reduced yield strength and no strain hardening, was included in the samples. As the loading process starts, before the Consid ere condition was reached, the plastic strain was shown to evolve nearly uniformly over the entire specimen with slight perturbations in the vicinity of the defects. With the continuation of the loading process, there was a magni cation of the strain uctuations, so that for the specimens with the smallest cross-section multiple localized necks formed right after the Consid ere strain was met. In contrast, for specimens with large length-to-thickness ratio, the calculations revealed the emergence of alarge post-uniform deformation regime, so that the localization, that took the form of multiple necking bands, developed when the average strain in the specimen was more than twice the Consid ere strain (which corresponds to the bifurcation strain for the sheet necking mode given by Hill (1952)). The in uence of defects on necking localization and the formation of multiple necks for strains well beyond the Consid ere condition was also the subject of study of Vaz-Romero et al. (2017), who performed nite element calculations of nonlinear elastic bars with spatially-localized defects of the strain rate eld and subjected to dynamic stretching. The computations of Vaz-Romero et al. (2017) showed that the post-uniform deformation regime that extends from the Consid ere strain until the formation of localized necks enlarges with the strain rate, bringing out the stabilizing e ect of inertia which delays localization. The nite element results of Vaz-Romero et al. (2017) also showed that, while the defects of the strain rate eld speed up the development of the necking pattern in the late stages of the localization process, the characteristic (average) distance between consecutive necks is largely independent of the defects for the defects' amplitudes investigated. The average neck spacing obtained with the nite elements was compared with the predictions of 1D and 3D dynamic stability analyses, nding good correlation for strain rates varying between 200 s1and 10000 s1. The agreement between numerical calculations and linear stability analyses was considered to be a proof that the multiple localization mechanisms cannot be e ectively explained using exclusively statistical theories. Similarly, it was also suggested that the distance between necks cannot be exclusively attributed to the random distribution of material properties and to the defects population (see Dequiedt (2015)). Following previous works of Fressengeas and Molinari (1994), Guduru and Freund (2002) and Mercier et al. (2010) among others, Vaz-Romero et al. (2017) indicated that there are dominant necking modes which de ne, at least up to some extent, the average neck spacing in multiple localization patterns, the speci c dominant modes being determined by the emergence of inertia e ects at high strain rates. The e ect of material defects on the formation of multiple necking patterns was also investigated by N'souglo et al. (2018), who performed nite element calculations of elasto-plastic bars with yielding modeled with Gurson porous plasticity criterion (Gurson, 1977; Tvergaard, 1981, 1982) and subjected to dynamic tension at strain rates varying from 103s1to 5104s1. The initial porosity, which in the Gurson model (Gurson, 1977; Tvergaard, 1981, 1982) intends to idealize material defects, was randomly varied at the element level, using various 5 upper and lower bounds for the maximum and minimum porosity variation, and di erent spatial distributions. The calculations of N'souglo et al. (2018) indicated that, for the lower strain rates, the distribution of neck spacings is heterogeneous and sensitive to the initial distribution of the porosity. In contrast, as the strain rate increases, the distribution of lengths between consecutive necks becomes more homogeneous and less dependent on the initial distribution of porosity. A possible explanation for these results was sought on the predictions of a linear stability analysis, which brought out that with the increase of the strain rate the dominant necking modes have a strong prevalence over the other growing modes, reducing the scatter in the distribution of neck sizes. Other authors who performed numerical calculations to investigate the in uence of material defects on multiple localization and fragmentation problems are Hiroe et al. (2008), Zhang and Ravi-Chandar (2010), Liu et al. (2013, 2016) and Cliche and Ravi-Chandar (2018). In these works, as in those mentioned above, material defects are included by allocating a reduced elastic limit to an arbitrary number of material points, or by assuming local variations in the failure strain. However, while this strategy to model material defects is easy to implement in numerical calculations, and thus it shows clear advantages from a practical standpoint, it is a dicult task to associate these idealized defects with actual microstructural heterogeneity or mechanical property variations, and hence to elucidate what is the role of actual material aws in the formation of multiple localization patterns and the dynamic fragmentation of metallic structures. Furthermore, most of the times the actual distribution of defects in the material is not known. These are precisely the gaps we intend to ll in this paper. For that purpose, we have devised a microstructurally- informed nite element analysis to determine the role of actual porosity, i.e. actual material defects, on the for- mation of multiple necks and fractures in metallic rings subjected to radial expansion. We have characterized by X-ray tomography the porous microstructure of 4 di erent additively manufactured materials (aluminium alloy AlSi 10Mg, stainless steel 316L, titanium alloy Ti 6Al4V and Inconel 718L) with initial void volume fractions ranging from0:0007% to 2%, and roughly spherical pores with diameter varying between 6m and 110m. The pores recorded in the X-ray tomograms are tted to a Log-normal statistical function that is later used to generate porosity distributions based on the actual microstructure of the material. These porosity distributions are then included in the nite element model of a ring subjected to rapid radial expansion so that the specimen contains actual voids. We have performed calculations for expansion velocities ranging from 50 m =s to 500 m=s, assuming that the matrix material follows von Mises plasticity. Speci cally, we assume that the matrix material is elastic perfectly-plastic in Section 4, and thermo-viscoplastic with strain hardening, strain rate hardening and thermal softening in Section 5. For simplicity, material fracture is assumed to occur when a critical e ective plastic strain is reached. Note that the objective of this paper is to determine the e ect of porous microstructure on dynamic localization and fragmentation, and we do not pay attention to the speci c mechanical response of the printed metals from which the microstructures are taken. The results of the nite element simulations provide new insights 6 into the role of porosity on the distributions of necks spacing and fragments size. While direct numerical simulation of specimens containing a large population of voids was performed by Becker and Callaghan (2018, 2020) to assess the load path and mean stress dependence of void growth, to the authors' knowledge, this is the rst paper ever that includes actual microstructural porosity distributions in large scale calculations of dynamic localization and multiple ductile fracture. 2. Porosity distribution In this section, we describe the methodology that we have developed to determine the distribution, shape and size of the pores present in several additively manufactured metals. Recall that in this paper we do not study the mechanical behavior of the materials investigated, the additively manufactured materials are solely employed to obtain representative porosity distributions to be used in the simulations of Sections 4 and 5. Cylindrical specimens with diameter and height of 6 mm were printed by Selective Laser Melting (SLM) by the company Materialise out of aluminium alloy AlSi 10Mg, stainless steel 316L, titanium alloy Ti 6Al4V and Inconel 718L. This printing technique often leads to residual porosity in the samples, generally lower than 5%, which can be controlled adjusting the major printing parameters (Aboulkhair et al., 2014; Kasperovich et al., 2016; Moussaoui et al., 2018). Two di erent printing directions of parallel and perpendicular to the longitudinal axis of the cylinder were chosen. We have analyzed 12 specimens (i.e. 12 porous microstructures) which are denoted as Al3Z, Al3XY, Al0 :5Z, Al0:5XY, SS5Z, SS5XY, SS0 :5Z, SS0:5XY, Ti0:5Z, Ti0:5XY, INC1Z and INC1XY, where Al, SS, Ti and INC stand for the material, the following numbers indicate the nominal void volume fraction (%) provided by the manufacturer, and Z and XY determine whether the sample was printed in parallel or perpendicular direction to the longitudinal axis of the cylinder. We performed X-ray Computed Tomography (XCT) of each sample using a Nanotom 160NF tomograph (General Electric-Phoenix) located at IMDEA Materials Institute. We used a 3 33 m3tomogram voxel size. The reconstruction of the tomograms was carried out using an algorithm based on the ltered back-projection procedure for Feldkamp cone beam geometry (e.g. see Stef et al. (2018)). Further analysis of 3D reconstructed images was performed using the open source software tool ImageJ (Schneider et al., 2012), so that for each sample we selected three cylindrical sub-volumes of 1 :5 mm diameter and 1 :5 mm height, with no overlapping between them, to obtain the distribution, shape and size of the voids in the specimens investigated. Analyzing three cylindrical sub-volumes for each sample, with the aforementioned dimensions, enables to record sucient number of voids to obtain statistically signi cant results which are representative of the actual porosity distribution in the materials (note that the 3D object counter plugin of ImageJ (Bolte and Cordeli eres, 2006) does not allow to analyze the AlSi 10Mg cylindrical samples of 6 mm height and 6 mm diameter in one step due to the very large number of voids they contain). 7 Figs. 1-4 show 2D XCT images of the middle cross-section of the samples perpendicular to the main axis of the cylinder for the 12 specimens investigated. The voids appeared in form of dark spots, with roughly circular area, which are randomly dispersed in the samples. Notice that there is a layer of larger voids close to the outer shell of the cylinders. Sukal et al. (2018) and Koutiri et al. (2018), among others, stated that the concentration of pores below the surface of SLM printed parts is caused by the inadequate connection of the hatching of the sample volume with the contour of the part. This layer of larger voids is excluded from the calculations used to characterize the porosity distribution of the materials in order to maintain a homogeneous microstructure. Notice that the number of voids in the AlSi 10Mg samples is greater than in the other materials tested (as mentioned before), with important di erences between the samples with nominal porosity 3% and 0 :5%, so that the specimens Al3XY and Al3Z contain more voids which are generally slightly larger (see also Table 1). Moreover, stainless steel 316L and titanium alloy Ti 6Al4V specimens have the lower levels of porosity. A more detailed quantitative analysis of the void volume fraction of the specimens and the distribution of void sizes is carried out below. (a) (b) (c) (d) Figure 1: X-ray computed tomography images of the middle cross-section of the samples perpendicular to the main axis of the cylinder for: (a) Al3Z, (b) Al3XY, (c) Al0 :5Z and (d) Al0 :5XY. The dark spots correspond to porosity. The pixel size is 3 m. 8 (a) (b) (c) (d) Figure 2: X-ray computed tomography images of the middle cross-section of the samples perpendicular to the main axis of the cylinder for: (a) SS5Z, (b) SS5XY, (c) SS0 :5Z and (d) SS0 :5XY. The dark spots correspond to porosity. The pixel size is 3 m. (a) (b) Figure 3: X-ray computed tomography images of the middle cross-section of the samples perpendicular to the main axis of the cylinder for: (a) Ti0 :5Z and (b) Ti0 :5XY. The dark spots correspond to porosity. The pixel size is 3 m. 9 (a) (b) Figure 4: X-ray computed tomography images of the middle cross-section of the samples perpendicular to the main axis of the cylinder for: (a) INC1Z and (b) INC1XY. The dark spots correspond to porosity. The pixel size is 3 m. Figs. 5-8 show 3D rendered images of the porosity distribution (in orange) in the cylindrical sub-volumes. Note that we only show 1 out of the 3 cylindrical sub-volumes with diameter and height of 1 :5 mm that we have analyzed per sample. Consistent with the 2D XCT images shown in Figs. 1-4, the 3D reconstructions make apparent that the AlSi 10Mg samples have the largest number of voids, being Inconel 718L the material with the next largest number of pores, followed by stainless steel 316L and titanium alloy Ti 6Al4V, for which the void volume fraction is the lowest. For all the specimens, the voids, which are roughly spherical (see Zhang et al. (2017)), are rather homogeneously/uniformly distributed in the bulk of the material (note that the Inconel 718L specimens also show attened pores, most likely due to lack of fusion, nevertheless, we assume that all voids can be considered spherical for the modeling purposes, see Section 3). A summary of the measurements obtained from the reconstructed 3D tomograms is stressed in Table 1. The void volume fraction and the number of voids per mm3are computed by averaging the results obtained with the three cylindrical sub-volumes that we have analyzed per sample to characterize materials porosity (the total material volume analyzed per sample is 8 mm3). Note that the measured void volume fractions are lower than the nominal values indicated by the supplier. This occurs, most likely, because the large voids near the outer shell of the cylinders and the voids with diameter less than 6m were excluded from the calculations (voids with diameter smaller than 6 m cannot be detected with the resolution of 3 33m3of voxel size). The XCT measurements of void size distributions are represented in Figs. 9-12 in form of relative frequency versus void diameter (black columns). The void distributions have been computed using all pores recorded in the three cylindrical sub-volumes that we have analyzed per sample. Note that we have considered a continuous probability density function, so that the histogram bins are just selected to illustrate the evolution of the number of voids with the voids size. The experimental measurements have been tted to the Log-normal distribution (solid 10 red line) with corresponding mean ( ) and standard deviation ( dev) values given in Table 1. The number of voids with diameter larger than 75 m is very small in comparison with the total number of voids, so that 75 m is the largest pore size considered in Figs. 9-12. The limitation of the void diameter to 75 m in the graphs is for illustrative purposes only, to not increase the scale of the plots. However, in the nite element models we include pores with diameter greater than 75 m (see Section 3), as recorded from the tomograms. Notice that the only microstructures containing pores with diameter greater than 75 m are Al3Z, Al3XY, SS5Z and INC1Z, see Table 1. For all materials and specimens investigated, the diameter of most voids lies within the range 5 m<d< 30m. The greatest and the smallest average pore sizes, 18 :45m and 6:41m, correspond to the specimens SS5Z and INC1XY, respectively. Moreover, the standard deviation of the distributions lies between 3 :81m for Al0:5Z and 9:85m for SS5Z. Notice that there is no clear e ect of the printing orientation on the average pore size. While for AlSi 10Mg and Ti 6Al4V the average diameter of the voids is smaller when the printing direction is parallel to the longitudinal axis of the cylinder, the opposite trend is obtained for stainless steel 316L and Inconel 718L. Similarly, there is no correlation between the printing direction and the standard deviation of the distributions of pores sizes, neither between the printing direction and the maximum size of the voids. 11 (a) (b) (c) (d) Figure 5: 3D reconstruction of cylindrical sub-volume with diameter 1 :5 mm and height 1 :5 mm for: (a) Al3Z, (b) Al3XY, (c) Al0 :5Z and (d) Al0 :5XY. Porosity distribution in orange. For interpretation of the