7 In this paper, we have investigated the eﬀect of the third invariant of the stress deviator on the formation 8 of necking instabilities in isotropic metallic plates subjected to plane strain tension. For that purpose, we have 9 performed ﬁnite element calculations and linear stability analysis for initial equivalent strain rates ranging from 10 10−4s−1to8·104s−1. The plastic behavior of the material has been described with the isotropic Drucker yield 11 criterion [11], which depends on both the second and third invariant of the stress deviator, and a parameter c 12 which determines the ratio between the yield stresses in uniaxial tension and in pure shear σT/τY. Forc= 0, 13 Drucker yield criterion [11] reduces to the von Mises yield criterion [32] while for c= 81/66, the Hershey-Hosford 14 (m= 6)yield criterion [19, 22] is recovered. The results obtained with both ﬁnite element calculations and linear 15 stabilityanalysisshowthesameoveralltrendsandthereisalsoquantitativeagreementformostoftheloadingrates 16 considered. In the quasi-static regime, while the specimen elongation when necking occurs is virtually insensitive 17 to the value of the parameter c, both ﬁnite element results and analytical calculations using Considère criterion 18 [10] show that the necking strain increases as the parameter cdecreases, bringing out the eﬀect of the third 19 invariant of the stress deviator on the formation of quasi-static necks. In contrast, at high initial equivalent strain 20 rates, when the inﬂuence of inertia on the necking process becomes important, both ﬁnite element simulations and 21 linear stability analysis show that the eﬀect of the third invariant is reversed, notably for long necking wavelengths, 22 with the specimen elongation when necking occurs increasing as the parameter cincreases, and the necking strain 23 decreasing as the parameter cdecreases. 24 Keywords: 25 Necking, Drucker yield criterion, Linear stability analysis, Finite elements, Inertia 26 ∗Corresponding author. Tel. +34 916249904; Fax: +34 916249430. E-mail address: jarmarti@ing.uc3m.es Preprint submitted to Meccanica February 11, 2021 2 1. Introduction 27 Signiﬁcant progress has been made over the last few decades in understanding and modeling the formation of 28 dynamic necking instabilities in ductile metallic materials subjected to diﬀerent stress states, including uniaxial 29 tension, plane strain tension and biaxial tension. 30 Fressengeas and Molinari [14] derived one-dimensional linear and non-linear solutions to study the inﬂuence 31 of inertia and thermal eﬀects on the formation of dynamic necks in bars subjected to simple tension. The linear 32 solutionswereobtainedusingalinearstabilityanalysis, whichevaluatesthegrowthofaninﬁnitesimalperturbation 33 superimposed on the homogeneous background solution of the problem at hand: if the perturbation grows faster 34 than the background solution, the plastic ﬂow is unstable and a neck-like deformation ﬁeld can exist. It was 35 shown that inertia stabilizes long wavelength perturbations, favoring the formation of shorter necks under dynamic 36 loadings. The nonlinear solutions, based on the long wavelength approximation, used a ﬁnite diﬀerences scheme 37 to solve a two-zone model which included a geometric imperfection in the bar to trigger necking localization. 38 It was shown that inertia/thermal softening delays/promotes necking formation, increasing/decreasing specimen 39 ductility. Few years later, Fressengeas and Molinari [15] extended the three-dimensional linear stability analysis 40 of Hutchinson et al. [23] to include inertia eﬀects and study the formation of necks in metallic sheets modeled 41 with a viscoplastic constitutive relation (no strain-hardening) and subjected to plane strain tension. It was 42 shown that at high strain rates, unlike what happens under quasi-static loadings, the stabilizing eﬀects of stress 43 multiaxiality and inertia on short and long wavelength perturbations promote the growth of an intermediate 44 wavelength (so-called the critical wavelength) for which the growth rate of the perturbation is maximum. This 45 critical wavelength was used by Fressengeas and Molinari [15] to compute the average spacing between necks 46 in the multiple localization patterns that emerge in copper shaped-charge jets, and satisfactory agreement was 47 found between theory and experimental data and nonlinear calculations available in the literature [28, 9]. Few 48 years later, Mercier and Molinari [31] developed a three-dimensional stability analysis to model the formation of 49 multiple necks in dynamically expanded rings. The salient feature of this model was to consider the curvature of 50 the specimens. The plastic behavior of the material was modeled with von Mises yield criterion [32], associated 51 ﬂow rule, and yield stress dependent on strain and strain rate. The predictions for the evolution of the number 52 of necks with the expansion velocity obtained by Mercier and Molinari [31] were compared with the experimental 53 data reported by Altynova et al. [2] for 6061aluminum rings tested at speeds ranging from 50m/s to300m/s, 54 and reasonable agreement was obtained between experiments and theoretical predictions for the whole range of 55 velocities considered. Moreover, Mercier et al. [30] extended the plane strain linear stability analysis developed 56 3 by Fressengeas and Molinari [15] to thermoviscoplastic materials with yielding based on the von Mises criterion 57 [32] to predict the critical conditions for the nucleation of necks and the spacing between necks in hemispherical 58 copper and tantalum shells expanded dynamically at strain rates of ≈104s−1. The predictions of the stability 59 analysis for the number of necks were in quantitative agreement with the experiments for both materials tested. 60 ShenoyandFreund[42]generalizedthequasi-staticperturbationanalysisoftheplanetensiontestdevelopedby 61 Hill and Hutchinson [21] including material inertia to model the nucleation of multiple necks in the ring expansion 62 experiments performed by Niordson [33], Grady and Benson [16] and Altynova et al. [2]. Notice that, unlike the 63 formulation of Mercier and Molinari [31], the model of Shenoy and Freund [42] did not account for the curvature 64 of the specimens. The material behavior was described with the rate-independent hypoelastic constitutive law 65 developed by Stören and Rice [44]. In agreement with Fressengeas and Molinari [15], the results of Shenoy and 66 Freund [42] showed that inertia leads to the development of a critical necking wavelength, which decreases with 67 the strain rate, explaining the experimentally observed increase in the number of necks with the ring expansion 68 velocity (see Niordson [33], Grady and Benson [16] and Altynova et al. [2]). Shortly after, Guduru and Freund 69 [17] adapted the perturbation analysis of Shenoy and Freund [42] to an extending cylinder in order to facilitate 70 direct comparison with the ring expansion experiments of Grady and Benson [16]. Assuming that necking occurs 71 when a critical value of the perturbation growth is reached, Guduru and Freund [17] obtained predictions for 72 the evolution of the number of necks and the necking strain with the expansion velocity. Despite the simpliﬁed 73 hypoelastic law used to model the material behavior, the stability analysis results obtained by Guduru and Freund 74 [17]wereinreasonableagreementwiththeexperimentaldatareportedbyGradyandBenson[16]forOFHCcopper 75 and1100−O aluminum rings tested at velocities ranging from 20m/s to 220m/s. Guduru and Freund [17] 76 complemented the stability analysis results with ﬁnite element calculations to simulate the fragmentation process 77 that followed the formation of multiple necks in the experiments of Grady and Benson [16]. The material behavior 78 was described with an elasto-plastic constitutive model with yielding based on the porous plasticity criterion 79 developed by Gurson [18], Tvergaard [45] and Tvergaard and Needleman [46]. Material fracture was assumed to 80 occur when a critical level of porosity is reached. The ﬁnite element predictions of Guduru and Freund [17] were 81 in quantitative agreement with the number of necks and fragments obtained in the experiments performed by 82 Grady and Benson [16] for both OFHC copper and 1100−O aluminum rings. 83 Zhou et al. [55] developed a one-dimensional linear stability analysis which accounts for stress multiaxiality 84 eﬀects onneckingformation using Bridgman approximation[4] for the hydrostatic stressesthat develop in anecked 85 section. The material was modeled using von Mises plasticity [32] with the yield stress dependent on strain and 86 4 strain rate. In agreement with Shenoy and Freund [42], the critical necking wavelength was found to decrease 87 with the strain rate and increase with the strain rate sensitivity. Zhou et al. [55] also showed that the strain 88 rate sensitivity slows down the growth of perturbations, stabilizing the material behavior and delaying necking 89 formation. The stability analysis of Zhou et al. [55] was later used by Zaera et al. [51] and Rodríguez-Martínez 90 et al. [41] to study the inﬂuence of strain induced martensitic transformation and dynamic recrystallization on the 91 formation of multiple necks in steel and titanium bars subjected to dynamic stretching. The main novelty of the 92 works of Zaera et al. [51] and Rodríguez-Martínez et al. [41] was to incorporate the thermomechanical coupling 93 into the formulation of Zhou et al. [55], including the heat release due to plastic dissipation and the exothermic 94 character of the microstructural evolutions. Shortly after, N’souglo et al. [36] extended the linear stability analysis 95 ofZhouetal.[55]toconsidermaterialswithyieldingbasedonGurson[18]criterion. Thepredictionsforthenecking 96 strain and the critical necking wavelength were compared with ﬁnite element simulations of long bars subjected 97 to dynamic stretching, and reasonable agreement was found for strain rates ranging from 1000s−1to50000s−1. 98 Xue et al. [50] developed a non-linear two-zone model based on the long wavelength approximation to describe 99 necking formation and development in metallic plates of 20mm thickness subjected to plane strain tension. The 100 theoretical predictions were compared with ﬁnite element simulations in which the plate was modeled as an array 101 of unit-cells with sinusoidal periodic geometric perturbations. The plastic behavior of the material was described 102 using von Mises criterion [32], with yield stress dependent on strain and strain rate. Two-zone model calculations 103 and ﬁnite element computations were performed for strain rates ranging from 100s−1to2000s−1. The two-zone 104 analysis captured the neck retardation due to inertia eﬀects, however, because stress multiaxiality eﬀects were 105 not included in the formulation of the model, short necking wavelengths were not suppressed, and the two-zone 106 analysis failed to predict the emergence of a critical necking wavelength for which the energy required to trigger a 107 neck is minimum. Very recently, Jacques [24] developed a non-linear two-zone model that extends the formulation 108 of Xue et al. [50] to consider any arbitrary in-plane loading and accounts for the hydrostatic stresses which develop 109 inside a necked region using Bridgman approximation [4]. The predictions of the non-linear two-zone model of 110 Jacques [24] for the necking strain were compared with the unit-cell ﬁnite element calculations of Rodríguez- 111 Martínez et al. [39], who extended the computational model of Xue et al. [50] to consider loading paths ranging 112 from plane strain to equibiaxial tension. Excellent agreement was found between theoretical model predictions 113 and ﬁnite element calculations, for all the loading paths considered, and strain rates ranging from 5000s−1to 114 50000s−1. Moreover, N’souglo et al. [34] have extended the non-linear two-zone model developed by Jacques [24] 115 to consider materials described with the anisotropic yield criterion of Hill [20], and constructed dynamic forming 116 5 limit diagrams for strain rates ranging from 100s−1to50000s−1. The formability limits at necking obtained 117 with the non-linear two-zone model were compared with unit-cell ﬁnite element calculations and with a linear 118 stability analysis speciﬁcally developed to model dynamic necking in anisotropic thin plates subjected to biaxial 119 stretching. The study of N’souglo et al. [34] encompassed ﬁve diﬀerent materials, two of them were model materials 120 with mechanical properties speciﬁcally tailored to bring to light the roles of inertia and orthotropy in dynamic 121 formability, and the other three were actual materials, namely, TRIP-780 steel, AA 5182-O and AA 6016-T4. 122 The results obtained with the three approaches –two-zone model, stability analysis and unit-cell ﬁnite element 123 calculations– showed qualitative agreement for the ﬁve materials considered, for the whole range of strain rates 124 investigated, and for loading paths ranging from plane strain tension to equibiaxial stretching. Consistent with 125 the quasi-static forming limit diagrams obtained by Sowerby and Duncan [43], Parmar and Mellor [37], Barlat [3] 126 and Butuc et al. [5] using the two-zone approach of Marciniak and Kuczyński [29], N’souglo et al. [34] showed that 127 if inertial eﬀects are considered, just like in the quasi-static case, the inﬂuence of the shape of the yield surface on 128 the forming limit strains is small for plane strain stretching while being important for biaxial stretching. 129 However, notice that most of the papers published so far on dynamic necking instabilities, including those 130 cited above, consider materials governed by a hypoelastic equation (e.g. refs. [42, 17]), or assume that the 131 material behavior in the plastic range obeys either von Mises [32], Gurson [18] or Hill [20] yield criterion (e.g. refs. 132 [17, 31, 50, 26, 27, 36, 34]), and therefore they do not account for the eﬀect of the third invariant of the stress 133 deviator on necking localization. An exception is the recent work of N’souglo et al. [35], who developed a linear 134 stability analysis and performed unit-cell ﬁnite element calculations to model the formation of dynamic necks in 135 plates subjected to plane strain tension and made of isotropic ductile materials displaying tension-compression 136 asymmetry. The plastic behavior of the materials was assumed to be governed by the isotropic Cazacu et al. [7] 137 yield criterion, which depends on all principal values of the stress deviator and a unique parameter k. Fork= 0, 138 the von Mises yield criterion is recovered, otherwise the criterion accounts for strength diﬀerential eﬀects, which 139 leads to a speciﬁc dependence of yielding on the third invariant of the stress deviator J3. It was shown that the 140 necking strain and the necking energy are signiﬁcantly greater for a material that displays a larger ﬂow stress 141 in uniaxial compression than in uniaxial tension ( k <0), than for a material modeled with von Mises plasticity. 142 In contrast, the critical necking wavelength was slightly dependent on the tension-compression asymmetry of the 143 material. 144 In this paper, we study the eﬀect of the third invariant J3on the formation of necking instabilities in isotropic 145 metallic materials with no tension-compression asymmetry. Speciﬁcally, we develop a linear stability analysis and 146 6 perform unit-cell ﬁnite element calculations to investigate the formation of necking instabilities in elasto-plastic 147 materials with yielding described by the Drucker criterion [11] and subjected to plane strain tension. This isotropic 148 yield criterion is pressure-insensitive, involves the second and third invariant of the stress deviator, and a unique 149 scalar material parameter c(see Section 2). For c/negationslash= 0, the criterion accounts for diﬀerent ratios between the yield 150 stress in uniaxial tension σTand the yield stress in pure shear τY, which in turn leads to a very speciﬁc dependence 151 of yielding on the third invariant of the stress deviator. If c= 0, the Drucker yield criterion reduces to the von 152 Mises yield criterion. It is also worth noting that another widely used isotropic yield criterion for BCC metals, the 153 Hershey-Hosford yield criterion [19, 22] with m= 6, is a particular case of Drucker model [11] corresponding to 154 c= 81/66(see the proof in Cazacu et al. [8] and further discussion later in this paper). In Section 3, we present the 155 linear perturbation analysis developed in this paper for materials modeled with Drucker criterion [11]. In Section 156 4, we present the unit-cell model used in the ﬁnite element simulations. Results of the unit-cell calculations 157 conducted using ABAQUS/Standard for low initial equivalent strain rates (10−4s−1)for materials characterized 158 by diﬀerent values of the parameter care discussed in Section 5. In Section 6, ﬁnite element simulations performed 159 with ABAQUS/Explicit for higher initial equivalent strain rates in the range 400−80000s−1are compared with 160 the stability analysis predictions. The main conclusions of this investigation are summarized in Section 7. 161 2. Constitutive framework 162 We begin with the presentation of the general equations governing elastic/plastic behavior for an isotropic 163 materialwithisotropichardening(Section2.1). Next, webrieﬂyreviewtheDrucker’syieldcriterion. Aspreviously 164 mentioned, to investigate the inﬂuence of J3on the plastic behavior and neck formation for both quasi-static and 165 dynamic conditions, theoretical and ﬁnite element analyses are presented for materials characterized by diﬀerent 166 values of the parameter c, describing the J3inﬂuence. The materials studied and key diﬀerences in their yielding 167 behavior are discussed (see Section 2.2). In addition, we show that for plane strain tension, it is possible to obtain 168 the plastic dissipation in closed form even for the case when yielding depends on J3(see Section 2.3). 169 2.1. General equations for elastic/plastic materials 170 We assume the additive decomposition of the total rate of deformation tensor dinto an elastic part deand a 171 plastic partdp: 172 d=de+dp(1) 7 where the elastic part of the rate of deformation tensor is related to the rate of the stress by the following linear 173 elastic law: 174 Oσ=C:de(2) withOσbeing an objective derivative of the Cauchy stress tensor, which corresponds to the Jaumann derivative 175 in ABAQUS/Standard and to the Green-Naghdi derivative in ABAQUS/Explicit, and Cbeing the fourth-order 176 isotropic elastic tensor deﬁned as: 177 C= 2GI/prime+K1⊗1 (3) where 1is the unit second-order tensor, and I/primethe unit deviatoric fourth-order tensor. Moreover GandKare 178 the shear and bulk modulus, respectively. 179 180 Moreover, the yield condition is expressed as: 181 f= ¯σ−σT= 0 (4) where ¯σis the equivalent stress associated to the Drucker yield criterion [11] (see equation (13)) and σTis the yield 182 stress in uniaxial tension which evolves with the equivalent plastic strain ¯εpfollowing a power-law type hardening 183 rule: 184 σT=σ0(ε0+ ¯εp)n(5) whereσ0,ε0, andnare material parameters. 185 186 The equivalent plastic strain is deﬁned as: 187 8 ¯εp=/integraldisplayt 0˙¯εpdτ (6) with ˙¯εpbeing the equivalent plastic strain rate. Note that a superposed dot denotes diﬀerentiation with respect 188 to time. 189 190 Moreover, assuming an associated plastic ﬂow rule, the plastic part of the rate of deformation tensor is: 191 dp=˙λ∂¯σ ∂σ(7) where ˙λis the rate of plastic multiplier. 192 Therefore, the work conjugacy relation: 193 σ:dp= ¯σ˙¯εp(8) leads to the identity: 194 ˙¯εp=˙λ (9) 2.2. Drucker isotropic yield criterion [11] and materials studied 195 According to the isotropic Drucker yield criterion [11], yielding is described as: 196 φ(σ) =J3 2−cJ2 3 (10) wherecis a material parameter, and J2=1 2/parenleftbigs2 1+s2 2+s2 3/parenrightbigandJ3=s1s2s3are, respectively, the second and 197 third invariant of the deviator of the Cauchy stress tensor, deﬁned as s=σ−1 3tr(σ)1, where tr denotes the trace 198 operator. Moreover, s1,s2ands3are the principal values of the stress deviator s. For the yield function to be 199 9 convex, the parameter cshould belong to the following range: −27/8≤c≤2.25(for proof, see Cazacu et al. [8]). 200 Note that the parameter cis expressible only in terms of the ratio between the yield stresses in uniaxial tension, 201 σT, and pure shear, τY, as: 202 c=27 4 1−/parenleftBigg√ 3τY σT/parenrightBigg6  (11) 203 Remark : It is worth noting that according to the von Mises yield criterion [32], irrespective of the material, the 204 ratio between the yield stresses in uniaxial tension and pure shear is the same, i.e. σT/τY=√ 3. In contrast, 205 Drucker criterion [11] allows to diﬀerentiate between isotropic materials. Namely, for materials withσT τY>√ 3the 206 parametercis positive, while for a material withσT τY<√ 3, the parameter cis negative. Note that for c= 0, the 207 von Mises yield criterion is recovered. It is also worth mentioning that very recently it was demonstrated that the 208 Hershey-Hosford yield criterion [19, 22] for isotropic BCC materials can be expressed in a very simple polynomial 209 form in terms of the second and third invariant of the stress deviator, as follows (see Cazacu [6] and Cazacu et al. 210 [8]): 211 φBCC(σ) = (s1−s2)6+ (s2−s3)6+ (s3−s1)6= 66J3 2−81J2 3 (12) which means that the Hershey-Hosford yield criterion [19, 22] for isotropic BCC materials is a particular case of 212 Drucker [11] corresponding to c= 81/66. 213 214 The equivalent stress associated to the Drucker yield criterion [11] is: 215 ¯σ=√ 3 /parenleftbigg 1−4c 27/parenrightbigg1/6/parenleftBig J3 2−cJ2 3/parenrightBig1/6(13) 216 10 2.2.1. Materials studied 217 The main objective of this investigation is to study the inﬂuence of the third invariant of the stress deviator on 218 the formation of necking instabilities in isotropic metal sheets with the same response in tension and compression 219 when subjected to plane strain extension. To this end, linear stability analysis and ﬁnite element calculations will 220 be performed for Drucker materials with the same elastic behavior, initial yield stress and hardening in uniaxial 221 tension but characterized by diﬀerent values of the parameter c. The numerical values of the elastic moduli 222 and hardening parameters in equations (3) and (5), are given in Table 1. Speciﬁcally, we analyze the inception 223 of necking instabilities in a material with σT/τY<√ 3in comparison with a von Mises material (c= 0)and 224 materials with σT/τY>√ 3(i.e.c>0). 225 226 To facilitate the discussion of the results, we denote as: 227 •Material 1: the von Mises material for which c= 0andσT/τY=√ 3. 228 •Material 2: it is characterized by σT/τY<√ 3andc=−1.5, may be representative of certain aluminum 229 alloys. 230 •Material 3: it is characterized by σT/τY>√ 3andc= 81/66. This is representative of an isotropic BCC 231 material obeying Hershey-Hosford [19, 22] yield criterion, see equation (12). 232 •Material 4: it is characterized by c= 2.25, which corresponds to the highest σT/τYratio that can be 233 accommodated by the Drucker yield criterion [11] (i.e. maximum value of cfor which the yield function 234 given by Eq. (10) is convex). 235 Fig. 1 shows the projection of the yield loci of these materials in the biaxial plane (i.e. plane corresponding to 236 one of the eigenvalues of the Cauchy stress tensor being equal to zero). We only show the tension-tension quadrant 237 because according to Drucker [11] the yielding response is the same in tension and compression. Note that, since 238 the Material 2is characterized by a negative value of c, its yield locus is exterior to the von Mises ellipse (c= 0). 239 On the other hand, the yield loci of Material 3and Material 4, which are characterized by positive values of c, 240 are interior to the von Mises ellipse. 241 2.3. Eﬀect of the third-invariant on the plastic dissipation according to Drucker yield criterion [11] 242 A key step in the stability analysis calculations is to obtain the plastic dissipation or equivalent plastic strain 243 rate, which allows to derive the fundamental solution of the problem to be perturbed, see Section 3. Indeed, it 244 11 Symbol Property and units Value ρ0 Initial density (kg/m3) 7800 GElastic shear modulus (GPa), Eq. (3) 115.4 K Bulk modulus (GPa), Eq. (3) 250 σ0Material parameter (GPa), Eq. (5) 1.8 ε0 Material parameter, Eq. (5) 0.0027 n Material parameter, Eq. (5) 0.1 Table 1: Numerical values of initial density, elastic constants and hardening parameters. Data after N’souglo et al. [35]. 00.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 σ1 / σTσ2 / σT plane strainMaterial 1: c=0 and σT/τY=1.732 Material 2: c=-1.5 and σT/τY=1.674 Material 3: c=81/66 and σT/τY=1.790 Material 4: c=2.25 and σT/τY=1.852 σ 2=σ 1 Figure 1: Plane stress ( σ3= 0) yield loci according to the isotropic Drucker yield criterion [11], Eq. (10), for Material 1(von Mises material,c= 0andσT/τY=√ 3), Material 2(c=−1.5andσT/τY= 1.674), Material 3(Hershey-Hosford for a BCC material, c= 81/66andσT/τY= 1.790) and Material 4(c= 2.25andσT/τY= 1.852). The tensile plane strain points are represented by symbols. 12 is well known that for non-quadratic yield criteria involving J3, for general loadings, the equivalent plastic strain 245 rate cannot be obtained in closed form. However, for plane strain tension, this is feasible, as it is shown below. 246 247 To simplify writing hereafter, we will denote the plastic strain rate tensor and the equivalent plastic strain 248 rate bydand ˙¯ε, respectively. Note that this notation is consistent with the fact that we neglect elasticity in the 249 linear stability analysis (see Section 3.2). For a material obeying Drucker yield criterion [11] and associated ﬂow 250 rule: 251 d=˙λ/bracketleftbigg 3J2 2s−2cJ3/parenleftbigg s2−2 3J21/parenrightbigg/bracketrightbiggA 6¯σ−5(14) tr/parenleftbigd2/parenrightbig 2=˙λ2¯σ−10/parenleftbiggA 6/parenrightbigg2/bracketleftBigg 9J3 2+/parenleftBigg 4c2 3−18c/parenrightBigg J2 3/bracketrightBigg J2 2 (15) 252 whereA=272 27−4c. For the problem studied, i.e. thin plates under plane strain stretching, relative to the 253 Cartesian coordinate system given in Fig. 2, dzz= 0andσyy= 0. Therefore, the principal values of the plastic 254 strain rate tensor are: 255 d1=dxx>0, d 2=dzz= 0, d 3=−d1=dyy<0 (16) Further making use of Eq. (14), we obtain that: 256 s2/bracketleftbigg3 4/parenleftBig s2 1+s2 2+s2 3/parenrightBig2−2c 3/parenleftBig −s2 1+ 2s2 2−s2 3/parenrightBig s1s3/bracketrightbigg = 0 (17) This leads to: 257 s2= 0 (18) 13 or 258 9/parenleftBig s2 1+s2 2+s2 3/parenrightBig2−8c/parenleftBig −s2 1+ 2s2 2−s2 3/parenrightBig s1s3= 0 (19) Sinceσyy= 0, Eq. (19) reduces to the following algebraic equation for b=σzz/σxx: 259 (4c−27)b4+ (54−8c)b3+ (−6c−81)b2+ (10c+ 54)b+ (4c−27) = 0 (20) For−27/8≤c≤2.25, which is the range of the parameter cfor which the Drucker yield surface is convex, 260 the above equation has no real solution. Consequently, d2= 0leads tos2= 0. It follows that the state of stress 261 is such that J3= 0,J2=s2 xxand¯σ=A1/6sxx. Further substitution into Eq. (14) leads to: 262 ˙¯ε=2 A1/6dxx=˙¯εMises/parenleftbigg 1−4c 27/parenrightbigg1/6 (21) wheredxxis the strain rate in the loading direction. The above expression shows that for materials with c <0, 263 ˙¯ε > ˙¯εMises, while for materials with c > 0,˙¯ε < ˙¯εMises. This diﬀerence in equivalent plastic strain rate and 264 consequently on plastic dissipation as compared to a von Mises material (c= 0)leads to diﬀerent localization 265 behaviors for both quasi-static and dynamic loadings (see Sections 5 and 6). 266 267 3. Statement of the problem and theoretical model 268 The statement of the problem is general. We use the notations customarily considered (e.g. see N’souglo et al. 269 [35]). We analyze a thin plate of initial thickness h0and initial length L0, see Fig. 2. Cartesian coordinates 270 (X,Y,Z )mark the location of a material point in the undeformed state, while the location of the same material 271 point in the deformed state is (x,y,z ). The plate is under plane strain constraint in the out-of-plane direction Z, 272 and the through-thickness direction Yis assumed of plane stress. Constant and opposed stretching velocities ±Vx 273 are applied at the ends of the plate X=±L0 2, as shown in Fig. 2. As anticipated in Section 2.3, we neglect the 274 elastic deformations and ¯ε= ¯εpwill be named hereafter equivalent strain and ˙¯ε=˙¯εpequivalent strain rate. The 275 14 kinematics of the problem can be found in Appendix A of N’souglo et al. [35]. We present the equations governing 276 the homogeneous deformation in Section 3.1, and the main features of the linear stability analysis in Section 3.2. 277 Figure 2: Schematic representation of the geometry and boundary conditions of the problem addressed: a plate of initial thickness h0 and initial length L0stretched at constant velocity Vx. The specimen is under plane strain constraint in the out-of-plane direction Z, and the direction Yis assumed of plane stress. This ﬁgure is taken from N’souglo et al. [35]. 3.1. Governing equations 278 The equations governing the problem are: 279   dxx=∂Vx ∂x Fxx=˙Fxx dxx FxxFyy= 1 h=h0Fyy ∂(hσavg xx) ∂X=ρh0∂Vx ∂t σavg xx=/parenleftbigg/radicalBig 1 +φ−1ln/parenleftbigg 1 + 2φ+ 2/radicalBig φ(1 +φ)/parenrightbigg −1/parenrightbigg σxx dxx=˙¯ε∂¯σ ∂σxx dzz=˙¯ε∂¯σ ∂σzz= 0(22a) (22b) (22c) (22d) (22e) (22f) (22g) (22h) 280 where expressions (22a) and (22b) are kinematic relations, equation (22c) is the incompressibility condition, the 281 continuity equation is given by expression (22d), equation (22e) is the balance of linear momentum in Lagrangian 282 description, relation (22f) gives the average stress in the loading direction, and the ﬂow rule is given by equations 283 (22g) and (22h). In this set of equations, Vxis the imposed velocity in the loading direction (as mentioned 284 before),dxxanddzzare the strain rates in the loading and the out-of-plane directions, respectively, FxxandFyy 285 are the deformation gradients in the loading and the through-thickness directions, his the current thickness of 286 the plate,ρis the material density (assumed constant), σxxandσzzare the uniform stresses in the loading and 287 the out-of-plane directions, respectively, σavg xxis the average stress in the loading direction [35] and φ=h 8∂2h ∂x2is 288 15 the argument of the Bridgman correction factor [4] which accounts for the axial hydrostatic stress that develops 289 in a necked zone (see Walsh [48] and Zhou et al. [55]). Note that equations (22g)-(22h) need to be specialized for 290 the speciﬁc yielding behavior considered. 291 292 Remark : The key novelty of this paper is to extend the theoretical analysis developed in N’souglo et al. [35] to 293 Drucker’s criterion [11]. Namely, specialization to a given plastic constitutive behavior is performed by completing 294 the system of governing equations (22) with the yield condition, the yield stress in uniaxial tension, the equivalent 295 strain and the equivalent stress; for the case of the isotropic Drucker yield criterion, see equations (4)-(6) and 296 (13). 297 298 The initial and boundary conditions are: 299 Vx(X,Y, 0) = ˙ε0 xxX;σij(X,Y, 0) = 0 ¯ε(X,Y, 0) = 0; h(X,Y, 0) =h0(23) Vx(L0/2,Y,t) =−Vx(−L0/2,Y,t) = ˙ε0 xxL0/2 (24) where ˙ε0 xxis the initial strain rate in the loading direction. 300 301 Remark : note that the applied loading is selected through the input value of the initial equivalent strain rate 302 ˙¯ε0. The rationale is that the equivalent strain rate enters into the nondimensionalization of the set of algebraic 303 equations resulting from the linear stability analysis, thus controlling the value of the inertia parameter (parameter 304 ˜L−1, see Eq. (31)). On the other hand, while the same rationale has been employed by other authors, e.g. Zaera 305 et al. [52] and N’souglo et al. [35], we are aware that it is also customary in the literature to impose the stretch rate 306 astheloadingcondition[49,34], sinceitmaybearcloserresemblancetotheactualboundaryconditionsindynamic 307 experiments like the expansion of hemispherical shells (see Mercier et al. [30]). Hence, stability analysis results 308 and ﬁnite element calculations obtained imposing the stretching rate, namely ˙ε0 xx= 4000s−1and˙ε0 xx= 80000 s−1, 309 are reported in Appendix A. On the other hand, for the case when yielding is described by the Drucker yield 310 criterion [11], according to equation (21), the relation between ˙¯ε0and ˙ε0 xxfor plane strain tension is: 311 16 ˙¯ε0=2 A1/6˙ε0 xx (25) Note that, for a given value of ˙¯ε0, as the parameter cincreases (recall that A=272 27−4c), i.e. as the ratio σT/τY 312 increases (see equation (11)), ˙ε0 xxincreases. 313 314 The solution of the problem at any given loading time, also called homogeneous or fundamental solution, is: 315 S1= (V1 x,d1 xx,F1 xx,F1 yy,h1,¯ε1,˙¯ε1,σ1 xx,σ1 zz,σavg,1 xx,¯σ1,σ1 T,φ1)T(26) 3.2. Linear stability analysis for an isotropic material with yielding according to Drucker criterion [11] 316 The stability of the fundamental solution is tested by introducing at some time t1a small perturbation δSof 317 the form: 318 δS(X,t)t1=δS1eiξX+η(t−t1)(27) with 319 δS1= (δVx,δdxx,δFxx,δFyy,δh,δ ¯ε,δ˙¯ε,δσxx,δσzz,δσavg xx,δ¯σ,δσT,δφ)T(28) whereδS1is the perturbation amplitude, ξis the wavenumber and ηis the growth rate of the perturbation at 320 timet1. Note that ηis usually referred to as the instantaneous instability index. Here ξandηare considered time 321 independent(frozencoeﬃcientsmethod),however,notethatatimedependentnumericalsolutionhasbeenrecently 322 developed by Xavier et al. [49] for von Mises materials subjected to plane strain tension. The perturbation, Eq. 323 (27), is imposed on lines X=constant because the preferential neck orientation for isotropic materials subjected 324 to plane strain tension is perpendicular to the loading direction (i.e. it corresponds to the direction of zero 325 extension, see refs. [30], [39] and [24]). 326 327 17 Therefore, the perturbed solution is given by: 328 S=S1+δS (29) with|δS|/lessmuch/vextendsingle/vextendsingleS1/vextendsingle/vextendsingle. A system of linear algebraic equations is obtained by substituting equation Eq. (29) into the 329 governing equations (22) specialized for Drucker yield criterion [11] and keeping only the ﬁrst-order terms in the 330 increments δS1. This system of equations, which is not explicitly shown in this paper for the sake of brevity, has 331 been adimensionalized by introducing the following dimensionless groups: 332 ˆVx=Vx h0˙¯ε1; ˆdxx=dxx ˙¯ε1; ˆFxx=Fxx; ˆFyy=Fyy; ˆh=h h0; ˆ¯ε= ¯ε;ˆ˙¯ε=˙¯ε ˙¯ε1; ˆσxx=σxx σ0; ˆσzz=σzz σ0; ˆσavg xx=σavg xx σ0; ˆ¯σ=¯σ σ0; ˆσT=σT σ0; ˆφ=φ; ˆη=η ˙¯ε1; ˆξ=ξh0 A non-trivial solution for δS1is obtained only if the determinant of the matrix of coeﬃcients of the system of 333 adimensionalized linear algebraic equations is equal to zero. This condition leads to the following cubic algebraic 334 equation for the growth rate of the perturbation ˆη: 335 B3(S1)ˆη3+B2(S1)ˆη2+B1(S1,ˆξ)ˆη+B0(S1,ˆξ) = 0 (30) with 336 B3(S1) = 8 ˆM2/parenleftBigˆd1 xx/parenrightBig2/parenleftBigˆF1 xx/parenrightBig3 B2(S1) = 8 ˆQ/parenleftBigˆF1 xx/parenrightBig3/parenleftBigˆM2ˆR1−ˆM1ˆR2/parenrightBig B1(S1,ˆξ) =2 3ˆh1ˆM2/parenleftBig˜Lˆξ/parenrightBig2/parenleftbigg/parenleftBigˆd1 xxˆh1ˆξ/parenrightBig2ˆσ1 xx+ 12/parenleftBigˆF1 xx/parenrightBig2/parenleftbigg ˆQ−/parenleftBigˆd1 xx/parenrightBig2ˆσ1 xx/parenrightbigg/parenrightbigg B0(S1,ˆξ) =2 3ˆh1ˆQ/parenleftBig˜Lˆξ/parenrightBig2/parenleftbigg −12ˆM2ˆd1 xx/parenleftBigˆF1 xx/parenrightBig2−ˆσ1 xx/parenleftBigˆM2ˆR1−ˆM1ˆR2/parenrightBig/parenleftbigg 12/parenleftBigˆF1 xx/parenrightBig2−/parenleftBigˆh1ˆξ/parenrightBig2/parenrightbigg/parenrightbigg(31) where1 ˜L2=ρ(h0˙¯ε1)2 σ0represents the inertial resistance to motion, ˆM1=σ0˙¯ε1∂dzz ∂σxx/vextendsingle/vextendsingle/vextendsingle t1and ˆM2=σ0˙¯ε1∂dzz ∂σzz/vextendsingle/vextendsingle/vextendsingle t1are the 337 18 non-dimensional derivatives of dzzwith respect to the components of the Cauchy stress tensor, ˆR1=σ0˙¯ε1∂dxx ∂σxx/vextendsingle/vextendsingle/vextendsingle t1338 and ˆR2=σ0˙¯ε1∂dxx ∂σzz/vextendsingle/vextendsingle/vextendsingle t1are the non-dimensional derivatives of dxxwith respect to the components of the Cauchy 339 stress tensor and ˆQ=1 σ0∂σT ∂¯ε/vextendsingle/vextendsingle/vextendsingle t1is the non-dimensional strain sensitivity of the material. 340 341 Equation (30) gives, for a certain value of the loading time t1, the value of the dimensionless instantaneous 342 instability index ˆηas a function of the dimensionless wavenumber ˆξ. Note also that the dimensionless wavenumber 343 is related to the normalized perturbation wavelength L0/h0as:L0/h0=2π ˆξ. The value of ˆηis an indicator of 344 the instantaneous stability of the solution. Equation (30) has only one real root, which is the physical solution of 345 the problem. The requisite for unstable growth of the perturbation is given by the real part of ˆη, denoted Re (ˆη), 346 to be positive ˆη+. Thus, if Re (ˆη)>0a neck-like deformation ﬁeld can exist. The stabilizing eﬀect of inertia 347 and stress multiaxiality on small and large wavenumbers, respectively, boosts the growth of a ﬁnite number of 348 intermediate modes. The mode with the greatest value of the instantaneous instability index ˆη+is characterized 349 by a wavenumber that is referred to as the critical instantaneous wavenumber, the critical instantaneous necking 350 mode, or the critical instantaneous perturbation mode ˆξˆη+ c. The critical instantaneous wavenumber, which evolves 351 with the loading time, and thus with the strain in the material, can be used to calculate the critical instantaneous 352 perturbation wavelength/parenleftbigL0/h0/parenrightbigˆη+ c(see Figs. 6 and 7 in Section 6.2). Moreover, an alternative indicator of 353 the stability of the solution was proposed by Fressengeas and Molinari [15], who used a cumulative instability 354 index,I=/integraltextt 0ˆη+˙¯ε1dtto track the evolution of the instantaneous growth rate of all the growing modes (see also 355 Petit et al. [38] and El Maï et al. [13]). To calculate I, we introduce the perturbation at diﬀerent times and 356 sum the instantaneous growth rate obtained for each loading time [35]. The mode with the greatest value of the 357 cumulative instability index Iis called the critical cumulative wavenumber, the critical cumulative necking mode, 358 or the critical cumulative perturbation mode ˆξI c. Like the critical instantaneous wavenumber, ˆξI calso evolves 359 with the loading time, i.e. with the strain, and it can be used to calculate the critical cumulative perturbation 360 wavelength/parenleftbigL0/h0/parenrightbigI c(see Figs. 6 and 7 in Section 6.2 and Fig. 16 in Section 6.3). 361 4. Unit-cell ﬁnite element model 362 This section presents the 2D ﬁnite element model developed in ABAQUS [1] to simulate necking localization 363 in Drucker [11] materials subjected to plane strain tension. Material points are referred to using a Cartesian 364 coordinate system with positions in the reference conﬁguration denoted as (X,Y ). The origin of coordinates is 365 located in the center of mass of the specimen, see Fig. 3. We use the same ﬁnite element model as N’souglo 366 19 et al. [35] who, following the earlier work of Xue et al. [50], modeled a plate subjected to plane strain tension and 367 with geometrical periodic perturbations as an array of unit-cells with sinusoidal spatial imperfections given by the 368 following expression: 369 h=h0−δ/bracketleftbigg 1 + cos/parenleftbigg2πX L0/parenrightbigg/bracketrightbigg (32) wherehis the perturbed thickness of the unit-cell and δis the amplitude of the imperfection. Consistent with the 370 notation used in the theoretical model presented in Section 3, L0andh0are the initial length and the unperturbed 371 thickness of the unit-cell, respectively. Due to the symmetry of the problem, only a quarter of the unit-cell is 372 analyzed, see Fig. 3. Finite element simulations will be conducted for several normalized cell lengths varying 373 within the range 0.5≤L0 h0≤6, with ∆ =2δ h0= 0.2%andh0= 2mm. The normalized cell length is the 374 counterpart of the normalized perturbation wavelength in the linear stability analysis, see Section 3.2, and it will 375 be also denoted as L0/h0. Hereinafter, the normalized cell length will be also referred to as cell size. Note that 376 the spatial imperfection is imposed on lines X=constant (see Section 3.2). The initial and boundary conditions 377 used in the ﬁnite element model are consistent with those used in the theoretical model (see Section 3.1 and Fig. 378 3). 379 Figure 3: Finite element model. Mesh and boundary conditions. A large imperfection amplitude has been shown in the ﬁgure for better illustration of the geometric perturbation. This ﬁgure is taken from N’souglo et al. [35]. The ﬁnite element model is meshed with four-node bilinear plane strain elements with reduced integration and 380 20 hourglass control (CPE4R in ABAQUS notation), with initial dimensions ≈25×25µm2. The constitutive model 381 presented in Section 2 has been implemented in ABAQUS [1] through UMAT and VUMAT user subroutines. The 382 UMAT has been used to obtain the low strain rate ﬁnite element results, ˙¯ε0= 10−4s−1, reported in Figs. 4, 5 383 and 15, and the VUMAT has been used to obtain all the other ﬁnite element calculations reported in this paper 384 for higher strain rates. The integration of the constitutive equations in both UMAT and VUMAT subroutines has 385 been performed using fully implicit algorithms. 386 5. Eﬀect of the third invariant on the formation of quasi-static necks in materials with the same 387 response in tension and compression and yielding according to Drucker criterion [11] 388 This section presents analytical calculations based on Considère criterion [10] and ﬁnite element results which 389 bring to light the inﬂuence of the third invariant J3described by the parameter c, on the formation of quasi- 390 static necks under plane strain stretching. Results are shown for Material 1(von Mises material, c= 0and 391 σT/τY=√ 3), Material 2(c=−1.5andσT/τY= 1.674), Material 3(Hershey-Hosford for a BCC material, 392 c= 81/66andσT/τY= 1.790) and Material 4(c= 2.25andσT/τY= 1.852), see Section 2.2. Recall the 393 relationship between the parameter cand the ratio σT/τY, Eq. (11), which shows that σT/τYincreases with c. 394 5.1. Analytical calculations based on Considère criterion 395 For plane strain tension, according to the Considère criterion [10], the onset of instability occurs when the 396 axial force in the specimen attains its maximum value. Making use of Eqs. (6), (13) and (21), we obtain the 397 following expressions for the necking strain ¯εneckand the necking stretch Fneck xx(i.e. the equivalent strain and the 398 axial stretch corresponding to the onset of necking): 399 ¯εneck=2n A1/6−ε0 (33) Fneck xx=eA1/6 2¯εneck(34) relations which are used to obtain the results presented in Table 2 for the materials studied. Note that both the 400 necking strain ¯εneckand the necking stretch Fneck xxincrease as the value of cdecreases (i.e. as the ratio between 401 21 the yield stresses in uniaxial tension and in pure shear decreases). However, while the value of ¯εneckis10%greater 402 in the case of σT/τY= 1.674than in the case of σT/τY= 1.852(these are the minimum and maximum values 403 considered for σT/τY), the variations in the necking stretch Fneck xxwithconly appear in the fourth decimal digit. 404 This is an original result of this investigation that, to the authors’ knowledge, has not been reported before. 405 Material Necking strain Necking stretch Material 1: von Mises material, c= 0andσT/τY=√ 3 0.1127 1.1025 Material 2:c=−1.5andσT/τY= 1.674 0.1166 1.1026 Material 3:c= 81/66andσT/τY= 1.790 0.1089 1.1025 Material 4:c= 2.25andσT/τY= 1.852 0.1052 1.1024 Table 2: Necking strain and necking stretch calculated for Material 1(von Mises material, c= 0andσT/τY= 1.732), Material 2 (c=−1.5andσT/τY= 1.674), Material 3(c= 81/66andσT/τY= 1.790) and Material 4(c= 2.25andσT/τY= 1.852) using Considère criterion, equations (33) and (34). Note that the Considère criterion [10] only provides reliable quantitative predictions for the necking strain 406 and the necking stretch in smooth specimens subjected to quasi-static loading conditions. If inertia eﬀects are 407 important, the necking localization occurs for equivalent strains and axial stretches greater than the ones reported 408 in Table 2, as it will be shown in Section 6. Moreover, for notched specimens, the hydrostatic stresses that develop 409 inside the notch also delay necking localization, as it will be discussed below. 410 5.2. Unit-cell ﬁnite element calculations 411 The ﬁnite element simulations are performed in ABAQUS/Standard at a low initial equivalent strain rate, 412 ˙¯ε0= 10−4s−1. Inertia eﬀects are not considered in the calculations. Recall that the normalized amplitude of the 413 imperfection is ∆ = 0.2%in all the ﬁnite element computations reported in this paper. 414 Fig. 4 shows the evolution of the equivalent strain ¯εwith the axial stretch Fxxfor two diﬀerent cell sizes, 415 L0/h0= 1andL0/h0= 3.5, and the various materials investigated in this paper. The axial stretch is calculated 416 asFxx= 1 + ˙ε0 xxt, with ˙ε0 xxbeing the initial strain rate in the axial direction and tthe loading time. Recall 417 from Section 3.1 the relationship between the initial strain rate in the axial direction ˙ε0 xxand the imposed initial 418 equivalent strain rate ˙¯ε0, see equation (25). On the other hand, the equivalent strain is measured in the thickest 419 section of the specimen, in the ﬁnite element located at the upper right corner of the cell, as indicated in Fig. 420 3 (see also refs. [39] and [35]). For both cell lengths investigated, L0/h0= 1in Fig. 4(a) and L0/h0= 3.5 421 in Fig. 4(b), the equivalent strain ﬁrst increases almost linearly with the axial stretch and then saturates to 422 a maximum. The saturation indicates that the ends of the cell are unloading, and that the plastic strain has 423 localized in the center of the specimen giving rise to a necking instability (see ref. [35]). Hence, we will refer 424 22 to the maximum value of the equivalent strain as necking strain ¯εneck. Similarly, the axial stretch for which the 425 saturation of the equivalent strain occurs will be referred to as necking stretch Fneck xx(the same notation we used 426 in Section 5.1). Namely, in the ﬁnite element calculations we assume that Fneck xxis the axial stretch for which the 427 condition d¯ε/dFxx= 10−3is met. Note that the necking stretch under quasi-static loading is hardly dependent on 428 the material considered (see Table 2 in Section 5.1). In contrast, the necking strain increases as the ratio between 429 the yield stresses in uniaxial tension and pure shear, σT/τY, decreases (see Table 2 in Section 5.1). The inﬂuence 430 of the value of con the necking strain arises from the diﬀerences in the equivalent strain rate for the 4materials 431 investigated, see equation (21) in Section 2.3. As the ratio σT/τYdecreases, i.e. as the parameter cdecreases, 432 the rate of accumulation of plastic deformation is greater, which rises the slope of the increasing portion of the 433 ¯ε−Fxxcurves. Since the necking stretch is approximately the same for the 4materials for this low strain rate (as 434 mentioned before), the greater the slope of the increasing portion of the ¯ε−Fxxcurves, the greater the necking 435 strain. Note that the diﬀerence in the necking strain obtained for c=−1.5andc= 2.25is approximately 10% 436 for both cell lengths investigated, see also Table 2 in Section 5.1. Moreover, the comparison between the results 437 forL0/h0= 1, Fig. 4(a), and L0/h0= 3.5, Fig. 4(b), shows that the necking strain and the necking stretch are 438 signiﬁcantly greater for the smaller cell size, for the 4materials considered. The larger values of ¯εneckandFneck xx439 obtained for L0/h0= 1are caused by the stabilizing eﬀect of stress multiaxiality on short cell sizes, see refs. [39] 440 and [35]. 441 00.10.20.30.40.5 1 1.2 1.4 1.6 1.8 2-1.5 0 1.227 2.249Equivalent strain, ε Axial stretch, FxxΔ=0.2% FEM: c=-1.5 FEM: c=0 FEM: c=81/66 FEM: c=2.25ε0=10-4 s-1 . L0/h0=1 (a) 00.10.20.30.40.5 1 1.2 1.4 1.6 1.8 2-1.5 0 1.227 2.249Δ=0.2%ε0=10-4 s-1 . L0/h0=3.5Equivalent strain, ε Axial stretch, FxxFEM: c=-1.5 FEM: c=0 FEM: c=1.227 FEM: c=2.25 (b) Figure 4: Finite element results obtained with Drucker yield criterion [11] for Material 1(von Mises material, c= 0andσT/τY=√ 3), Material 2(c=−1.5andσT/τY= 1.674), Material 3(Hershey-Hosford for a BCC material, c= 81/66andσT/τY= 1.790) and Material 4(c= 2.25andσT/τY= 1.852). The initial equivalent strain rate is ˙¯ε0= 10−4s−1. Inertia eﬀects are not considered in the ﬁnite element simulations. The amplitude of the imperfection is ∆ = 0.2%. Equivalent strain ¯εversus axial stretch Fxx. (a)L0/h0= 1 and (b)L0/h0= 3.5 23 Fig. 5 shows the necking strain ¯εneckand the necking stretch Fneck xxversus the cell size L0/h0. The necking 442 strain and the necking stretch decrease with the cell size so that the ¯εneck−L0/h0andFneck xx−L0/h0curves 443 display a concave-upward shape for the 4materials considered. The decrease rate of ¯εneckandFneck xxslows down 444 asL0/h0increases so that for long cell sizes the necking strain and the necking stretch are practically constant. 445 Similar ﬁnite element results were reported by Xue et al. [50] who showed that, under quasi-static loading, 446 long wavelength imperfections are the most deleterious, and the necking strain shows little dependence on the 447 wavelength if L0/h0>3. As mentioned before, the increasing necking strains obtained as the cell size decreases 448 are caused by the stabilizing eﬀects of stress multiaxiality on short wavelengths. Moreover, note that the necking 449 strain is greater as cdecreases for all the wavelengths, i.e. the trend noted in Fig. 4 for L0/h0= 1andL0/h0= 3.5 450 is the same for the whole range of cell sizes investigated. Similarly, the necking stretch is practically independent 451 of the material for all wavelengths. Note that for L0/h0= 6, the necking strain and the necking stretch calculated 452 using ﬁnite elements for the 4materials investigated are very close to the theoretical predictions obtained using 453 the Considère criterion [10], see Table 2. In other words, for long wavelengths and quasi-static loading, the eﬀect 454 of the imperfection on the necking strain and the necking stretch is very small. 455 00.10.20.30.40.5 0 1 2 3 4 5 6-1.5 0 1.227 2.249 L0/h0Necking strain, εneck Δ=0.2% FEM: c=-1.5 FEM: c=0 FEM: c=81/66 FEM: c=2.25ε0=10-4 s-1 .zoom 0.0750.15 3 4 5 6 (a) 11.21.31.5 0 1 2 3 4 5 6-1.5 0 1.227 2.249 L0/h0FEM: c=-1.5 FEM: c=0 FEM: c=81/66 FEM: c=2.25zoom 3 4 5 61.15 1.075Necking stretch, Fneck xx Δ=0.2%ε0=10-4 s-1 . (b) Figure 5: Finite element results obtained with Drucker yield criterion [11] for Material 1(von Mises material, c= 0andσT/τY=√ 3), Material 2(c=−1.5andσT/τY= 1.674), Material 3(Hershey-Hosford for a BCC material, c= 81/66andσT/τY= 1.790) and Material 4(c= 2.25andσT/τY= 1.852). The initial equivalent strain rate is ˙¯ε0= 10−4s−1. Inertia eﬀects are not considered in the ﬁnite element simulations. (a) Necking strain ¯εneckversusL0/h0. (b) Necking stretch Fneck xxversusL0/h0. The amplitude of the imperfection is ∆ = 0.2%. 24 6. Eﬀect of the third invariant on the formation of dynamic necks in materials with the same 456 response in tension and compression and yielding according to Drucker criterion [11] 457 In this section, we investigate the inﬂuence of inertia on the necking strain and the necking stretch for dynamic 458 conditions. For that purpose, ﬁnite element calculations are performed for initial equivalent strain rates ranging 459 from 400s−1to80000s−1, for the four materials obeying Drucker yield criterion [11]. Note that the largest strain 460 rates investigated exceed the regular experimental capabilities. For instance, the dynamic expansion of thin-walled 461 cylinders and hemispherical shells, which is frequently used to investigate the formation of dynamic necks, can 462 rarely be performed for strain rates higher than 20000s−1, see refs. [53] and [30]. However, exploring loading rates 463 greater than 20000s−1helps to bring out and understand the eﬀect of the third invariant of the stress deviator on 464 the inception of dynamic necks under plane strain tension. For the given range of initial equivalent strain rates, 465 the parameter describing the inertial resistance to motion ˜L−1varies between 0.00166and0.333(see equation 466 (31)). Note that previous numerical results reported by N’souglo et al. [36, 35] and Zheng et al. [54] indicated 467 that dynamic eﬀects start to play an important role in the inception of necks for ˜L−1in the range 0.06−0.1. The 468 ﬁnite element results are compared with the predictions of the linear stability analysis. 469 6.1. Calibration of the linear stability analysis 470 The comparison between ﬁnite elements and linear stability analysis requires to deﬁne a criterion for the 471 perturbation mode to turn into a necking mode. For that purpose, Dudzinski and Molinari [12] introduced the 472 concept of eﬀective instability , which is based on the assumption that a neck is triggered when the instability 473 index reaches a critical value. This critical value is known as the critical instantaneous instability index ˆη+ cor the 474 critical cumulative instability index Ic, depending on whether it refers to the instantaneous instability index or to 475 the cumulative instability index (see Section 3.2). 476 Based on a previous work of Vaz-Romero et al. [47], N’souglo et al. [35] has recently developed a methodology 477 to determine Icwith ﬁnite element simulations. Using the unit-cell model presented in Fig. 3, N’souglo et al. 478 [35] obtained, for a von Mises material ( c= 0) described with the parameters given in Table 1, the evolution of 479 the necking strain ¯εneckwith the cell size L0/h0reported in Fig. 6 (black solid markers). The calculations were 480 performed in ABAQUS/Explicit, accounting for inertia eﬀects. The amplitude of the imperfection was ∆ = 0.2% 481 (as in the ﬁnite element calculations presented in this work), and the imposed initial equivalent strain rate was 482 ˙¯ε0= 40000 s−1. Note that, unlike the quasi-static loading results reported in Fig. 5(a), the ¯εneck−L0/h0curve 483 for˙¯ε0= 40000 s−1shows a minimum for an intermediate cell size, with the corresponding strain being referred 484 25 to as the critical necking strain ¯εneck c, see Fig. 16(b). The minimum is the result of the stabilizing eﬀect of stress 485 multiaxiality and inertia on short and long cell sizes, respectively (we will further elaborate on this issue in Section 486 6.3). The value of the minimum necking strain is 0.475, and the corresponding cell size, referred to as the critical 487 cell size in Rodríguez-Martínez et al. [40, 39], is/parenleftbigL0/h0/parenrightbigFEM c≈2.2. Using this minimum of the necking strain as 488 the equivalent strain for which the cumulative index is calculated, the maximum value of I, which corresponds to 489 the critical cumulative perturbation wavelength, is 3.75, see Fig. 7. N’souglo et al. [35] took this value of Ias the 490 cumulative instability index required to trigger a neck, i.e. as the critical cumulative instability index Ic= 3.75. 491 Following an analogous procedure for the instantaneous index, in this work we have also obtained the critical 492 instantaneous instability index ˆη+ c= 19(see Fig. 7). Note that for the determination of ˆη+ candIcwe use as input 493 data only the necking strain corresponding to the critical cell size obtained from the ﬁnite element calculations 494 (i.e. the critical necking strain ¯εneck c, see Fig. 16(b)). Predictions of the linear stability analysis for wavelengths 495 other than the critical are presented in Section 6.2. 496 6.2. Comparison between critical instantaneous instability index and critical cumulative instability index 497 Fig. 6 shows a comparison for a von Mises material ( c= 0) between the equivalent strains calculated with the 498 stability analysis using the criteria I=Ic(green line) and ˆη+= ˆη+ c(red line), and the necking strains calculated 499 with the ﬁnite element simulations for a wide range of wavelengths L0/h0. The equivalent strains corresponding to 500 the criteria I=Icandˆη+= ˆη+ care the counterpart of the necking strains ¯εneckin the ﬁnite element calculations 501 and thus will be denoted the same way. The perturbation analysis results obtained with both criteria show 502 qualitative agreement with the ﬁnite element calculations: the value of ¯εneckﬁrst decreases with L0/h0, reaches 503 a minimum ¯εneck c, and then increases. However, the predictions obtained with the critical cumulative instability 504 indexIcﬁnd better quantitative agreement with the ﬁnite element results. For example, the critical cumulative 505 perturbation wavelength/parenleftbigL0/h0/parenrightbigI c≈2.1, which corresponds to the minimum of the ¯εneck−/parenleftbigL0/h0/parenrightbigcurve obtained 506 forIc= 3.75, is very similar to the critical cell size/parenleftbigL0/h0/parenrightbigFEM c≈2.2obtained in the ﬁnite element calculations. 507 In contrast, the critical instantaneous perturbation wavelength is signiﬁcantly smaller/parenleftbigL0/h0/parenrightbigˆη+ c≈1.3. Note that 508 the diﬀerences between the predictions obtained with the critical instantaneous instability index ˆη+ c= 19and the 509 ﬁnite element results are particularly important for long wavelengths. For/parenleftbigL0/h0/parenrightbig>/parenleftbigL0/h0/parenrightbigˆη+ c, the criterion 510 ˆη+ c= 19predicts a severe increase of ¯εneckwhich is not shown by the ﬁnite element results. 511 A key outcome of this paper is to demonstrate that at high strain rates the critical cumulative instability 512 indexIcprovides predictions closer to the ﬁnite element results than the critical instantaneous instability index 513 ˆη+ c. Note that additional comparisons between ﬁnite element results and stability analysis predictions which 514 26 support this statement are presented in Appendix B for a von Mises material ( c= 0) and ﬁve diﬀerent imposed 515 equivalent strain rates: 400s−1,4000s−1,10000s−1,20000s−1and80000s−1. Hence, most of the stability 516 analysis results presented in Section 6.3 are obtained with Ic= 3.75(we only show selected results with ˆη+ c= 19). 517 As N’souglo et al. [35], we assume that this critical value of the cumulative instability index is the same for all 518 the initial equivalent strain rates and materials investigated in this work. We are aware that this is a rather crude 519 assumption, however, considering Ic=constant facilitates the comparison between stability analysis predictions 520 and ﬁnite element simulations, and the interpretation of results, while providing trends for the necking strain and 521 the necking stretch that are in qualitative agreement with the unit-cell calculations. Nevertheless, for the sake 522 of completeness, in Appendix C we show stability analysis predictions obtained for a von Mises material ( c= 0) 523 with a critical cumulative index which depends nonlinearly on the strain rate. The comparison with ﬁnite element 524 calculations shows that accounting for the strain rate dependence of Icimproves the predictions of the stability 525 analysis, yet at the expense of needing additional calibration data. 526 00.20.40.60.811.2 0 1 2 3 4 5 6FEMInst Int L0/h0Necking strain, εneckε0=40000 s-1 .Δ=0.2% c=0 (von Mises)FEM ηc=19^+ Ic=3.750.475 (L0/h0)I c(L0/h0)η c+^(L0/h0)FEM c Figure 6: Comparison between ﬁnite element results (FEM) and linear stability analysis predictions (LSA) for Material 1(von Mises material,c= 0andσT/τY=√ 3). Necking strain ¯εneckobtained using ﬁnite element simulations (FEM) and linear stability analysis (LSA) versus L0/h0. Linear stability analysis results are shown for both criteria Ic= 3.75and ˆη+ c= 19. In the ﬁnite element simulations the amplitude of the imperfection is ∆ = 0.2%. The initial equivalent strain rate is ˙¯ε0= 40000 s−1. (For interpretation of the