6 In this paper we have investigated, using ﬁnite element calculations performed in ABAQUS/Explicit [1], 7 the eﬀect of ab initio geometric imperfections in the development of multiple necking patterns in ductile rings 8 subjected to dynamic expansion. Speciﬁcally, we have extended the work of Rodríguez-Martínez et al. [33], 9 who studied the formation of necks in rings with sinusoidal spatial perturbations of predeﬁned amplitude 10 and constant wavelength, by considering specimens with random distributions of perturbations of varying 11 amplitude and wavelength. The idea, which is based on the work of El Maï et al. [4], is to provide an idealized 12 modeling of the surface defects and initial roughness of the rings and explore their eﬀect on the collective 13 behavior and spacing of the necks. The material behavior has been modeled with von Mises plasticity and 14 constant yield stress, and the ﬁnite element simulations have been performed for expanding velocities ranging 15 from 10m/s to1000m/s, as in ref. [33]. For each speed, we have performed calculations varying the number 16 of imperfections in the ring from 5to150. In order to obtain statistically signiﬁcant results, for each number 17 of imperfections, the computations have been run with ﬁve random distributions of imperfection wavelengths. 18 For a small number of imperfections, the variability in the wavelengths distribution is large, which makes the 19 imperfections play a major role in the necking pattern, largely controlling the spacing and growth rate of the 20 necks. As the number of imperfections increases, the variability in the wavelengths distribution decreases, 21 giving rise to an array of more regularly spaced necks which grow at more similar speed. A key outcome is 22 to show that, for a large number of imperfections, the number of necks formed in the ring comes closer to 23 the number of necks obtained in the absence of ab initio geometric imperfections. 24 Keywords: 25 Ring expansion, Finite elements, Multiple necking, Geometric imperfections, Inertia, Stress multiaxiality 26 ∗Corresponding author. Tel. +34 916249904; Fax: +33 916249430. E-mail address: jarmarti@ing.uc3m.es Preprint submitted to International Journal of Impact Engineering July 17, 2020 2 1. Introduction 27 The eﬀect of material and geometric imperfections in the fragmentation of metallic structures subjected 28 to dynamic loading has been the subject of recurrent debate within the Solid Mechanics community for more 29 that 70years. In the 40’s of the last century, Mott [26, 24, 22, 23, 25] published what is still considered 30 as a reference model to explain the basic physical mechanisms underlying dynamic fragmentation of ductile 31 materials. The theory of Mott, that was primarily developed during World War II to describe the process of 32 fragmentation resulting from the explosive rupture of cylindrical shell casings, is basically a statistical one- 33 dimensional model that considers the onset of fractures as a random process that responds to the inherent 34 variability of the fracture strain of metallic materials. The underlying idea is that the fragmentation occurs 35 due to the activation of weak points of the material, such as defects or geometric imperfections, which are 36 distributed throughout the specimen and lead to the formation of multiple necks (in ductile materials) which 37 eventually develop into fractures. The distribution of neck spacings and fragment sizes is determined by 38 the propagation of the unloading waves which emanate from each necked and fracture site, and lead to the 39 development of obscured (unloaded) zones in which additional necks cannot nucleate and fractures cannot 40 occur. 41 The original theory of Mott [26, 24, 22, 23, 25] was extended years later by Grady and collaborators 42 [10, 17, 8, 11] to account for the dissipation of energy associated with the fracture process. These authors 43 derived expressions for the nominal fragment size, fracture time, and dynamic fracture strain that found 44 reasonable agreement with experiments and numerical computations, e.g. refs. [17, 18, 11]. Indeed, with 45 the development of Computational Mechanics, the ﬁnite element method has been extensively used over the 46 last3decades to simulate multiple necking and dynamic fragmentation problems, e.g. refs. [13, 39, 31, 36, 47 12, 40, 34, 33]. The popularity of computer simulations to study fragmentation problems is partly due to 48 the elevated cost of fragmentation experiments, and the fact that there are only a few laboratories in the 49 world with the equipment and skills required to perform such tests (e.g. high-velocity expansion of ring 50 [28, 9, 2, 47, 15, 3], thin-walled cylinders [35, 14, 7, 48, 16] and hemispherical shells [19]). In addition, the 51 ﬁnite element calculations have the potential to provide information about the mechanisms which control 52 multiple necking and fragmentation that is not accessible by experimentation. 53 For instance, ﬁnite element simulations have been extremely useful to obtain insights into the role of 54 geometricimperfectionsonthedominantandarrestednecks, thefragmentationpatterns, andthedistribution 55 3 of fragment sizes. Namely, Han and Tvergaard [13] analyzed using 2D numerical simulations the formation 56 of multiple necks in plane-strain cylinders expanding dynamically under prescribed body forces. The 57 material,consideredrateandtemperatureindependent,wasdescribedusingvonMisesplasticityandisotropic 58 hardening. The authors included in the specimen an array of sinusoidal geometric imperfections to break the 59 symmetry of the problem and trigger necking localization. They showed that, due to inertia eﬀects, multiple 60 necks can be formed at locations other than the prescribed initial thickness imperfections. Sørensen and 61 Freund [36] studied numerically the formation and growth of necks in thin-walled metal tubes undergoing 62 high-rate radial expansion under plane strain conditions. The necking pattern was triggered including in 63 the ﬁnite element model sinusoidal periodic imperfections of constant wavelength and amplitude. A main 64 diﬀerence with previous work of Han and Tvergaard [13] is that the plastic behaviour of the material was 65 described using Gurson plasticity, taking into account the nucleation and growth of microvoids, and the 66 temperature and strain rate sensitivities of the material. These authors showed that, for long wavelength 67 imperfections of small amplitude, an array of regularly spaced necks appeared around the circumference of 68 the ring. As in the numerical results of Han and Tvergaard [13], the spacing of the necks showed little 69 correlation with the initial imperfections distribution. Moreover, Guduru and Freund [12] simulated in 70 ABAQUS/Explicit [1] the ring expansion experiments performed by Grady and Benson [9] with 1100−0 71 aluminum and OFHC copper specimens. As in Sørensen and Freund [36], the material was modeled with 72 Gurson plasticity, and fracture was considered to occur when a critical value of porosity was reached. The 73 ﬁnite element model consisted of a long cylindrical bar subjected to dynamic stretching and with initial 74 conditions consistent with the expanding ring. The authors performed calculations in which the radius of 75 the bar was given a small sinusoidal geometric imperfection and showed, consistently with the results of Han 76 and Tvergaard [13] and Sørensen and Freund [36] for long wavelength imperfections, that the amplitude 77 of the imperfection had no signiﬁcant inﬂuence on the number of necks and fragments. The authors 78 also noted that in ABAQUS/Explicit [1], unlike in the in-house code used by Han and Tvergaard [13], 79 no imperfection is needed to trigger the formation of necks, since the numerical perturbations introduced 80 by the software are suﬃcient to cause the instability when the critical conditions are reached. The ﬁnite 81 element results obtained with and without geometric imperfection found reasonable agreement with the 82 experimentsofGradyandBenson[9]forthenumberofnecksandthefragmentationstatistics. Thenumerical 83 calculations were also compared with a linear stability analysis which provided accurate predictions for the 84 4 increase in the number of necks with the extension velocity. The linear stability analysis suggested that 85 the suppression of both short and long necking wavelengths due to stress multiaxiality (i.e. triaxiality) 86 and inertia, respectively, favors the growth of an intermediate necking wavelength which determines the 87 average size of the necks at high loading velocities [5, 6, 20, 21]. Moreover, Rodríguez-Martínez et al. [33] 88 performed ﬁnite element simulations with ABAQUS/Explicit [1] of the dynamic expansion of rings with 89 periodic geometric imperfections of constant amplitude and wavelength. The main diﬀerence with previous 90 works of Han and Tvergaard [13], Sørensen and Freund [36] and Guduru and Freund [12] is that Rodríguez- 91 Martínez et al. [33] explored a wider range of imperfection wavelengths such that the number of imperfections 92 included in the ring was varied from 5to150(i.e. from short to long wavelength imperfections). The ﬁnite 93 element calculations of Rodríguez-Martínez et al. [33] conﬁrmed the linear stability analysis predictions 94 reported in refs. [5, 6, 12, 20, 21]: while for intermediate wavelength imperfections every imperfection evolved 95 intoaneck, forshortand longwavelengthimperfections theneckingpatternshowedlittlecorrelation with the 96 imperfections distribution. The suppression of short and long wavelength imperfections led to the emergence 97 of a dominant necking pattern with the same average spacing obtained in ﬁnite element calculations in 98 which no geometric imperfection was included (and the necking pattern was triggered by the numerical 99 perturbations introduced by the software). 100 In this paper we extend the ﬁnite element analyses of Han and Tvergaard [13], Sørensen and Freund [36], 101 Guduru and Freund [12] and Rodríguez-Martínez et al. [33] by considering expanding rings with random 102 distributions of geometric imperfections of varying amplitude and wavelength. As in ref. [33], the material 103 is modeled with von Mises plasticity and constant yield stress. The numerical calculations are performed in 104 ABAQUS/Explicit [1] for expanding velocities ranging from 10m/s to1000m/s. For each speed, we have 105 performed calculations varying the number of imperfections in the ring from 5to150. In order to obtain 106 statistically signiﬁcant results, for each number of imperfections, the computations have been run with ﬁve 107 random distributions of imperfection wavelengths. The ﬁnite element results are compared with: (i) the 108 calculations reported by Rodríguez-Martínez et al. [33] for rings with imperfections of constant amplitude 109 andwavelength, (ii)thenumericalsimulationsperformedbyGuduruandFreund[12]withABAQUS/Explicit 110 [1]forstretchingbarsinwhichtheneckingpatternistriggeredbythenumericalperturbationsofthesoftware, 111 (iii) the predictions of a one-dimensional linear stability analysis developed by N’souglo et al. [30] (which is 112 based on the earlier work of Zhou et al. [50]) and (iv) the experiments of Grady and Benson [9] with 1100−0 113 5 aluminum and OFHC copper rings. The ﬁnite element calculations performed in this paper show that, if the 114 variability in the wavelengths of the perturbations is large (small number of imperfections), both number 115 and growth rate of the necks are mostly controlled by the imperfections. However, if the variability in the 116 wavelengths of the perturbations is small (large number of imperfections), the number of necks obtained in 117 the ﬁnite element simulations show qualitative and quantitative agreement with the results obtained in refs. 118 [33, 12, 30, 9]. 119 2. Finite element model 120 This section describes the 3D ﬁnite element model developed in ABAQUS/Explicit [1] to study the eﬀect 121 of geometric imperfections on the formation of multiple necks in ductile rings subjected to dynamic radial 122 expansion. 123 The model, shown in Fig. 1, is based on previous works, see refs. [34, 41, 33]. Material points are referred 124 to using a Cartesian coordinate system with positions in the reference conﬁguration denoted as {X,Y,Z}. 125 The origin of the coordinate system is located at the center of mass of the specimen. The ring is considered 126 to have a constant inner radius and variable radial thickness in the initial conﬁguration, with the outer 127 radius being deﬁned by an array of Nsinusoidal imperfections with varying amplitude and wavelength (the 128 imperfections are also called geometric perturbations everywhere in this paper). Motivated by the recent work 129 of El Maï et al. [4], the goal is to provide an idealized modeling of the surface defects and initial roughness 130 of the ring and explore their eﬀect on the collective behavior and spacing of the necks. The wavelengths of 131 the imperfections are generated using a Gaussian probability density function: 132 N/summationdisplay n=12niπRext=N/summationdisplay n=1λi (1) whereRext= 16mm is the maximum outer radius of the ring (i.e. the radius without imperfection), niis 133 the random number ∈(0,1)with/summationtextN n=1ni= 1, andλiis the wavelength of the i-th imperfection. We have 134 generated ﬁve random distributions of imperfection wavelengths denoted as RDIW iwithi= 1,...,5. Fig. 2 135 shows histograms with the number of imperfections Nas a function of the wavelength of the imperfections λ. 136 The results corresponding to RDIW 2(black blocks) and RDIW 3(red blocks) are included in each histogram. 137 ForN= 10, Fig. 2(a), the distribution of wavelengths is very heterogeneous with values of λranging between 138 1.7mm and 21.6mm, being the average wavelength λavg≈10mm. The variability in the wavelengths 139 6 distribution is gradually reduced as the number of imperfections increases. For instance, for N= 50and 140 N= 100, Figs. 2(b) and 2(c), the span of wavelengths is 0.004mm≤λ≤4.84mm and 0.002mm≤ 141 λ≤2.09mm, respectively. Notice that the average wavelength of the imperfections for N= 50and100is 142 λavg≈2mm and≈1mm, respectively. 143 Figure 1: Finite element model. Mesh, geometry and boundary conditions. Note that, due to the symmetry of the specimen, only half of the ring is shown. The number of imperfections is N= 40and the amplitude of the imperfections is δ= 0.01mm. Moreover, the inner radius and the axial thickness of the ring are Rint= 15mm ande0= 1mm, 144 respectively (see Fig. 1). These dimensions are taken from Rodríguez-Martínez et al. [33], and they are 145 similar to those used in the experiments of Grady and Olsen [11] and Zhang and Ravi-Chandar [47]. The X 146 andYcoordinates of the outer perimeter of the ring are calculated as: 147 X=/braceleftBigg Rext−δi 2/bracketleftBig 1−cos/parenleftBigg2πRextθ λi/parenrightBigg/bracketrightBig/bracerightBigg/bracketleftBig cos (θ)/bracketrightBig (2a) Y=/braceleftBigg Rext−δi 2/bracketleftBig 1−cos/parenleftBigg2πRextθ λi/parenrightBigg/bracketrightBig/bracerightBigg/bracketleftBig sin (θ)/bracketrightBig (2b) 148 whereδiis the amplitude of the i-th imperfection and θis the radial angle, see Fig. 1. Let us denote by 149 7 02468 0 5 10 15 20 25A BNumber of imperfections, N Imperfection wavelength, λ (mm)RDIW2 RDIW3N=10 (a) 01020304050 0 5 10 15 20 25A B N=50RDIW2 RDIW3Number of imperfections, N Imperfection wavelength, λ (mm) (b) 050100150 0 5 10 15 20 25A BNumber of imperfections, N Imperfection wavelength, λ (mm)N=100RDIW2 RDIW3 (c) Figure 2: Histograms showing the number of imperfections Nas a function of the imperfection wavelength λ. Results are presented for two diﬀerent random distributions of imperfection wavelengths: RDIW 2(black blocks) and RDIW 3(red blocks). The height of a colored block within a bar of the histogram marks the number of imperfections within a ﬁxed λinterval for a given distribution of imperfection wavelengths. Results are shown for diﬀerent number of imperfections: (a) N= 10, (b) N= 50and (c)N= 100. For interpretation of the