In this paper, we have investigated, using finite element calculations, the effect of initial texture on the formation of multiple necking patterns in ductile metallic rings subjected to rapid radial expansion. The mechanical behavior of the material has been modeled with the elasto-viscoplastic single crystal constitutive model developed by Marin (2006). The polycrystalline microstructure of the ring has been generated using random Voronoi seeds. Both 5000 grain and 15000 grain aggregates have been investigated, and for each polycrystalline aggregate three different spatial distributions of grains have been considered. The calculations have been performed within a wide range of strain rates varying from 1 .66·104s−1to 3.33·105s−1, and the rings have been modeled with four different initial textures: isotropic texture, ⟨001⟩ ∥Θ Goss texture, ⟨001⟩ ∥R Goss texture and ⟨111⟩ ∥Z fiber texture. The finite element results show that: (i) the spatial distribution of grains affects the location of the necks, (ii) the decrease of the grain size delays the formation of the necking pattern and increases the number of necks, (iii) the initial texture affects the number of necks, the location of the necks, and the necking time, (iv) the development of the necks is accompanied by a local increase of the slip activity. This work provides new insights into the effect of crystallographic microstructure on dynamic plastic localization and guidelines to tailor the initial texture in order to delay dynamic necking formation and, thus, to improve the energy absorption capacity of ductile metallic materials at high strain rates. Keywords: Dynamic necking, Inertia, Crystal plasticity, Texture, Finite elements 1. Introduction The ring expansion experiment developed by Niordson (1965) has become a reference benchmark problem to investigate dynamic necking localization and fragmentation of ductile metallic materials. The test consists of a ∗Corresponding author. E-mail address: jarmarti@ing.uc3m.es Preprint submitted to Mechanics of Materials January 17, 2023 2 thin circular specimen with square/rectangular cross-section which is expanded radially at high strain rates using electromagnetic loading schemes, explosive charges or mechanical loading systems such as gas guns (Fyfe and Rajendran, 1980; Goto et al., 2008; Wood et al., 2021; Gant et al., 2021), so that multiple necks are formed throughout the circumference of the sample at large strains, leading to the final fragmentation of the ring. The principal advantage of the rapidly expanding ring experiment –in comparison with the tensile impact testing of linear specimens (slender bars)– is that the radial symmetry of both specimen and loading nearly eliminates the propagation of stress waves along the hoop direction of the sample before necking localization, thus revealing the true dynamic properties of the material. The experiments of Grady and Benson (1983) with OFHC copper and aluminum 1100-O rings having inner diameter of 32 mm and square cross-section of 1 mm2, tested at radial velocities ranging from about 20 m /s to 200 m /s, showed that the number of necks, the proportion of necks developing into fractures, and the fracture strain, increase with the applied velocity. Following Mott (1947) postulates, the distribution of neck spacings and fragment sizes was attributed to the spatial variability of the necking and fracture strains throughout the circumference of the ring, and to the unloading waves released from growing necks and early fractures which remove the driving force from nearby sections of the specimen before fracture is completed. Moreover, Altynova et al. (1996) carried out ring expansion tests with aluminum 6061 and OFHC copper samples, for radial speeds ranging between 50 m /s and 300 m /s. The rings had inner diameter of 30 .5 mm, radial thickness of 1 mm, and different widths varying from 1 mm to 2 mm. For expansion velocities above 200 m /s, the ductility of aluminum 6061 and OFHC copper rings increased with respect to the quasi-static value by 60% and 250%, respectively. The improved ductility at high strain rates was attributed to inertia effects which diffuse deformation throughout the specimen, leading to stabilization against neck growth and delaying failure. Ring expansion experiments on U6N specimens were carried out by Grady and Olsen (2003). The samples had inner diameter of 30 mm and square cross-section of 0 .75×0.75 mm2, and were submitted to radial velocities ranging from 50 m /s to 300 m /s. The number of fragments, and the distributions of fragment sizes and masses, were compared with the fragmentation theories of Mott (1947) and Kipp and Grady (1985) which consider the fragmentation patterns of metallic samples subjected to intense impulsive tensile loads to be the result of the activation of weak points of the material (e.g., material defects like pores, cavities and cracks formed during materials processing, and microstructural heterogeneities like soft phases, grain boundaries) which are scattered throughout the specimen and determine the distributions of necks and fracture sites. Zhang and Ravi-Chandar (2006) tested aluminum 6061-O specimens 3 with inner diameter of 30 .5 mm, and rectangular cross-section of thickness 0 .5 mm and length 1 mm. The radial loading speeds investigated ranged between 80 m /s and 200 m /s. Consistent with previous works of Grady and Benson (1983) and Grady and Olsen (2003), the experiments of Zhang and Ravi-Chandar (2006) showed that the neck and fragment distributions shift towards smaller neck spacings and smaller fragments, and towards narrower distributions, with increasing loading rate. It was concluded that as the strain rate increases, the propagation of the release wave emanating from growing necks and early fractures is limited to shorter times, so that the unloading wave does not travel far enough quickly to inhibit further nucleation of necks and fractures at neighboring locations. An incomplete list of additional papers including ring expansion experiments on metallic samples is (Perrone, 1968; Stepanov and Babutskii, 1984; Janiszewski, 2012; Cliche and Ravi-Chandar, 2018). The ring expansion problem has been also extensively investigated over the last two decades using finite element calculations. For instance, Guduru and Freund (2002) carried out simulations in ABAQUS/Explicit of the ring expansion experiments performed by Grady and Benson (1983), modeling the mechanical behavior of the material using Gurson porous plasticity, and assuming that fracture occurs when a critical value of porosity is achieved. The finite element model consisted of a long rod submitted to rapid stretching and with initial conditions consistent with the expanding ring. The finite element results yielded qualitative agreement with the experiments for the number of necks, the fragmentation statistics, and the increase in the fracture strain with the loading rate. Moreover, Rusinek and Zaera (2007) performed ring expansion finite element simulations modeling the material behavior with von Mises plasticity, different constitutive relations for the evolution of the yield stress, and a fracture criterion which assumed the material to fail when a predefined value of plastic strain is reached. The numerical predictions for the number of fragments were compared with the experimental data reported by Grady and Benson (1983), Altynova et al. (1996) and Diep et al. (2004) for aluminum 1100-0, aluminum 6061 and steel 34CrNiMo6, respectively, and satisfactory quantitative agreement was obtained for the whole range of loading velocities considered. Moreover, additional calculations with baseline material parameters corresponding to mild steel showed that the number of fragments increases with the decrease of the strain hardening and with the increase of the material density. Shortly after, Zhang and Ravi-Chandar (2008) simulated ring expansion experiments on aluminium 6061-O specimens with different cross-section sizes using von Mises plasticity and a power law strain hardening to model the mechanical behavior of the material. A random distribution of material defects (modeled as elements of the finite element grid with low yield stress) was included in the finite element model to break the symmetry of the problem and trigger plastic localization. The loading condition was a pulse pressure applied in the inner surface of the ring, 4 leading to a maximum expansion velocity of 250 m /s, corresponding to a strain rate of 104s−1. The calculations showed, in agreement with the experiments, that increasing the cross-section of the ring delays localized necking. N’souglo et al. (2018) carried out finite element calculations of elasto-plastic bars with yielding modeled with Gurson plasticity and submitted to dynamic tension at strain rates ranging from 103s−1to 5·104s−1. An initial velocity field compatible with the loading condition was applied on the specimen to minimize the intervention of waves before plastic flow localization, so that the deformation field during the bar extension was consistent with the ring expansion problem. The main difference with respect to the calculations of Guduru and Freund (2002) is that the initial porosity –which in the Gurson model intends to idealize material defects– was randomly varied at the element level, using predefined upper and lower bounds for the porosity variation, and different spatial distributions. The results of N’souglo et al. (2018) showed that, for the lower strain rates, the distribution of neck spacings is heterogeneous, and the porosity defects act as preferential sites for the nucleation of necks. On the other hand, as the strain rate increases, the distribution of neck spacings becomes more homogeneous due to the regularizing effect of inertia. Moreover, Vaz-Romero et al. (2019) performed a comparative study on the formation of multiple necks and fragments in elasto-plastic and hyperelastic rings subjected to expansion velocities ranging between 25 m /s and 600 m /s. The elasto-plastic material and the hyperelastic material were modeled with constitutive equations which provide nearly the same stress-strain response during monotonic uniaxial tensile loading, and fracture was assumed to occur at the same level of deformation energy. The calculations predicted virtually the same number of necks for the elasto-plastic and the hyperelastic rings, however, the mechanisms controlling the development of the necking pattern and the final fragmentation of the specimens were shown to be different. In the elasto-plastic rings several necks were arrested due to the release waves that travel throughout the circumference of the specimen after the localization process has started, so that the number of fractures was significantly lower than the number of necks. On the other hand, the release waves did not arrest the growth of any neck in the hyperelastic rings, on the contrary, the elastic energy released from the sections of the ring which are unloading during the localization process fueled the development of the necks, so that most of the necks developed into fractures. More recently, Marvi-Mashhadi et al. (2021) performed finite element calculations of expanding rings which included explicit representation of the porous microstructure of different additive-manufactured metals. The mechanical behavior of the material was modeled with von Mises plasticity, and a critical value of effective plastic strain was used as fracture criterion. The calculations, carried out for expansion velocities ranging from 50 m /s to 500 m /s, yielded individualized correlations between the number of necks and fragments, and the main features of the porous microstructure, including the initial void volume fraction and the maximum void size. The following 5 is an incomplete list of additional finite element works in which the formation of multiple necks and fragments in metallic rings subjected to dynamic expansion is modeled (Pandolfi et al., 1999; Becker, 2002; Rodr´ ıguez-Mart´ ınez et al., 2013b; Vaz-Romero et al., 2017; Marvi-Mashhadi and Rodr´ ıguez-Mart´ ınez, 2020; El Ma¨ ı et al., 2022). Notice that most of the works cited in previous paragraph focused the attention on the effect that mechanical behavior of the material and material defects have on the necking and fragmentation patterns. However, all these papers modeled the material response using macroscopic constitutive equations, paying no attention to the polycrystalline microstructure of the metallic rings, although the microstructure is known to be responsible for inherent material variations at the micromechanical level that favor plastic localization. This is precisely the gap we intend to fill in this paper, in which crystal plasticity finite element simulations of metallic rings subjected to dynamic expansion have been performed using the elasto-viscoplastic constitutive model developed by Marin (2006). The polycrystalline microstructure of the ring has been generated using two different random Voronoi tessellations containing 5000 and 15000 grains, respectively. The grains of the microstructure have been assigned with an initial crystal orientation corresponding to one of the four different textures investigated: isotropic texture, ⟨001⟩ ∥Θ Goss texture, ⟨001⟩ ∥R Goss texture and ⟨111⟩ ∥Z fiber texture. A salient feature of this investigation is that the finite element results show that the initial texture can be tailored to delay the formation of dynamic necks, improving the energy absorption capacity of ductile metallic materials at high strain rates. Note that, while Dequiedt and Denoual (2021) were probably the first researchers to perform crystal plasticity finite element simulations in order to study the effect of material microstructure on dynamic plastic localization, to the authors’ knowledge, this is the first paper ever providing a systematic analysis on the effect of initial texture on the formation of multiple necking patterns. 2. Crystal plasticity model The single crystal constitutive model used to perform the calculations of Section 4 is the elasto-viscoplastic formulation developed by Marin (2006), specialized to the case of small elastic strains. The main features of the formulation are presented in what follows, while the reader is referred to the technical report of Marin (2006) to obtain additional details. The model formulation follows the classical works by Hill and Rice (1972) and Asaro and Needleman (1985) as concerns the kinematics description and the use of the rate-dependent power law for slip. Accordingly, the constitutive model considers that crystallographic slip is the dominant deformation mechanism. 6 The single crystal kinematics rely on the multiplicative decomposition of the deformation gradient Finto an elastic component Feand a plastic component Fp: F=FeFp=VeReFp=VeˆFp(1) where Fpdescribes the motion of dislocations on crystallographic planes (leaving the crystal lattice unchanged), while ReandVerepresent the rotation and the elastic stretching of the lattice, respectively. The decomposition (1) introduces two intermediate configurations between the reference B0and the current Bconfigurations, which are denoted as ¯Band ˜B, respectively. Note that Fpbrings the crystal from the reference B0to the intermediate configuration ¯B, while ˆFpconnects B0with the intermediate configuration ˜B, which is used to express the crystal constitutive equations. Moreover, assuming small elastic strains, Veis approximated by: Ve=1+ϵe(2) where 1is the unit second-order tensor and ϵeis the small elastic strain tensor with ∥ϵe∥ ≪1, so that the current configuration Bdiffers from the intermediate configuration ˜Bonly by an infinitesimal amount. Hence, the rate of deformation tensor dand the spin tensor ware expressed as: d=▽ϵe+˜Dp w=−skew ( ˙ϵeϵe) +˜Ωe+˜Wp(3) with▽ϵe=˙ϵe+ϵe˜Ωe−˜Ωeϵeand ˜Ωe=˙Re(Re)Tbeing the elastic part of the rate of deformation tensor and the elastic lattice spin tensor, respectively. Note that the superscript ()Trefers to the tensor transpose,▽ () is the material Jaumann rate, and˙() denotes differentiation with respect to time. Moreover, ˜Dpand ˜Wpare the plastic part of the rate of deformation tensor and the plastic spin tensor given by: 7 ˜Dp=NX α=1˙γαsym( ˜Zα) ˜Wp=NX α=1˙γαskew( ˜Zα)(4) where ˙ γαis the plastic shear rate on the αslip system, Nis the number of slip systems, and ˜Zα=˜sα⊗˜mαis the Schmidt tensor with ˜sα=Resαand ˜mα=Remαbeing the slip direction and the normal to the slip plane in the current configuration, respectively. The plastic shear rate is defined with a power law flow rule: ˙γα= ˙γ0|τα| κα1 m sign ( τα) (5) where ˙ γ0is a reference shear strain rate and mis the rate sensitivity of slip. Moreover, καis the αslip system hardness governed by the following modified Voce-type relation: ˙κα=h0κs−κα κs−κ0 ˙γ (6) where h0is the initial hardening rate and κ0is the initial slip strength, while κs, the saturation slip strength, is given by: κs=κs0˙γ ˙γs0m′ (7) with ˙ γ=PN α=1|˙γα|being the net shear strain rate within the crystal, and κs0, ˙γs0andm′being the slip system hardening parameters which, similarly to h0andκ0, are taken to be the same for all slip systems. Moreover, in equation (5), ταis the resolved shear stress on the αslip system: 8 τα=τ: sym( ˜Zα) (8) where τis the Kirchhoff stress tensor defined by the elasticity relationship: τ=˜Ce:ϵe(9) with ˜Cebeing the fourth order crystal elasticity tensor. The calculations presented in Section 4 have been performed with the UMAT user subroutine developed by Marin (2006) for ABAQUS/Standard. The integration of the crystal plasticity constitutive model has been carried out using an implicit numerical integration procedure, see Marin (2006) for details. The material investigated is aluminum with FCC structure. The twelve {111}⟨110⟩slip systems in FCC crystal structure are shown in Table 1. The anisotropic elasticity constants (see Eq. (10)), taken from Marin (2006), and the parameters for the flow rule (see Eq. (5)) and the hardening law (see Eq. (6)), taken from Beaudoin et al. (1994), are shown in Table 2. The initial density used in the calculations is ρ= 2710 kg /m3. Note that the mild strain rate sensitivity of the material presumably heightens the influence of texture on the multiple necking patterns. It is expected that increasing the value of the strain rate sensitivity parameter will decrease the relative influence of texture on necking ductility (strain rate sensitivity stabilizes the material behavior, see Mercier and Molinari (2003) and Zhou et al. (2006)). Nevertheless, validation of this hypothesis is left for future work. 3. Finite element model This section describes the finite element model developed to study the formation of multiple necks in polycrys- talline thin rings subjected to rapid radial expansion, see Fig. 1. Material points are referred to using cylindrical coordinates ( R,Θ, Z). The origin of the coordinate system is located at the center of mass of the ring. The inner and outer radii of the specimen are Ri= 15 mm and Re= 15.5 mm, respectively, and the axial and radial thicknesses are e= 0.5 mm and h=Re−Ri, see Fig. 1a and 1b. Similar specimen dimensions were 9 αmαsα 1 (111) [01 ¯1] 2 (111) [10 ¯1] 3 (111) [1 ¯10] 4 ( ¯111) [01 ¯1] 5 ( ¯111) [101] 6 ( ¯111) [110] 7 ( ¯1¯11) [011] 8 ( ¯1¯11) [101] 8 ( ¯1¯11) [1 ¯10] 10 (1 ¯11) [011] 11 (1 ¯11) [10 ¯1] 12 (1 ¯11) [110] Table 1: Slip systems ( mα,sα) in FCC crystal structure. Symbol Value C11 108.2 GPa C12 61.3 GPa C44 28.5 GPa ˙γ0 1 s−1 m 0.05 h0 200 MPa κ0 210 MPa κs0 330 MPa ˙γs0 5×1010s−1 m′0.005 Table 2: Material parameters for aluminum. The anisotropic elasticity constants are taken from Marin (2006), and the parameters for the flow rule and the hardening law from Beaudoin et al. (1994). 10 used in the experiments performed by Zhang and Ravi-Chandar (2006). In order to reduce the computational time, only one eighth of the ring has been analyzed, so that, to maintain the axial symmetry of the problem, the circumferential displacement of the lateral surfaces of the specimen has been impeded, see Fig. 1b. The loading condition is a radial velocity VRapplied in the inner surface of the ring which remains constant during the entire analysis (Rusinek and Zaera, 2007; Vadillo et al., 2012; Vaz-Romero et al., 2019), whereas the initial condition is a radial velocity of the same value applied to all material points (Marvi-Mashhadi and Rodr´ ıguez-Mart´ ınez, 2020; Marvi-Mashhadi et al., 2021). The application of this initial condition minimizes the propagation of waves through the radial thickness of the ring caused by the abrupt motion of the inner surface at t= 0, precluding instantaneous plastic localization due to the velocity loading condition (Needleman, 1991; Xue et al., 2008; Vaz-Romero et al., 2019). The calculations are performed for loading velocities ranging between 250 m /s and 5000 m /s, leading to nominal strain rates varying from 1 .66·104s−1≤˙ε0≤3.33·105s−1. While the largest strain rates investigated exceed the regular experimental capabilities (depending on the specimen size, the maximum strain rate attained in ring expansion laboratory experiments is ≈5·104s−1, see Zhang and Ravi-Chandar (2006) and Janiszewski (2012)), exploring such a wide range of loading rates helps to enlighten the interplay between inertia effects and crystallographic microstructure. Moreover, similar range of loading velocities and strain rates was considered in the numerical simulations performed by El Ma¨ ı et al. (2022) to investigate the effect of surface roughness on the formation of multiple necks in round bars subjected to dynamic stretching, and in the ring expansion calculations performed by Rodr´ ıguez-Mart´ ınez et al. (2013a) to determine the effect of strain rate on the average neck spacing in multiple necking patterns (see Section 1). Note that the largest strain rates investigated are characteristic of hypervelocity impacts (Hassani et al., 2020) and high-energy forming operations (Golovashchenko et al., 2013). The polycrystalline microstructure of the rings has been created with the open-source software NEPER (Quey et al., 2011), so that the specimens have been tessellated into an aggregate of grains using random Voronoi seeds (grains are also referred to as crystals in this paper). The color map in Fig. 1a depicts the grains with different orientations. Two polycrystalline aggregates of 5000 and 15000 grains have been generated (the number of grains will be denoted as Ng), and for each polycrystalline aggregate up to three different spatial distributions of seed points, D1, ..., D3, have been considered (note that each spatial distribution of seed points amounts to a different spatial distribution of grains with different orientations). The resulting microstructures are denoted by N5000- D1,..., N5000-D3, and N15000-D1, ..., N15000-D3, respectively. The idea is to investigate the effects of grain size and spatial distribution of grains on the formation of the necking pattern. Note that the microstructures with 11 5000 and 15000 grains have in average 5 and 8 grains throughout the thicknesses of the ring, which leads to an average grain size of 100 µm and 62 .5µm, respectively. While the grains are relatively large, similar grain sizes have been reported for different commercial aluminum alloys (Shankar et al., 2005; Patil et al., 2014; Padmanabhan et al., 2019). Performing calculations with smaller grain size led to excessive increase of the computational cost of the calculations (see next paragraphs). Moreover, each grain of the polycrystalline aggregate is allotted with a crystal orientation taken from one of the four different textures that have been generated with an in-house Wolfram Mathematica code developed for this purpose, namely, isotropic texture (randomly oriented grains which indeed lead to isotropic texture, see Fig. 3), ⟨001⟩ ∥Θ Goss texture (see Fig. 5), ⟨001⟩ ∥R Goss texture (see Fig. 7) and⟨111⟩ ∥Z fiber texture (see Fig. 9). The anisotropic textures were generated perturbing the corresponding ideal orientations Q0according to the formula ( Q0andQiare rotation matrix transforming the local basis of the cylindrical frame ( R,Θ, Z) to the crystal frame): Qi= ∆Qi(n(ϕ1, ϕ2), ϕ3)Q0 with i= 1, . . . , N g (10) where ∆ Qiis the rotation matrix of perturbation specified by n(ϕ1, ϕ2) –the misorientation axis defined by two angles ϕ1,ϕ2polled randomly from the intervals (0 ,2π) and (0 , π), respectively, assuming uniform distribution in the interval. Moreover, ϕ3is the misorientation angle also polled randomly assuming a normal (Gaussian) distribution with the mean value equal zero and 1 .5◦standard deviation. The resulting orientation Qiis then assigned to the grain iof the ring microstructure. Recall that Ngis the number of grains. All the microstructures have been generated using a specific seed (random number seed), allowing for replicable analysis. The goal is to investigate the effect of initial texture on the formation of the necking pattern (the orientation of the crystals can change during loading). Note that the radial symmetry of the problem is broken by the different orientation of the grains, leading to perturbations in the field variables and ultimately to the formation of necks (there is no need to introduce geometrical or material defects to break the symmetry of the problem, e.g., see Zhang and Ravi-Chandar (2008)). The ring is discretized using four node linear tetrahedral elements (C3D4 in ABAQUS notation), as they are particularly suitable to describe the complex geometry of the grains, see Fig. 1a. For the microstructures with 5000 and 15000 grains, the average number of elements and the standard deviation for the three distributions of seed points are 582098 ±1018 and 1760760 ±7108, respectively, with the average number of elements per grain being 12 116 for both microstructures. Moreover, we have performed a mesh sensitivity analysis and checked that increasing the number of elements hardly affects the strain fields, the number of necks and the necking time predicted by the finite element simulations (the results of the mesh sensitivity analysis are not shown for the sake of brevity). We are aware that using tetrahedral elements to discretize the ring introduces artificial stiffness in the model which may delay necking. Nevertheless, it is difficult to provide a quantitative measure of the effect that using this type of element has on the finite element results. Note that we have also used C3D4 elements to discretize rings and plates containing explicitly resolved pores and subjected to dynamic loading (Marvi-Mashhadi et al., 2021; Nieto-Fuentes et al., 2022), and the necking predictions were consistent with experiments and numerical calculations available in the literature, which reinforces the idea that the trends presented in this paper are hardly affected by the mesh. The calculations have been performed with the dynamic implicit solver of ABAQUS/Standard (2019), using 1 core of a workstation Intel Xeon Gold 6128 @ 3 .4 GHz. The time increment was set by the code. The computational cost of each simulation ranged between 1 and 17 days, depending on the microstructure considered and the applied velocity. Increasing the loading velocity has a twofold impact on the computational time, decreasing the necking time because the loading process is shorter (i.e., reducing the computational cost), but increasing the necking strain due to inertia effects (i.e., increasing the computational cost). Note that the finite element results have been processed by first converting the ABAQUS ODB files to VTK format for ParaView using the Python script developed by Liu et al. (2017). 4. Results The presentation of results is divided into three sections: the effects of grain size, spatial distribution of grains and texture development on the necking pattern are investigated in Section 4.1 for the initially isotropic material, Section 4.2 shows the influence of initial texture on the necking time, on the necking pattern and on the distribution of grain orientations after necking localization, and Section 4.3 illustrates the effect of loading velocity on the number of necks for different initial textures. 4.1. Salient features The calculations presented in this section are performed with the initially isotropic texture (grains randomly oriented), for a loading velocity of VR= 1000 m /s, which corresponds to a nominal strain rate of ˙ ε0= 6.66·104s−1. Fig. 2 shows the normalized circumferential logarithmic strain ˆ εθθversus the normalized perimeter of the ring ˆP=4Θ π. Recall from Section 3 that only one eighth of the ring is modeled in the finite element simulations. The normalized perimeter of the ring is also referred to as normalized radial angle. The results correspond 13 ℎZoomMesh in the zoomed area   Θ (a) (b) Figure 1: Finite element model. (a) Polycrystalline microstructure N5000-D1 and mesh discretization. The origin of the cylindrical coordinate system ( R,Θ, Z) is located at the center of mass of the whole specimen. The color map depicts grains with different orientations. (b) Geometry, dimensions and boundary conditions. The yellow dashed line is the measurement path of the circumferential logarithmic strain shown in Figs. 2, 4, 6, 8 and 11-14. 14 to aggregates containing 5000 and 15000 grains, generated with three different spatial distributions of grains: N5000-D1,..., N5000-D3, and N15000-D1, ..., N15000-D3. The notation used to designate the microstructures was introduced in Section 3. The normalized circumferential logarithmic strain is defined as ˆ εθθ=εθθ εb θθ, where εθθis the circumferential logarithmic strain measured along the path shown in Fig. 1b, and εb θθ= ln Ri+VRt Ri approximates the background circumferential logarithmic strain in the ring, where trefers to the loading time. The background strain corresponds to the fundamental theoretical solution of the problem, which is computed assuming homogeneous deformation of the specimen, so that before plastic localization occurs ˆ εθθ≈1 (see Fig. 3a in Marvi-Mashhadi and Rodr´ ıguez-Mart´ ınez (2020)). The normalized circumferential logarithmic strain facilitates to perform one-to-one comparisons of necking patterns obtained from calculations carried out at different loading velocities and with different initial textures, for which the formation of necks occurs at (very) different logarithmic strains (see Marvi-Mashhadi and Rodr´ ıguez-Mart´ ınez (2020)). The ˆ εθθ−ˆPcurves of Fig. 2 correspond to the necking time and display a succession of excursions of strain which stand for sections of the ring with localized plastic deformation. Following N’souglo et al. (2021) and Jacques and Rodr´ ıguez-Mart´ ınez (2021), the necking time is the loading time when the condition ˙¯ε <10−3s−1is met for the first time for any material point of the path indicated in Fig. 1b. Note that ˙¯ε=q 2 3˜Dp:˜Dpis a scalar measure of plastic strain rate. The necking criterion corresponds to the specimen unloading, indicating that plastic strain has localized, giving rise to the formation of necking instabilities. The necking pattern is assumed to be (fully) formed at the necking time. Moreover, note that, while 10−3s−1is an arbitrary value of plastic strain rate for the necking condition, we have checked that the necking time is largely independent of this threshold value, provided that it is small enough as compared to the nominal strain rate, so that the sections of the ring outside the excursions of strain are unloaded. Similarly to Marvi-Mashhadi and Rodr´ ıguez-Mart´ ınez (2020), the excursions of strain that meet the condition ˆ εθθ≥1.2 at the necking time are taken to be the necks of the localization pattern (the results in Figs. 2, 4, 6, 8 and 11-14 correspond to the necking time). This criterion is relaxed by 5% since, as the necking condition is approached, the simulations yield a field output every 0 .25−1µs (it is difficult to obtain more field and history outputs due to the increase in the size of the ODB file), which determines the error in the identification of the necking time and thus in the number of necks (the actual condition for the necking time may be met between the selected field output included in Figs. 2, 4, 6, 8 and 11-14, and the next one corresponding to ≈0.25−1µs later). This explains that necks 3 and 4 in Figs. 4a and Fig 6a are taken to be necks, while the peak strain is slightly below 1 .2. (the necking time and the number of necks for all calculations reported in the paper are included in Appendix A). The necks for the microstructures N5000-D1 and N15000-D1 are indicated with black numbers in Figs. 2a and 2b, respectively. 15 Moreover, Marvi-Mashhadi and Rodr´ ıguez-Mart´ ınez (2020) showed that this criterion to determine the number of necks is not generally very sensitive to the cut-off value of ˆ εθθchosen (1 .2 in this paper), providing numerical predictions for the number of necks which are consistent with the experimental evidence (the reader is referred to the works of Guduru and Freund (2002) and El Ma¨ ı et al. (2022) to find alternative criteria for the identification of necks in multiple necking patterns). Note that the roughness of the ˆ εθθ−ˆPcurves, which show many little peaks of strain of smaller size than the necks, see the zoomed area in Fig. 2b, is due to the fact that adjacent grains with different orientations are subjected to different levels of strain, which leads to sharp transitions of strain at the grain boundaries. Recall from Section 1 that Dequiedt and Denoual (2021) performed crystal plasticity calculations to study multiple necking formation in thin plates subjected to dynamic biaxial stretching. Similarly to the results shown in Fig. 2, Dequiedt and Denoual (2021) obtained strain profiles with fluctuations of strain from grain to grain that they also attributed to differences in the local slip resistance of the grains because of their different lattice orientation. Fig. 2a includes ˆ εθθ−ˆPcurves for the three microstructures generated with 5000 grains (recall that all the ˆ εθθ−ˆP curves shown in this paper correspond to the necking time). Attending to the necking criterion (introduced in the paragraph above), 6 necks are formed in the ring for N5000-D1 and 7 for N5000-D2 and N5000-D3. The location of the necks varies with the spatial distribution of grains, which makes apparent the effect of the microstructure on the morphology of the localization pattern. The necking time also shows a slight variation with the microstructure, being≈9% smaller for N5000-D2 than for N5000-D1 and N5000-D3 (note that this difference lies within the error in the determination of the necking time based on the criterion given in previous paragraphs). Fig. 2b shows results for the microstructures with 15000 grains. The ˆ εθθ−ˆPcurves are qualitatively the same than for calculations with 5000 grains. For N15000-D1 the necking pattern consists of 8 necks, while 7 necks are formed for N15000-D2 and N15000-D3. While these results are not conclusive, it seems that there is a small increase in the number of necks with the increase in the number of grains (for the grain sizes investigated in this work). Note that the results for the number of necks show limited statistical significance since only one-eight of the ring is modeled. On the other hand, the necking time is ≈8% greater in average than for the calculations with 5000 grains, suggesting that decreasing the grain size delays the formation of the necking pattern (yet, 8% difference lies within the error in the determination of the necking time based on the criterion given in previous paragraphs). Note that, while the mesh is finer for the microstructures with 15000 grains, the differences in the number of necks and the necking time between the microstructures with 5000 and 15000 grains do not seem to be attributed to the discretization, as decreasing the element size generally leads to the opposite effect (i.e., it leads 16 to earlier formation of necks, see Nieto-Fuentes et al. (2022)). The results obtained in this paper are consistent with the calculations performed by Dequiedt and Denoual (2021) with 4000 and 50000 grains which showed that decreasing the grain size delays necking formation in thin plates subjected to biaxial stretching. The increasing necking time with decreasing grain size can be attributed to a decrease of the perturbation amplitude introduced by the tessellation with the increasing number of grains. This observation seems to be consistent with the results of El Ma¨ ı (2014) –see equation (4.10) therein– which showed that increasing the amplitude of perturbations leads to a decrease of the necking time in thin rods with imposed geometrical perturbations and subjected to dynamic stretching. (a) (b) Figure 2: Initial isotropic texture. Normalized circumferential logarithmic strain ˆ εθθversus normalized perimeter of the ring ˆPfor calculations performed with loading velocity VR= 1000 m /s. Results corresponding to the necking time for microstructures: (a) N5000-D1, N5000-D2 and N5000-D3 and (b) N15000-D1, N15000-D2 and N15000-D3. The horizontal yellow dashed line corresponds to ˆεθθ= 1.2. For interpretation of the