We develop a comparative analysis of the processes of dynamic necking and fragmentation in elasto-plastic and hyperelastic ductile rings subjected to rapid radial expansion. For that purpose, nite element simulations have been carried out using the commercial code ABAQUS/Explicit. Expanding velocities which range be- tween 25m=s and 600m=s have been investigated. The elasto-plastic material and the hyperelastic material are modelled with constitutive equations which provide nearly the same stress-strain response during mono- tonic uniaxial tensile loading, and fracture is assumed to occur at the same level of deformation energy. The computations have revealed that, while the number of necks nucleated in the elasto-plastic and hyperelastic rings is similar, the mechanisms which control their development are signi cantly di erent. In the elasto-plastic rings several necks are arrested due to the stress waves which travel the specimen after the localization process has started, and thus the number of fractures in the ring is signi cantly lower than the number of incepted necks. On the contrary, these stress waves do not stop the development of any neck in the hyperelastic rings. The elastic energy released from the sections of the ring which are unloading during the localization process fuels the development of the necks. Hence, for the whole range of investigated velocities, the proportion of necks that develop into fracture sites is much greater for the hyperelastic rings than for the elasto-plastic ones. The comparison between the numerical results obtained for both materials brings to light the roles of elastic unloading and plastic dissipation in multiple necking and fragmentation processes. Keywords: Dynamic necking, Fragmentation, Numerical simulations, Plasticity, Hyperelasticity 1. Introduction The mechanics of multiple necking and fragmentation in ductile metals submitted to high strain rates have been a subject of investigation over the last 70 years. The theoretical framework developed by Mott (1947), Corresponding author. Tel. +33 387315369; Fax: +33 387315366. E-mail address: alain.molinari@univ-lorraine.fr Preprint submitted to Mechanics of Materials January 9, 2019 2 which aimed to describe the process of fragmentation resulting from the explosive rupture of cylindrical casings, was the rst attempt to identify the mechanisms which control multiple fracture in metallic structures. The theory of Mott describes the onset of fractures as random processes that respond to the inherent variability in the fracture strain of ductile materials. When fracture occurs at one point, stress waves (named as Mott's waves) propagate away from the fracture sites releasing the stress in the neighbourhood. Within the regions of the specimen subjected to the action of the release waves, the material does not continue to stretch and thus failure is precluded. The size of the fragments is then determined by the distance travelled by the release wave. However, an assumption of Mott's theory is that fractures occur instantaneously and thus the fracture energy was not accounted for. Nevertheless, the proposed theory was able to predict the distributions of fragment sizes observed in the fragmentation of plastically deforming casings. Grady and co-workers (Grady et al., 1984; Kipp and Grady, 1985; Grady, 2002) enhanced the theory of Mott by including the dissipation of energy associated with the fracture process. Grady (2006) argued that some degree of work must be expended, and some fracture energy overcome, in opening the cracks delineating the fragment boundaries produced in the fragmentation event . In this way, the stress release analysis developed by Mott (1947) was extended by considering the time history of plastic release waves emanating from sites of fracture. These authors were able to predict the average fragment size observed in experiments, and showed that the fracture energy scales the fragment size. Over the last two decades, the ideas of Mott (1947) and Grady and co-workers (Grady et al., 1984; Kipp and Grady, 1985; Grady, 2002) have been frequently applied to interpret and model the distributions of necks and fragment sizes observed in ring expansion experiments. The ring expansion test, originally developed by Niordson (1965), consists of an electromagnetic device which drives the radial expansion of a thin ring at velocities as large as 300 m=s. The success of this experiment lies in the radial symmetry of specimen and loading conditions. This symmetry practically eliminates the propagation of stress waves along the hoop direction of the sample before localization, and allows to reveal the true dynamic properties of the material. Grady and Olsen (2003) performed experiments on rings of uranium-6%-niobium (U6N) with 30 mm in diameter, and square cross section with 0 :75mm in thickness. The expanding velocities covered during the experiments ranged between 50 m=s and 300m=s. More recently, Ravi-Chandar and co-workers (Zhang and Ravi-Chandar, 2006, 2008; Cliche and Ravi-Chandar, 2018), carried out experiments with specimens made of Al 6060-O, Al 1100-H14, Cu-101 and AZ31B-O, for expanding velocities which ranged between 40 m=s and 225m=s. The thickness of the samples was 0 :5mm, their cross-sectional length was varied from 0 :5mm to 2:286mm, and their diameter from 30 mm to 33mm. In all these works, the formation of fragments 3 was analysed using Mott's and Grady's probabilistic models for fragmentation. The authors stated that the fragmentation which follows the formation of multiple necks in ductile metallic rings can be described in terms of the Mott's release waves. To gain additional insight into these experimental results, Ravi-Chandar and Triantafyllidis (2015) de- veloped a 1D nite di erence model to simulate incompressible non-linear elastic bars subjected to dynamic stretching. Localized perturbations of strain and velocity were included in the bars at di erent locations. The authors suggested that the distance between necks, and thus the size of the fragments in the subsequent fragmentation process, is in uenced by the interaction between these perturbations that travel along the bar as the loading time evolves. This idea nds resemblance with the theory of Mott who, as mentioned before, assumed that the distribution of fractures responds to the inevitable imperfections present in all materials. However, the numerical calculations of Ravi-Chandar and Triantafyllidis (2015), being based on the method of characteristics, were restricted to strains below the one obtained from the Consid ere criterion. Since loading conditions prevailed at all points of the bar during the calculations, it was not possible to investigate the role of unloading waves in the development of the necking pattern. In order to complement the computations of Ravi-Chandar and Triantafyllidis (2015), Vaz-Romero et al. (2017) carried out 3D nite element simulations in ABAQUS/Explicit of non-linear elastic rods subjected to dynamic stretching. In the simulations of Vaz-Romero et al. (2017), the strains attained in the bars were above the one obtained from the Consid ere criterion. It was shown that, due to inertia e ects, the core of the multiple necking process occurs during the post-uniform deformation regime of the bar. Note that, for elasto-plastic materials under dynamic loading, the role of in- ertia in the formation of the multiple necking pattern during the post-Consid ere regime has been elucidated in various aspects by theoretical stability analyses, see the works of Fressengeas and Molinari (1994), Mercier and Molinari (2003, 2004), Zhou et al. (2006), Xue et al. (2008), Rodr guez-Mart nez et al. (2013a, 2015) and El Ma  et al. (2014). The importance of accounting for elastic unloading in order to simulate the formation and development of the necking pattern was also noted by Vaz-Romero et al. (2017), who made clear that their results were speci c for non-linear elastic bars and, in principle, could not be extrapolated to elasto-plastic specimens. Motivated by this statement of Vaz-Romero et al. (2017), in the present paper we carry out nite ele- ment simulations using ABAQUS/Explicit (2013) of elasto-plastic and hyperelastic rings subjected to dynamic expansion at velocities varying from 25 m=s to 600m=s. The elasto-plastic material and the hyperelastic material are modelled with constitutive equations which provide nearly the same stress-strain response during monotonic uniaxial tensile loading. The simulations conducted for both materials bring to light the in uence of 4 plastic dissipation and elastic unloading in the development of the necking pattern, and provides new insights into the role of the release waves that emanate from nucleated necks and fracture sites in the fragmentation of elasto-plastic and hyperelastic specimens. 2. Constitutive frameworks Two di erent isotropic constitutive frameworks are used to model the mechanical behaviour of the material in ring expansion simulations: (1) an elasto-plastic model and (2) a hyperelastic model. Both constitutive models are laid out in order to provide nearly the same stress-strain response during loading. The di erence sets in the unloading process, as further shown in Sections 2.1 and 2.2. The objective of the present work is to provide a deeper understanding on the respective e ects on necking and fragmentation of (i) plastic dissipation (elasto-plastic case) and (ii) stored and released elastic energy (hyperelastic case). 2.1. Elasto-plastic constitutive model The constitutive model, pre-implemented in ABAQUS/Explicit (2013), follows the standard principles of von Mises plasticity. The strain rate tensor dis decomposed as the sum of an elastic deand a plastic part dp: d=de+dp(1) The relation between the elastic strain rate and the stress rate is given by the following hypo-elastic law: _=C:de(2) where _is the Green-Naghdi objective derivative of the Cauchy stress tensor and Cis the tensor of isotropic elastic moduli given by: C= 2G~I+K1 1 (3) whereGis the elastic shear modulus, Kis the bulk modulus, 1is the second order unit tensor and ~Iis the fourth order deviatoric unit tensor. Moreover, the plastic part of the strain rate tensor is de ned by the J2 ow law: 5 dp=3 2_"ps (4) where _"p=q 2 3dp:dpis the equivalent plastic strain rate and  =q 3 2s:sis the equivalent von Mises stress. Moreover, s=h1denotes the deviatoric Cauchy stress tensor and h=1 3:1is the mean stress. The yield function fis given by: f= y (5) whereyis the yield stress of the material de ned as: y=B("p+"0)n(6) whereB,"0andnare material constants and  "p=Rt 0_"p( )d is the accumulated plastic strain. Thermal and viscous e ects are not considered in the constitutive behaviour of the material so that the e ects of inertia, stress multiaxiality and strain hardening control the formation of necks in the ring. On the one hand, we are aware that thermal and viscous e ects may be important in the behaviour of elasto-plastic materials subjected to high strain rates. On the other hand, note that similar constitutive model was used by Zhang and Ravi-Chandar (2010) to investigate the mechanisms of dynamic fragmentation in metallic tubes subjected to rapid radial expansion. In addition, neglecting thermal and viscous e ects allows to identify, by comparison with the results obtained with the non-linear elastic material presented in section 2.2, the speci c roles of plastic dissipation, stored elastic energy and unloading waves in the necking and fragmentation behaviors of ductile rings subjected to high velocity radial expansion. Moreover, note that, during unloading, the response of the elasto-plastic material is given by the linear elastic law de ned by Eq. (2), see Fig. 1. 2.2. Hyperelastic constitutive model The goal is to model a non-linear elastic material which, under monotonic uniaxial tensile loading, replicates the response of the elasto-plastic material described in Section 2.1. Following Ogden (1997), we use the isochoric-volumetric decomposition of the strain energy density: 6 (~I1;J) = iso(~I1) + vol(J) (7) where ~I1is the rst invariant of the isochoric left Cauchy-Green strain tensor ~b=J2=3FFT, whereFis the deformation gradient tensor and Jis the determinant of F. The Cauchy stress tensor is then given by the hyperelastic constitutive relation: =1 J@ (~I1;J) @FFT(8) which leads to (e.g. Bergstrom (2015)): =iso+vol=2 J@ iso(~I1) @~I1~b+K(J1)1 (9) where we have assumed, following for instance Bergstrom (2015), that vol(J) =K 2(J1)2. Moreover, note that ~b=~b1 3~I11. Recall that Kis the bulk modulus. As shown by Gough et al. (1999) and Marlow (2003), since iso(~I1) depends exclusively on the invariant ~I1, the isochoric strain energy can be completely determined from a single mechanical test. Thus, let us consider a uniaxial tension test de ned by the following deformation gradient tensor: F=2 6666410 0 020 0 033 77775(10) where1is the stretch in the direction of loading and 2=3are the stretches in the perpendicular directions. The component of the stress in the direction of the loading is recorded so that a relation exists between11and the longitudinal stretch 1or, equivalently, between 11and ln(1): 11=H(ln(1)) (11) As an example, for a material with Norton type behavior, the function His of the form H(x) =B(x+xo)n. The isochoric left Cauchy-Green tensor is: 7 ~b=2 66664~2 10 0 0~2 20 0 0 ~2 23 77775(12) where ~2 1=J2=32 1and~2 2=J2=32 2. Since det ~b= 1, we have that ~2=~1=2 1, and therefore the rst invariant of ~bis: ~I1=~2 1+ 2~1 1 (13) Hence, the tensor ~bis de ned as: ~b=2 666642 3 ~2 1~1 1 0 0 01 3 ~2 1~1 1 0 0 0 1 3 ~2 1~1 13 77775(14) Therefore, under uniaxial tension conditions (assuming that the stress state in the specimen is homoge- neous), the components of the Cauchy stress tensor in the loading and perpendicular directions, based on Eq. (9), are obtained: 11=4 3J@ iso(~I1) @~I1 ~2 1~1 1 +K(J1) (15) 22=33=2 3J@ iso(~I1) @~I1 ~2 1~1 1 +K(J1) (16) For a uniaxial tension test, the lateral surface of the specimen is traction free, which implies that 22= 33= 0. Thus, from Eq. (16) we have: K(J1) =2 3J@ iso(~I1) @~I1 ~2 1~1 1 (17) Using Eqs. (15) and (17), the component of the Cauchy stress tensor in the loading direction becomes: 11=2 J@ iso(~I1) @~I1 ~2 1~1 1 (18) For any material subjected to uniaxial tension, all the variables of the problem can be expressed in terms 8 of the longitudinal stretch 1or, analogously, of ln( 1). Therefore, the Jacobian can be expressed as J(ln(1)). This allows to obtain a relationship between the longitudinal stretch 1and the isochoric longitudinal stretch ~1such that1=(~1) with1(1) =fJ(ln(1)g1 31. Therefore, knowing the tensile response of the hyperelastic material Eq. (11), it is possible to obtain a relationship between 11and~1: 11=H(ln(1)) =F(ln(~1)) (19) Note that, if the material is incompressible, the functions HandFare identical since J= 1 and therefore ~1=1. It follows from the equation (18) that the derivative of the isochoric part of the strain energy is solely depending upon the tensile response of the material, as proposed by Marlow (2003). @ iso(~I1) @~I1=J 2" F(ln(~1)) ~2 1~1 1# (20) Previous expression shows that the derivative of the isochoric strain energy is related to ~1. Nevertheless, from equation (13), a link between ~1and~I1can be obtained through a third-order polynomial equation in ~1: ~3 1~I1~1+ 2 = 0 (21) Indeed, based on the Cardano's formula, only one root is real and two are imaginary. The real root is the physical solution: ~1=31=3~I1+ 9 +q 813~I3 12=3 32=3 9 +q 813~I3 11=3(22) Therefore, inserting (22) into (20) proves that the derivative of the isochoric strain energy has an explicit expression which is related to the rst invariant ~I1, and to the uniaxial stress-strain response of the hyperelastic material given by the function F(ln(~1)). Finally Eqs. (9), (20) and (22) provide the expression of the Cauchy stress tensor for the considered hyperelastic material: =F(ln(~1(~I1))) ~1(~I1)2  ~1(~I1)1~b(~I1) +K(J1)1 (23) with ~1(~I1) given by equation (22). Therefore, the general form of the constitutive law has been iden- ti ed from a single tensile test. The hyperelastic constitutive model, Eq. (23), has been implemented in 9 ABAQUS/Explicit (2013) through a VUMAT subroutine. In order to have the same loading response for the elasto-plastic and the hyperelastic materials, Eq. (6) is used with"0= 0 for the hyperelastic material (i.e. F(ln(~1)) =Bln(~1)n) and"0= 0:0001 for the elasto-plastic material. Thus, the initial yield stress of the elasto-plastic material is 263 MPa , see Fig. 1. Other material parameters, which are taken from the works of Ravi-Chandar and Triantafyllidis (2015) and Vaz-Romero et al. (2017), are provided in Table 1. As mentioned before, the material is considered nearly incompressible. The value ofKreported in Table 1 corresponds to a Poisson's ratio of 0 :495. Symbol Property and units Value 0 Initial density ( kg=m3) 7740 G Elastic shear modulus ( GPa ) 87.9 K Bulk modulus ( GPa ) 8766 B Material parameter ( GPa ) 2 "0 Material parameter 0.0001 n Material parameter 0.22 Table 1: Values of the parameters used in the elasto-plastic and hyperelastic constitutive models. Fig. 1 compares loading and unloading stress-strain curves obtained from both constitutive models under uniaxial tension. Note that "11= ln(1) is the logarithmic strain. The loading 11"11curves are superimposed, also for strains greater than the one obtained from the Consid ere criterion (0 :22 for this material), see Fig. 1. This is important for dynamic applications in which, due to the stabilizing e ect of inertia, the necking pattern is formed during the post-Consid ere deformation regime, see for instance the work of Vaz-Romero et al. (2017). As mentioned before, neglecting the strain rate and temperature dependencies of both materials facilitates to obtain good agreement between their respective loading curves, which is required to identify the speci c roles of elastic stored energy and plastic dissipation in the necking and fragmentation behaviours of non-linear elastic and elasto-plastic rings. The only di erence between the loading curves, that can hardly be noticed in the gure, even in the zoomed area shown in the left bottom part of Fig. 1, occurs below the initial yield stress of the elasto-plastic material when it features a linear stress-strain response, see Eq. (2), while the hyper-elastic material already features a non-linear behaviour. Nevertheless, we assume that this di erence in the response of both materials under monotonic uniaxial tensile loading is not playing any signi cant role in the results presented in Section 4. On the other hand, as mentioned before, the unloading 11"11curves are very di erent, see the arrows depicted in Fig. 1. For the elasto-plastic material, the stress goes to zero following 10 the linear elastic law de ned by Eq. (2). For the hyperelastic material, the stress goes to zero following the path taken during loading. Moreover, we are aware that non-linear elastic materials (e.g. some polymers) do not feature, under uniaxial tension, the concave-downward shape shown in Fig. 1. In many of these materials the nominal stress under uniaxial tension is an increasing function of the strain, see Treolar (1944). Thus the Consid ere criterion is never ful lled, which precludes the formation of necks. In some others, the nominal stress rst presents a peak, then drops until reaching a minimum, and then increases again to levels well above the initial peak. As shown by Hutchinson and Neale (1983), a neck initiates in these materials in the same way as in the materials presented in Fig. 1. However, the upturn of the uniaxial stress-strain curve forces the neck to spread along the entire size of the specimen, instead of remaining localized as in the materials presented in Fig. 1 which are described by a power-law, see equation (6). Therefore, the non-linear elastic material used in this work should be understood as a model material that, by comparison with the elasto-plastic material, allows to provide new insights into the role of elastic unloading on the formation of multiple necking and fragmentation patterns in ductile metals. Note that Ravi-Chandar and Triantafyllidis (2015) used similar constitutive model to investigate the necking and fragmentation behaviours of metallic rings subjected to rapid radial expansion. 0500100015002000 0 0.1 0.2 0.3B DAxial Cauchy stress, σ 11 (MPa) Axial logarithmic strain, ε 11Elasto-plastic HyperelasticUniaxial tension loading unloadingunloading loading zoom 0750 0 0.005 Figure 1: Axial Cauchy stress 11versus axial logarithmic strain "11for the elasto-plastic material and the hyperelastic material. While the loading path is almost identical for both materials, elastic unloading is di erent, see the arrows included in the plot. A zoom illustrating response of the material for low values of "11is included in the left bottom part of the plot. 11 3. Finite element model A 3D nite element model has been developed in ABAQUS/Explicit (2013) to simulate multiple necking and fragmentation in ductile rings subjected to rapid radial expansion. The numerical model, shown in Fig. 2, is based on previous works (Becker, 2002; Rusinek and Zaera, 2007; Vadillo et al., 2012; Rodr guez-Mart nez et al., 2013b,a). LetfR;;Zgdenote the cylindrical Lagrangian coordinates of a point in the ring in the undeformed state, andfr;;zgthe current position at time t, see Fig. 2. The initial thickness of the square cross section of the ring is e0= 0:75mm, and its initial inner and outer radii are Ri= 15mm andRo= 15:75mm, respectively. These dimensions are taken from Rodr guez-Mart nez et al. (2013a), and are similar to those used in the experiments of Grady and Olsen (2003) and Zhang and Ravi-Chandar (2006, 2008). The boundary and initial conditions applied in the simulations are: Boundary conditions : the following velocity boundary condition has been applied to the inner surface of the ring. VR(Ri;;Z;t) =V whereVRis the radial particle velocity. The outer and lateral surfaces of the ring are traction free. Initial conditions . The specimen is initially undeformed and unstressed. An initial velocity eld is prescribed to all material points. VR(R;;Z;0) =Vr Ri R;V(R;;Z;0) = 0;VZ(R;;Z;0) =V 2r Ri R3Z whereVandVZare the material particle velocities along the circumferential and axial directions, respec- tively. This velocity eld is calculated assuming material incompressibility (which is an approximation for the nearly incompressible materials used in the calculations). Note that, the above initial velocity eld is included to avoid, at the initial time, a velocity jump at the boundary that would be conducive to parasite wave propagation e ects a ecting the localization pattern, (Needleman, 1991; Xue et al., 2008). The ring is meshed with 63680 eight-node tri-linear brick elements with reduced integration (C3D8R in ABAQUS notation). The reduced integration helps to prevent mesh locking and provides accurate solutions 12 in nearly or fully incompressible materials. All the calculations presented in this paper are performed with 1 processor (no parallelization). The mesh shows radial symmetry. Five elements are placed through the thickness of the specimen. The aspect ratio of the elements is close to 1 : 1 : 1 (with dimensions 0:15 0:150:15 mm3), see Fig. 2. The integral viscoelastic approach available in ABAQUS/Explicit (2013) has been used to prevent hourglass deformation modes, with the scale factor used for all hourglass sti nesses being equal to one. Neither geometrical nor material imperfections were introduced in the nite element model. The numerical perturbations introduced by the software are sucient to perturb the eld variables, breaking the symmetry of the problem and thus allowing for ow localization (Rusinek and Zaera, 2007; Vadillo et al., 2012). It was discussed in the works of Rodr guez-Mart nez et al. (2013a,b, 2017) and Vaz-Romero et al. (2017) that, while the numerical perturbations are not controlled by the user, at high strain rates they do not dictate the average distance between necks but they just allow for the localization to occur. It was shown that the localization pattern obtained in ABAQUS/Explicit calculations, in non-linear elastic and elasto-plastic materials, is controlled, to a large extent, by the constitutive behaviour of the material, inertia e ects and stress multiaxiality. Moreover, we have also checked that the numerical perturbations have small in uence in the fragmentation results reported in this paper. In fact, we performed simulations with di erent versions of ABAQUS/Explicit (versions corresponding to di erent years) and di erent number of processors (from 1 to 6), and the number of necks and fragments generally varied less than 5%. The loading times corresponding to the beginning and end of the fragmentation process also showed a maximum variation of 5%7%. These results suggest that the trends shown in this paper represent the intrinsic material behavior, and that the inherent variability associated to the numerical calculations (numerical integration, discretization, parallelization, etc.) plays a secondary role. Material failure is accounted for in the simulations using the element deletion technique available in ABAQUS/Explicit (2013). For the elasto-plastic material, we consider that failure occurs when a critical value of the accumulated plastic strain, denoted by  "f, is reached. Neglecting the contribution of the elastic strains, the mechanical energy per unit volume expended in the deformation of the elasto-plastic material until failure isEep f=R"f 0d"p. Based on equations (5) and (6), the failure energy is: Eep f=B n+ 1 ("f+"0)n+1("0)n+1 (24) For the hyperelastic material, we consider that failure occurs when the value of the strain energy function (~I1;J) equalsEep f, where ( ~I1;J) can be obtained after integration of equation (20) using equation (22). 13 Figure 2: Finite element model. Mesh, geometry and boundary conditions. While only half of the specimen is shown for the sake of clarity, calculations are performed using a model of the whole ring. Considering that both materials fail for the same level of the deformation energy is a strong assumption. Nev- ertheless this allows us to compare, on energy basis, the fragmentation behaviour of a ring for an elasto-plastic material and a hyperelastic model material. Note that, while Eep fand ( ~I1;J) are energies per unit volume in the deformed and undeformed con gurations, respectively, they can be compared because the materials are considered nearly incompressible. 4. Results In this Section we report numerical results obtained with the nite element model presented in Section 3, and the two constitutive models developed in Section 2. Applied velocities which range between 25 m=s and 600m=s are investigated. These velocities correspond to initial strain rates ranging from 1666 :67s1to 40000s1. While the largest velocities (and associated strain rates) exceed the regular experimental capabilities (ring expansion tests can rarely be performed for velocities higher than 300 m=s, see Grady and Olsen (2003) and Zhang and Ravi-Chandar (2006, 2008)), exploring such a wide range of loading rates helps to enlighten the di erences between the necking and fragmentation behaviours of elasto-plastic and hyperelastic materials. Recall that the objective of this paper is to shed light into the speci c roles played by plastic dissipation, stored elastic energy and unloading waves. 14 Fig. 3 shows the normalized maximum principal logarithmic strain ^ "Iversus the normalized angular position P= 2for the two materials. The normalized strain is de ned as ^ "I="I "b, where"Iis the maximum principal logarithmic strain measured along the path shown in Fig. 2. Moreover, "b= lnRo+Vt Ro approximates the background logarithmic strain in the outer surface of the ring (in absence of strain localization). Therefore, during the homogeneous deformation of the specimen, before the localization pattern is formed, the normalized maximum principal logarithmic strain is ^ "I1. The ^"IPcurves show a succession of peaks and valleys, exposing the multiple necking process which develops in the rings. For the sake of clarity, we only cover values ofPranging from 0 to 0 :3, i.e. we show results for an angular section of 108. Showing 30% of the perimeter of the ring allows to include enough necks to obtain a representative sample of the entire localization pattern, without impairing the clarity of the graph, for all the expansion velocities investigated. In Fig. 3(a) results are displayed for V= 100m=s. For the elasto-plastic material they correspond to t= 98s, and for the hyperelastic material to t= 73s. The loading times are selected such that the maximum value of ^ "Ireaches1:9. This value has been chosen such that the necking pattern is fully formed, yet the strains are not so large that the nite element grid becomes signi cantly distorted. We consider that a neck is formed when, for a given excursion of strain, the ratio between the peak and the neighbouring valleys is greater than 1 :1. The same criterion was used, for instance, in N'souglo et al. (2018). Strain excursions occur in the sections of the ring which develop a strain signi cantly larger than the neighbourhood. The total number of necks in a calculation is determined once the localization pattern is fully developed i.e. further increase in strain does not lead to the formation of new necks. Based on this criterion, in Fig. 3(a) both the elasto-plastic and the hyperelastic ring show 10 necks within the range of Pinvestigated. Nevertheless, the formation and development of the necking pattern is di erent depending on the material type considered. The condition ^"I1:9 is reached earlier in the hyperelastic ring: the development of the necking pattern appears to be faster for the hyperelastic material. In the case of the elasto-plastic ring, one excursion in strain, indicated with a blue arrow in Fig. 3(a), has been arrested before forming a neck (the ratio of strains between the peak and the neighbouring valleys is smaller than 1 :1). The strain corresponding to the valleys which surround the excursions is nearly the same for all valleys, i.e. largely independent of the degree of development of the excursions to which they surround. This is indicated with a dashed yellow line which represents the accumulated plastic strain (background plastic strain) prior to full necking localization. In the case of the hyperelastic ring all the excursions in strain reach the neck criterion, none of them is arrested before. Moreover, the more developed is the excursion in strain, the smaller is the strain in the neighbouring valleys (note that for the hyperelastic material the unloading process only ends when the strain reaches zero). This is because the elastic energy 15 released from the valleys fuels the development of the necks. This idea will be further elaborated when the results of Figs. 4 and 5 will be discussed. In Fig. 3(b) the applied velocity is four times greater V= 400m=s. For the elasto-plastic material the results correspond to t= 48s, and for the hyperelastic material to t= 36s. These loading times are selected such that, based on the criterion used in Fig. 3(a), the maximum value of ^ "Iis1:9. In agreement with the results presented in Fig. 3(a), the necking pattern develops faster for the hyperelastic material. In the case of the elasto-plastic material there are 15 excursions in strain, one of them, indicated with a blue arrow, is arrested before the necking criterion is reached. In the case of the hyperelastic material, there are 17 excursions in strain, all of them reaching the neck criterion. These results suggest that for the hyperelastic material the necking pattern grows faster and all the excursions in strain develop into necks. Fig. 4 shows the normalized maximum principal logarithmic strain ^ "Iversus the normalized angular position Pfor both materials investigated and di erent loading times. The idea is to complement the results presented in Fig. 3, and explore the temporal evolution of the localization pattern. The applied velocity is V= 100m=s. Fig. 4(a) shows results for the elasto-plastic material and three loading times: t= 88s,t= 93sand t= 98s. Note that the data corresponding to t= 98swere already shown in Fig. 3(a). The localization pattern is clearly visible for the three loading times considered. Note that the three ^ "IPcurves intersect with each other several times. If the ^ "IPcurve is shifted upwards with increasing time (e.g. the solid red curvet= 93sis above the dashed black curve t= 88s), the maximum principal logarithmic strain "Ihas increased more than the background strain "b, which indicates an on-going strain localization process. Similarly, if increasing the loading time shifts the ^ "IPcurve downwards (e.g. the solid red curve is below the dashed black curve), the material is unloading elastically. Thus, we observe that all the valleys which surround the excursions in strain are being unloaded within the time interval 88 98s: the dashed green curve is below the solid red curve and this red curve, in turn, is below the dashed black curve. The value of ^ "I is nearly the same for all valleys for each time investigated (as mentioned before). This value is determined by the accumulated plastic strain in the material prior to full necking localization (which remains constant while "bincreases with loading time). On the other hand, some of the excursions in strain progress during the time interval 8898sand some other are arrested. For the excursions 1, 8 and 10, see the orange numbers in Fig. 4(a), the curve for 88 sis above the curve for 93 sand this, in turn, is above the curve for 98 s. This order of the ^ "IPcurves indicates that excursions 1, 8 and 10 do not grow for loading times greater than 88s. For the excursions 2, 3, 6, 9 and 11, see the blue numbers in Fig. 4(a), the curve for 98 sis above 16 00.40.81.21.62 0 0.05 0.1 0.15 0.2 0.25 0.3plastic 9.6 e-5 elastic 7.3e-5Elasto-plastic - Loading time: t=98 μs Hyperelastic - Loading time: t=73 μsV=100 m/sNormalized maximum principal logarithmic strain, εI^ Normalized angular position, P (a) 00.40.81.21.62 0 0.05 0.1 0.15 0.2 0.25 0.3Plastic 4.8 e-5 Elastic 3.6e-5 Normalized angular position, PV=400 m/s Elasto-plastic - Loading time: t=48 μs Hyperelastic - Loading time: t=36 μsNormalized maximum principal logarithmic strain, εI^ (b) Figure 3: Normalized maximum principal logarithmic strain ^ "Iversus normalized angular position P= 2. Comparison between elasto-plastic and hyperelastic materials. (a) Applied velocity V= 100m=s. Loading time for the elasto-plastic material: t= 98s. Loading time for the hyperelastic material: t= 73s. (b) Applied velocity V= 400m=s. Loading time for the elasto-plastic material:t= 48s. Loading time for the hyperelastic material: t= 36s. The yellow dashed line, which indicates the value of ^"Iin the valleys corresponding to the elasto-plastic material, represents the accumulated plastic strain prior to full necking localization. The blue arrows indicate excursions of strain that have been arrested before forming a neck. (For interpretation of the