. The present paper is devoted to the analysis of st rain-rate history effects on neck formation under dynamic loading. For materials pres enting strain-rate history effects, two different strain-rate sensitivities should be disti nguished: the instantaneous strain-rate sensitivity and the work-hardening strain-rate sens itivity. We have analysed the relative contributions of these two kinds of strain-rate sen sitivities to neck retardation for two different configurations: a bar under impact tension and a dy namically expanding ring. For this purpose, we have developed finite element models and, for th e second configuration, an analytical model based on the linear stability analysis. The obtaine d results show that strain-rate history effects have a significant influence on the onset and devel opment of necking. The reason of this phenomenon is that, contrary to the instantaneous s train-rate sensitivity, the work-hardening strain-rate sensitivity does not contribute to dela y the neck formation. Keywords: Necking; Dynamic loading; Plastic flow localisatio n; Critical impact velocity; Viscoplasticity; Strain-rate history effects. 1. Introduction It is well established that the occurrence and deve lopment of necking instabilities is strongly related to the work hardening and strain-rate sensi tivity of the material. Woodford (1969) analysed results of uniaxial tensile tests for diff erent materials and observed a correlation between the total elongation at failure and the str ain-rate sensitivity of the materials. Several * Corresponding author. Tel.: +33 2 98 34 89 36; fax: +33 2 98 34 87 30. E-mail address: nicolas.jacques@ensta-bretagne.fr (N. Jacques) 2 theoretical and computational models have been prop osed to rationalise the effect of strain-rate sensitivity on necking (Marciniak et al., 1973; Ghosh, 1977a; Hutchinson a nd Neale, 1977; Chung and Wagoner, 1988; Wagoner and Wang, 1983; Au doly and Hutchinson, 2019) . These studies have shown that a rather small amount of st rain-rate sensitivity can significantly delay the occurrence of localised necking. In this regard , it should be noted that the predictions of localisation analyses are generally very sensitive to the adopted constitutive relation. For instance, when a simple power-law relation (for whi ch the strain-rate sensitivity is constant) is employed, the necking strain is independent of the prescribed strain-rate (Hutchinson and Neale, 1977) . With more advanced constitutive laws, an increase in strain-rate may either increase or decrease the ductility (Xue et al., 2008; Ghosh, 1977b) . At high strain rate, inertia causes an additional s tabilising effect on necking (Fressengeas and Molinari, 1985; Altynova et al., 1996) . Both mechanisms (inertia and strain-rate sensitiv ity) contribute to hinder neck development and delay the occurrence of plastic flow localisation (Mercier and Molinari, 2003; Xue et al., 2008) . However, the interplay between inertia and strain-rate sensitivity is complex and, in some cas es, increasing rate sensitivity may lead to a less stable material behaviour (Vadillo et al., 2012; Jacques, 2020) . In some circumstances, inertia may have a detriment al effect on ductility, mainly because of wave propagation effects. For instance, in linear s pecimens initially at rest and subjected to dynamic impact tension, the location of the necks i s controlled by the intervention of stress waves (Nemes and Eftis, 1993; Vaz-Romero et al., 2015; Ri ttel et al., 2014; Rotbaum et al., 2015; Needleman, 2018) . Moreover, wave propagation effects are responsibl e for the existence of a critical impact velocity (CIV) which defines t he maximum energy absorbed by the sample and the maximum sample ductility in the dynamic ten sion test (von Karman and Duwez, 1950; Wood, 1967; Klepaczko, 2005; Rusinek et al., 2005; Rodríguez-Martínez et al., 2013b) . The CIV corresponds to the lowest impact-velocity which induces sample failure close to the impacted end due to plastic waves trapping. When th e velocity applied at the end of the specimen exceeds the CIV, plastic deformation does not spread along the gauge of the specimen, it is “trapped” near the impacted end of the sample. If the applied velocity is further increased, the energy absorbed by the specimen drop s rapidly. Note that several numerical studies showed a relation between CIV and material strain-rate sensitivity (increasing strain- rate sensitivity leads to a larger CIV) (Hu and Daehn, 1996; Vaz-Romero and Rodríguez- Martínez, 2019) . In contrast, no critical velocity is observed in experiments on dynamic expansion of rings (Niordson, 1965; Altynova et al., 1996; Zhang and R avi-Chandar, 2006, 3 2018) and cylinders (Wood, 1967; Zhang and Ravi-Chandar, 2010; Jeanson et al., 2016) . In these experiments, the radial symmetry of the speci men and the loading enables to obtain a uniform strain field (prior to the occurrence of pl astic flow instabilities), even at very high loading velocity. Moreover, it should be mentioned that most of the t heoretical and numerical works devoted to the analysis of the influence of strain-rate sensit ivity on necking instabilities consider “standard” viscoplastic models in which the flow st ress of the material depends on the current values of the effective plastic strain and strain-r ate. However, many materials notably Face Centred Cubic (FCC) and Hexagonal Close Packed (HCP ) metals exhibit strain-rate history effects (Nicholas, 1971; Klepaczko, 1975; Senseny et al., 1 978; Lea, 2018) . This means that the flow stress does not only depend on the instantaneo us (actual) value of the strain rate, but also on its prior evolution. Fig. 1 shows schematic stress-strain curves illustrating strain-rate history effects (as typically observed in FCC and HCP mater ials, see for e.g. (Klepaczko, 1975) and (Senseny et al., 1978) ). The two solid lines correspond to the responses of the material at constant strain rates  and  (with >  ) and the dotted line to the case when a sudden strain-rate increment from  to  occurs during the test. Notice that the flow stres s after the strain-rate jump is significantly lower than if the material had been deformed at  since the beginning of the test. Fig. 1. Schematic stress-strain curves illustrating strain rate and strain-rate history effects for FC C and HCP materials (adapted from the experimental re sults presented in (Klepaczko, 1975) and (Senseny et al., 1978) ). Strain Stress  →  4 Strain-rate history effects are related to the fact that the microstructural evolution of a material with the deformation may be dependent on the applie d strain rate 1. For instance, Chiem and Duffy (1983) carried out observations of the dislocation substr ucture in single-crystal aluminium specimens deformed at quasi-static and hi gh strain rates and detected that for a given strain level the dislocation density is larger when the material is deformed at high strain rate. Therefore, for materials with strain-rate history e ffects, it is necessary to distinguish the instantaneous strain-rate sensitivity (related to t he dependence of flow stress on strain rate at given microstructure) and the strain-rate sensitivi ty of work hardening (related to the influence of strain rate on the microstructure evolution). Th e strain-rate sensitivity of work hardening is particularly marked for FCC metals (Klepaczko and Chiem, 1986; Lea, 2018) . A variety of constitutive models have been proposed to describe strain-rate history effects (Estrin and Mecking, 1984; Klepaczko and Chiem, 198 6; Follansbee and Kocks, 1988; Tong et al., 1992; Bodner and Rubin, 1994; Molinari and Rav ichandran, 2005; dos Santos et al., 2016; Lea, 2018) . These models can be either based on physical cons iderations (e.g. dislocation density evolution) or phenomenological. However, a common feature of these models is that hardening is described by an internal mnesic variable whose evolution is strain-rate dependent. Walley et al. (2000) presented a comparison between experimental result s of Taylor-impact tests on XM copper and numerical simulations based on two different constitutive models: the standard Zerilli-Armstrong (1987) viscoplastic relation and an extended version of t his model which takes strain-rate history effects into accoun t (Gould and Goldthorpe, 2000) . It was found that using the history-dependent model significantl y improves the accuracy of the numerical results. This shows that accounting for strain-rate history effects may be important to describe the response of structures under dynamic loading. A s mentioned at the beginning of this introduction, the predictions of plastic flow insta bilities are generally highly dependent on the constitutive model assumed, see for e.g. (Xue et al., 2008; Vadillo et al., 2012; Jacques, 2 020) . Therefore, one may wonder what is the role played b y strain-rate history effects in dynamic necking problems and what are the relative contribu tions of the instantaneous and work- hardening strain-rate sensitivities. These question s, that to the best of the authors’ knowledge have not been addressed in the literature, are the subject of the present study. Two different problems are considered: the impact tension test an d the ring expansion test. As mentioned 1 Note that the microstructure evolution may be also temperature dependent (Klepaczko and Chiem, 1986) . 5 previously, there is an important difference betwee n these two problems: owing to the radial symmetry of the ring expansion test, the deformatio n state in the specimen is nearly homogeneous until multiple necks are nucleated in t he circumference of the specimen, whereas the tension impact test is generally affected by wa ve propagation effects, leading to non- homogeneous strain and stress fields (Hu and Daehn, 1996) . The two problems are investigated using finite element simulations in which the mater ial is modelled with a constitutive relation that accounts for strain-rate history effects. More over, an analytical model (linear stability analysis) is developed for the ring expansion probl em. 2. A constitutive model with strain-rate history ef fects In this section, an elastic-viscoplastic model with strain-rate history effects is presented. As mentioned in the introduction, several models takin g strain-rate history effects into account have been proposed in the literature, see for e.g. (Estrin and Mecking, 1984; Klepaczko and Chiem, 1986; Follansbee and Kocks, 1988; Tong et al ., 1992; Bodner and Rubin, 1994; Molinari and Ravichandran, 2005; dos Santos et al., 2016, Lea, 2018) . However, we have developed a specific constitutive relation for the present study. This model is well-suited for parametric studies as the relative contribution of instantaneous strain-rate sensitivity and work- hardening strain-rate sensitivity to material respo nse is controlled by a single parameter. Moreover, the model has been specifically designed to facilitate the comparison of the necking behaviour of different materials having the same st ress-strain response at constant strain rate, but different strain-rate history effects. The constitutive framework relies on the classical assumption of the additive decomposition of the rate of deformation tensor and the  flow theory. The total rate of deformation tensor  is the sum of an elastic and a plastic part:  =  +  (1) The elastic rate of deformation  is related to the Cauchy stress tensor by a hypoelastic relation of the form  =  + tr   (2) 6 where  is an objective time derivative of the Cauchy stre ss tensor,  and  are the Young modulus and the Poisson ratio, respectively,  is the second order unit tensor and tr denotes the trace operator. The plastic rate of deformation is given by the  flow law:  =  (3) where is the deviatoric stress tensor, = − tr  , ! is the von Mises effective stress, ! ="  : , and  is the effective plastic strain rate, ="  :  . The yield function $ associated to the model is $ = ! − ! %&,  (4) where ! % is the flow stress of the material ! %(&, ) = ! *+ &. ,1 + 1 − . . /()0 . 1() (5) The functions /() and 1() are defined as /() = 2. 34 4567 (6) 1() = 89 (4) 9 (4):: 9(4); (7) In Eqs. (5)-(7), !*, 2, <, = and . are constants, while & is the hardening variable. The evolution of & is strain-rate dependent: >? >4%= ! =. @.̅B. ,1 + .. /()0 . 1() (8) with ̅ = C. dEF * being the effective plastic strain. Moreover, != is a reference stress and @ is the hardening exponent. In the following, we will a ssume the initial value of the hardening variable is equal to zero ( & = 0 for ̅ = 0 ). Eq. (8) describes the strain-rate sensitivity of work hardening. Indeed, it is apparent from Eq. (8) that the value of & at a given time (or strain level) depends on the preceding strain-rate evolution. Next, let us discuss the role played by two key fea tures of the constitutive model: the function /() and the parameter . (see Eqs. (5)-(8)). The function /() characterises the global (or overall) strain-rate sensitivity of the material. B y global, we mean the rate-sensitivity observed 7 from several tests performed at constant strain rat e. Indeed, when  is constant with & = 0 for ̅ = 0 , the flow stress can be expressed in the following form: ! % = ! *+ ! =. ̅B. ,1 + /()0 (9) The parameter . characterises strain-rate history effets. Its infl uence is illustrated in Fig. 2 , which displays the response to a plastic strain-rate jump (from ̅ to ̅) for several values of .. For . = 0 , the hardening variable & can be expressed as a function of the effective pl astic strain, & = ! =. ̅B. Therefore, the flow stress depends on the current values of the plastic strain and strain-rate (but not on the previous strain rat e history). The material does not have rate history effects or, in other words, only shows inst antaneous strain-rate sensitivity. Consequently, the stress after the strain-rate jump is the same as if the material had always been deformed at ̅ (Fig. 2 ). On the other hand, for . = 1 , the flow stress is no longer dependent on the current value of the plastic strain rate (the e xpression of the flow stress reduces to ! % = ! *+ &). The rate sensitivity of the material response is only due to the influence of strain rate on the evolution of its microstructure. The material has p ure work-hardening strain-rate sensitivity (or equivalently no instantaneous strain-rate sensitivi ty). In this case, a plastic strain-rate jump does not lead to a jump of the flow stress ( Fig. 2 ). Intermediate values of . correspond to materials with both instantaneous strain-rate sensitivity and work-hardening strain-rate sensitivity. A sudden increase in plastic strain rate causes a flo w stress jump, whose magnitude is proportional to 1 − . . The present elastic-viscoplastic model has been imp lemented in the finite element software ABAQUS/Explicit. For this purpose, a user material subroutine (VUMAT) has been developed. The numerical integration of the constitutive equat ions is based on the radial return algorithm, see for e.g. (Simo and Hughes, 1998) . 8 Fig. 2. Illustration of the role of the parameter . of the constitutive model presented in section 2. The solid lines show the stress-strain curves for const ant values of plastic strain-rate, ̅= 0.001 s and ̅= 1000 s. The dashed lines show the stress evolution caused by a plastic strain-rate jump from ̅ to ̅ at ̅ = 0.1 for several values of parameter ., see Eqs. (5), (8). Other parameters are given in table 1. Young modulus  124000 MPa Poisson ratio  0.34 Initial mass density J* 8960 kg/m 3 Initial yield stress !* 35 MPa Reference stress != 300 MPa Hardening exponent @ 0.3 Reference strain rate = 1000 s -1 Strain-rate sensitivity exponent < 0.05 Strain-rate sensitivity coefficient 2 0.7 Strain-rate history effect parameter . Ranging between 0 and 1 Table 2. Material parameters representative of copp er used in the numerical simulations and the linear stability analysis. 050 100 150 200 250 300 350 400 0 0.05 0.1 0.15 0.2 0.25 0.3 Effective plastic strain 1000 s-1 α=0 α=0.5 α=0.75 α=1 0.001 s-1 ̅= 1000 s ̅→ ̅ | . = 0 ̅→ ̅ | . = 0.5 ̅→ ̅ | . = 0.75 ̅→ ̅ | . = 1 ̅= 0.001 s Effective plastic strain, N%Flow stress, O(Mpa) 9 3. Round bar under impact tension 3.1 Finite element model Lagrangian finite element computations have been pe rformed using ABAQUS/Explicit (version 2017) to analyse the influence of strain-r ate history effects on neck formation in a linear specimen with circular cross-section subject ed to dynamic uniaxial tensile loading. The geometry and the dimensions of the specimen are sho wn in Fig. 3 . The finite element model developed in ABAQUS/Explicit is axisymmetric, and t he mesh consists of 14250 quadrilateral elements with bilinear interpolation functions, red uced integration and hourglass control (CAX4R in ABAQUS notation). The element size in the gauge of the specimen is equal to 40×30 µm 2 (the largest value being the element dimension in the radial direction). The specimen is initially at rest: P=Q, R, 0 = P SQ, R, 0 = 0, Q, R, 0 = T, &Q, R, 0 = 0 (10) where P= and PS are the radial and axial components of the velocit y vector, respectively, is the Cauchy stress tensor and & is the hardening variable used in the constitutive model (see section 2). The bottom surface of the specimen is clamped: P=Q, 0, U = P SQ, 0, U = 0 (11) On the top surface, a constant velocity is applied in the axial direction: PSQ, 0, U = P V7 , P=Q, 0, U = 0 (12) where PV7 is the magnitude of the applied velocity, hereinaf ter also referred to as the impact velocity. 10 Fig. 3. Geometry of the specimen considered in the simulations of impact tests presented in section 3. The figure also shows a close view of the mesh in t he gauge region (note that only the part on the rig ht side of the symmetry axis is modelled in ABAQUS). 3.2 Results Fig. 4 presents the deformed geometry of the specimen and the distribution of effective plastic strain obtained for an impact velocity of 20 m/s an d three values of the parameter . that characterises strain-rate history effects: . = 0 (pure instantaneous strain-rate sensitivity), . = 0.5 and . = 1 (pure work-hardening strain-rate sensitivity), see also Fig. 2 . For each value of ., the plastic strain distribution presented in Fig. 4 corresponds to a time such that the plastic strain at the centre of the neck is close to 1.5. T his value was chosen because the neck is fully formed (plastic deformation concentrates in the nec ked region) for the three values of .. Moreover, this value is close to the fracture strai n of annealed copper under uniaxial tension (French and Weinrich, 1975) . Hereinafter, the time for which the plastic strai n at the neck centre reaches 1.5 will be referred to as the localisation time. It is apparent from Fig. 4 that the elongation of the specimen at the localisation time depends on the value of . used in the 11 simulations. The largest elongation is obtained for the material having a pure instantaneous strain-rate sensitivity ( . = 0 ). When strain-rate history effects are taken into account ( . = 0.5 and . = 1 ), the localisation time and the corresponding elon gation of the specimen are significantly reduced. These results suggest that t he formation of the neck is affected by strain- rate history effects. To further illustrate this po int, Fig. 5 displays the time history of the plastic strain (a) and the plastic strain-rate (b) at the n eck centre for several values of . and for a rate- independent material (i.e. when the strain-rate sen sitivity coefficient is 2 = 0 , see Eq. (6)). In Fig. 5-a , we have also plotted the strain evolution calcula ted assuming that the strain in the gauge of the specimen is homogeneous (generally ref erred to as homogeneous deformation solution). The neck formation leads to a deviation of the strain from the homogenous deformation solution and a rapid increase in strain rate in the necked region. When the value . is decreased, the deviation of the plastic strain e volution from the homogenous solution is delayed. Moreover, notice that the results obtained with . = 1 are rather close to those obtained with the rate-independent material. Fig. 5-a also shows the instability strain associated to th e maximum force Considère criterion of the rate-depen dent materials 2. Recall that the Considère criterion characterises the onset of necking in rat e-independent materials under uniaxial tension. We observe that for the material with a pure work-h ardening strain-rate sensitivity ( . = 1 ) the necking instability develops rapidly when the Consi dère strain is met. This result is consistent with the force evolutions presented in Fig. 6 . With . = 1 , the force drops rapidly after it reaches the peak value. When the value of . is decreased (corresponding to greater contributio n of the instantaneous strain-rate sensitivity), the formati on of the neck and the resulting force drop are significantly delayed. As discussed in the introduction, for materials tha t exhibit strain-rate history effects, it is important to distinguish between the instantaneous strain-rate sensitivity and the work- hardening strain-rate sensitivity. From the results presented in Figs. 5 and 6, it appears that only the instantaneous strain-rate sensitivity leads to neck retardation effect. When the relative contribution of the work-hardening strain-rate sens itivity is increased, the neck retardation effect is significantly decreased (for the same glo bal strain-rate sensitivity). For the material with a pure work-hardening strain-rate sensitivity ( . = 1 ), the development of the necking instability is very close to that of the rate-indep endent material. 2 The Considère strain was computed using the condit ion > O >4%= ! % and considering the initial strain-rate. It is equal to 0.273, irrespectively of the value of .. 12 Fig. 4. Deformed geometry of the bar specimen and d istribution of plastic strain for several values of the strain-rate history effect parameter . and an impact velocity PV7 = 20 m/s . For each value of ., the time is chosen so that the maximum effective pl astic strain is ̅~1.5 . Other material parameters are given in table 1. (a) 13 (b) Fig. 5. Evolution of the (a) effective plastic stra in and (b) effective plastic strain rate at the cen tre of the neck for several values of the strain-rate hist ory effect parameter . and an impact velocity PV7 = 20 m/s . The results obtained for a rate-independent mater ial (K=0 in Eqs. (6)) are also presented. In Fig. 5-a, both the strain evolution obtained assumi ng a homogeneous deformation in the gauge region of the specimen and the Considère strain (for the r ate-dependent materials) are shown. Other material parameters are given in table 1. 00.2 0.4 0.6 0.8 11.2 1.4 1.6 0 0.1 0.2 0.3 0.4 0.5 Time (ms) Homogeneous deformation Considère strain Rate-Independent α=1 α=0.7 α=0.5 α=0 Vimp = 20 m/s Effective plastic strain, N% 010000 20000 30000 40000 0 0.1 0.2 0.3 0.4 0.5 Time (ms) Rate-Independent α=1 α=0.7 α=0.5 α=0 Vimp = 20 m/s Effective plastic strain rate, N%(s -1) 14 Fig. 6. Evolution of the axial force acting on the clamped surface of the specimen for several values of the strain-rate history effect parameter . and an impact velocity PV7 = 20 m/s . Other material parameters are given in table 1. Fig. 7-a presents the evolution of the effective stress wit h the plastic strain at the centre of the neck obtained for several values of the strain-rate history effect parameter . and an impact velocity of 20 m/s. For comparison purpose, a refer ence stress-strain curve corresponding to a constant plastic strain rate of 1666 s -1 (the nominal strain rate ( PV7 [\⁄) for the considered applied velocity) is also plotted. The Considère st rain of the materials is also displayed. Some differences between the curves obtained with the di fferent values of . are observed for large strains, higher than the Considère strain. At these strain levels, the strain is no longer homogeneous along the gauge length of the specimen. The necking development leads to a concentration of the strain and an increase in stra in rate inside the necked region. In fact, the differences between the stress-strain responses mea sured at the centre of the neck and this reference curve are related to the combined effect of this increase in strain rate caused by the formation of the neck and the strain-rate sensitivi ty of the different materials. The evolution of the effective plastic strain rate at the neck centr e as a function of the plastic strain is shown in Fig. 7-b . It is interesting to observe that for a given val ue of plastic strain, the strain rate is larger when the parameter . is increased (corresponding to more pronounced str ain-rate history effects) as the necking instability develops faster . However, the resulting stress increase is lower. When . is increased, the role of the work-hardening strai n-rate sensitivity becomes 0200 400 600 800 0 0.1 0.2 0.3 0.4 0.5 0.6 Force at the clamped end (N) Time (ms) α=0 α=0.5 α=0.7 α=1 Vimp = 20 m/s 15 predominant. Contrary to the instantaneous strain-r ate sensitivity, the work-hardening strain- rate sensitivity is a cumulative mechanism. Its inf luence on the mechanical response of the material does not only depends on the current value of the strain rate, but on the previous strain rate history. For this reason, a rapid increase in strain rate does not immediately leads to an increase in the stress level. This is probably the reason why the work-hardening strain-rate sensitivity does not significantly slow down the fo rmation of the neck (contrary to the instantaneous strain-rate sensitivity). It is interesting to draw a parallel between the re sults presented in Fig. 7 and the locking phenomenon reported in recent experimental studies (Mirone et al., 2019, 2020) . These authors observed from split Hopkinson bar tensile tests tha t even if the strain-rate strongly increases after the onset of necking, this steep increase doe s not cause a significant amplification of the flow stress. This locking phenomenon could be relat ed to strain-rate history effects. (a) (b) 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Effective plastic strain α=0 α=0.5 α=0.7 α=1 eps-dot=1333 s-1 Considère strain Vimp = 20 m/s α = 0 α = 0.5 α = 0.7 α = 1 ̅= 1333 s Considère strain Effective plastic strain, N%Flow stress, O(MPa) 16 Fig. 7. (a) Flow stress versus effective plastic at the centre of the neck for several values of the s train- rate history effect parameter .. For comparison purpose, the stress-strain curve f or a constant strain rate of 1666 s -1 (corresponding to the nominal strain rate for the considered impact velocity) and the Considère strain are also displayed. The beginning of the stress-strain curves (for stresses lower tha n 200 MPa) is not represented to make the differences between the curves more visible. (b) Effective plastic strain rate versus effective plastic strain relation observed at the centre of the neck for se veral values of strain-rate history effect parameter .. The applied velocity is PV7 = 20 m/s . Other material parameters are given in table 1. We now investigate the influence of the applied vel ocity. Figs. 8-a and 8-b present respectively the evolution of the normalised localisation time ( defined as Ûabc = P V7 Uabc [\⁄ with Uabc being the time for which the peak plastic strain reaches 1.5) and the energy dissipated at localisation as a function of the impact velocity for several va lues of strain-rate history effect parameter .. One notices that both quantities follow the same ev olution and two different regimes are apparent. When the applied velocity is lower than ~ 70 m/s, the normalised localisation time and the dissipated energy show little dependence on the applied velocity. Moreover, in this range of velocities, the localisation time and the energy dissipated at localisation are both significantly affected by the nature of strain-rate sensitivity e ffects. They are maximal for . = 0 (pure instantaneous rate sensivity) and decrease signific antly for larger values of .. This finding is consistent with the observations made previously th at showed that only the instantaneous strain- rate sensitivity delays necking formation. However, we observe that the differences between 010000 20000 30000 40000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Effective plastic strain α=1 α=0.7 α=0.5 α=0 Considère strain Vimp = 20 m/s Effective plastic strain, N%Effective plastic strain rate, N%(s -1) 17 the localisation time and energy obtained for the t hree values of . tend to decrease gradually with increasing impact velocity. When the velocity exceeds ~70 m/s, both the localisation time and the energy drop rapidly and become independent of the value of ., so that they are no longer influenced by the strain-rate history effect s. This velocity of ~70 m/s is the critical impact velocity (CIV) that according to Mann (1936) , Clark and Wood (1950) and Klepaczko (2005) , among others, determines the maximum energy absorbe d by the sample and the maximum sample ductility (which corresponds to the maximum normalised localisation time) in the dynamic tension test. The CIV also leads to a chang e of the mode of localisation, see Rusinek et al. (2005) , so that for PV7 ≤ CIV localisation is induced by a plastic flow instabil ity (leading to a gradual destabilisation of a nearly homogeneou s deformation state), while for PV7 > CIV , localisation is due to plastic wave trapping (von Karman and Duwez, 1950; Klepaczko, 2005) . In this case, localisation takes place near the imp acted end of the bar right after the application of the loading, the rest of the bar remaining virtu ally free of plastic deformation, see Fig. 9 . A very interesting observation from Fig. 8 is that the critical impact velocity is roughly the same for the three materials ( . = 0 , . = 0.5 and . = 1 ), meaning that it is largely insensitive to strain-rate history effects. This phenomenon is rel ated to the fact the propagation of plastic waves through the specimen is almost unaffected by strain-rate history effects (see infra). Fig. 8-c shows that the value of the rate history effect pa rameter . generally does not have a significant influence on the location of the main n eck. It has been shown in the literature that the location of necking in linear specimens under i mpact loading is controlled by the mechanisms of wave propagation, reflection and inte raction in the sample (Clark and Duwez, 1948; Nemes and Eftis, 1993; Osovki et al., 2013; V az-Romero et al., 2015) . From the results of the finite element computations, we have observe d that strain-rate history effects have little influence on the propagation of plastic waves in th e specimen (results not presented). This explains why neck location is almost unaffected by strain-rate history effects (or in other words by the relative contributions of the instantaneous and work-hardening strain-rate sensitivities). 18 (a) (b) (c) 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 20 40 60 80 100 120 α=0 α=0.5 α=1 CIV Impact velocity, hijk (m/s) Normalised localisation time, lmnop 0246810 0 20 40 60 80 100 120 Energy dissipated at localisation (J) α=0 α=0.5 α=1 CIV Impact velocity, hijk (m/s) 19 Fig. 8. Evolution of (a) the normalised localisatio n time, (b) the energy dissipated at localisation a nd (c) the normalised Lagrangian neck location as a fu nction of the impact velocity for several values of the strain-rate history effect parameter .. Other material parameters are given in table 1. Fig. 9. Deformed geometry of the bar specimen and d istribution of plastic strain for several values of the strain-rate history effect parameter . and an impact velocity PV7 = 100 </q . For each value of 00.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 Normalised main neck location α=0 α=0.5 α=1 Near impact end Near clamped end CIV Impact velocity, hijk (m/s) 20 ., the time is such that the maximum effective plast ic strain is ̅~1.5 . Other simulation parameters are given in table 1. 4. Ring expansion 4.1 Finite element model Three-dimensional numerical simulations of ring exp ansion have been performed using ABAQUS/Explicit (version 2017). The ring internal r adius, external radius and height are QVBF = 15.5 mm , Q rF = 16.5 mm , respectively, and & = 1 mm . Initially, the ring occupies the domain QVBF ≤ Q ≤ Q rF , 0 ≤ t ≤ 2u , 0 ≤ R ≤ & (Q, t and R being the cylindrical coordinates). A constant radial velocity P is applied on the internal surface of the specimen : P=QVBF , t, R, U = P (13) Other surfaces of the ring are traction free. The f ollowing initial velocity field is prescribed (Rodríguez-Martínez et al., 2013a) : P=Q, t, R, 0 = P , PvQ, t, R, 0 = 0, PSQ, t, R, 0 = 0 (14) The finite element mesh consists of 12550 hexahedra l elements with trilinear interpolation functions, reduced integration and hourglass contro l (C3D8R in ABAQUS notation). The mesh has a radial symmetry and the dimensions of the ele ments are close to 0.2×0.2×0.2 mm3 (Fig. 10). The development of necking instabilities is known t o be very sensitive to imperfections (Hutchinson and Neale, 1977; Han and Tvergaard, 199 5; Xavier et al., 2020; Marvi-Mashhadi and Rodríguez-Martínez, 2020) . In the present simulations, a random material imp erfection has been used to break the symmetry of the problem and favour necking localisation. The imperfection consists in a perturbation of the init ial yield stress. A similar approach was used by Bishop and Strack (2011) , Petit (2013) and El Maï (2014) . In each element, the perturbed yield stress is defined as ! w*xy = ! *(1 + z {y− 0.5 ) (15) where !* is the reference yield stress (see section 2 and t able 1), z is the imperfection magnitude, y is the element number and {y is random number of uniform probability density 21 ranging between 0 and 1. Note that for a given impe rfection amplitude, it is possible to generate several initial yield stress distributions (varying the seed of the random number generator). The influence of stochastic uncertainties is outside th e scope of the present paper. Therefore, all results presented in section 4.3 have been obtained with the same imperfection with z = 0.02 (Fig. 10 ). However, we have carried out some simulations fo r other realisations of the imperfection and found almost the same necking char acteristics (localisation time and strain, number of necks), see appendix. Fig. 10. Finite element mesh used in the ring expan sion simulations. The bottom picture illustrates th e distribution of initial yield stress (material impe rfection). 4.2 Linear stability analysis The stability analysis developed in this paper adap ts the one-dimensional formulation of Zhou et al. (2006) to the constitutive model derived in section 2. As in Zhou et al. (2006) , the elastic deformations are neglected, and the hydrostatic str esses that develop inside a necked section are modelled using Bridgman (1952) approximation. The ring expansion problem is approximated by a cylindrical bar with initial radi us |*, cross-section area }*= u|* (the area }* is taken identical to that of the ring), and lengt h [*= 2uQ % (Q% being the mean ring radius, 22 Q%==~ =  ) subjected to axial velocity P= 2uP (P being the radial expansion velocity, see Eq. (13)) and spatially-uniform effective strain ra te ̅. The equations of the problem are (see Zhou et al. (2006) and Vadillo et al. (2012) for further details): • Kinematic relations > > = ̅ 4% (16) ̅=>4% >F (17) • Momentum balance >(  %) > = J> >F (18) • Average axial stress ! = ! %  (19) • Bridgman (1952) correction factor = ,1 + 0 ln 1 +   (20) with  = |>; >r;= % 3 ,>4% > 0 − 2 ,>;4% >;0 where  is the Lagrangian coordinate,  is the Eulerian coordinate,  is the axial velocity, and | and } are the current radius and cross-sectional area, r espectively. The governing equations are completed with expressions (5) and (8), see section 2, which describe the evolution of the yield stress ! % and the hardening variable &, and with the following initial and boundary condi tions: , 0=  0, U= 0 [*, U= P  (21) The homogeneous (or fundamental) solution of the pr oblem is denoted by = (, ̅, ̅, ! , ! % , & , ). At some time U, a perturbation z , U= z  V ¡ FF ¢ is inserted into the fundamental solution, so that the perturbed solution is  =  + z , where z=z, z̅, z̅ , z! , z! % , z&, z £ is the perturbation amplitude, { is the perturbation wavenumber (also called perturbation mode) and ¤ is the perturbation growth rate (also called instantaneous instability index). Assuming that |z |≪||, the following linearized equations are obtained: • Kinematic relations z + y{ 4%¢z̅+ ̅z̅= 0 (22) ¤z̅ − z̅ = 0 (23) 23 • Momentum balance J¤z + y{! 4%¢z̅ − y{ 4%¢z! = 0 (24) • Average axial stress z! − z! % − ! %z = 0 (25) • Bridgman (1952) correction factor z − ¦{  4%¢z̅ = 0 (26) • Yield stress evolution z! % −> O >? § F¢z& −> O >4%§ F¢z̅= 0 (27) • Hardening variable evolution z& −>? >4%§ F¢z̅ = 0 (28) A solution for z is obtained when the determinant of the matrix of coefficients of equations (22)-(28) is zero. This condition leads to: ¤̂+  ¤̂ +  *= 0 (29) with = 1, = 1 + ¨ ©Λ[©{«, *= Λ[©{«,¬©Q©− ! m +­©; O® 0, ¤̂ =¡ 4%¢, {«= {. | *, Λ = 4%¢, [©="  ¯4%¢;;, ¬©=> O >? § F¢, ¨©=4%¢ > O >4%§ F¢, Q©= >? >4%§ F¢, ! m =  § F¢ and ! %©= O § F¢ Equation (29) gives, for a given time U, the value of the dimensionless instantaneous inst ability index ¤̂ as a function of the dimensionless perturbation mo de {«, being the latter related to the dimensionless perturbation wavelength as [*|*⁄ = ©. Fressengeas and Molinari (1994) proposed the cumulative instability index ° = C¤̂̅dEF * as indicator of the stability of the solution. The wavelength of the mode having the lar gest cumulative instability index ° is the critical cumulative wavelength, and the maximum val ue of ° is the critical cumulative instability index °c. The critical cumulative instability index will be used in section 4.3 to compare the results obtained with the linear stability analysis set out in this section, and the finite element model presented in section 4.1. 24 4.3 Results Fig. 11 shows linear stability analysis results for the evo lution of the critical cumulative instability index °c as a function of the effective strain ̅ for several values of the strain-rate history effect parameter α and two expansion velocities: P = 70 m/s and P = 300 m/s (corresponding to initial strain rates of 4375 s and 18750 s, respectively). The critical cumulative index increases monotonically with the e ffective strain because thinning of the specimen promotes the formation of necking instabil ities (Zaera et al., 2014) . Moreover, increasing the applied velocity shifts downwards th e °c− ̅ curves due to the increase of inertia effects, slowing down the growth of the per turbation and stabilising material behaviour (Fressengeas and Molinari 1994, Rodríguez-Martínez et al. 2013a). Notice that the differences between the results obtained for the three differen t values of α are reduced as the expansion velocity increases. Notice also that the greater th e strain-rate history effects (i.e. the greater the value of .), the greater the gap between the results obtained for 70 m/s and 300 m/s , and thus the effect of inertia slowing down the growth of th e perturbation. In contrast, the instantaneous strain-rate sensitivity, while reducing the role of inertia, leads to greater material stabilisation. For instance, the °c− ̅ curve for α = 0 and P = 70 m/s is below the curve obtained for . = 0.5 and P = 300 m/s . These results make apparent that the relative con tributions of inertia and rate dependence on material stabilisation depen d on the specific nature of the strain-rate sensitivity, with the instantaneous strain-rate sen sitivity playing a dominant role on the growth rate of the perturbation, and thus on the stability of the material and the formation of necking instabilities for the range of strain rates investi gated in this work. 25 Fig. 11. Linear stability analysis results: evoluti on of the critical cumulative instability index °c with the effective strain ̅ for several values of the strain-rate history effe ct parameter . and two expansion velocities: P = 70 </q and P = 300 </q (corresponding to initial strain rates of 4375 q and 18750 q, respectively). Other material parameters are give n in table 1. The greater stabilising role of instantaneous strai n-rate sensitivity, in comparison with the work- hardening strain-rate sensitivity, is further illus trated in Fig. 12, which shows finite element results for the normalised plastic strain rate ̅« (ratio between the current and the background effective plastic strain rate) along the normalised circumferential coordinate, ¬©= t 2π ⁄, for . = 0 , 0.5 and 1. The ̅«− ¬© curves show a series of peaks and valleys. The pea ks correspond to developing necks, while the valleys are the unlo ading portions of the ring surrounding the localising sections (the plastic strain rate is low er outside the necked sections). Recall from Fig. 5(b) that when a neck is formed the plastic strain rate increases drastically, leading to a heterogeneous distribution of plastic strains in th e specimen. Therefore, the presence of numerous peaks in the ̅«− ¬© curves indicates multiple plastic localisations. T his phenomenon is commonly observed in dynamic ring expansion expe riments (Altynova et al., 1996; Zhang and Ravi-Chandar, 2006, 2008) . Results for P = 70 m/s and P = 300 m/s are shown in Figs. 12(a) and 12(b) , respectively, with the corresponding loading time s being 130 µs and 53.7 µs (the associated background strains are 0.45 and 0.7 ). The loading times have been chosen so that the maximum value of the normalised plastic st rain rate for . = 1 is ̅«≈ 1.65 . The 0246810 0.2 0.3 0.4 0.5 0.6 0.7 α=1 - Ve=70 m/s α=0.5 - Ve=70 m/s α=0 - Ve=70 m/s α=1 - Ve=300 m/s α=0.5 - Ve=300 m/s α=0 - Ve=300 m/s α=1 –Ve=70 m/s α=0.5 –Ve=70 m/s α=0 –Ve=70 m/s α=1 –Ve=300 m/s α=0.5 –Ve=300 m/s α=0 –Ve=300 m/s LSA Effective strain, N%Critical cumulative instability index, ¶p 26 heterogeneity in the distribution of plastic strain rates is significantly more pronounced in the case of . = 1 , which shows that the material with pure work-hard ening strain-rate sensitivity is prone to plastic localisation, noticeably more t han the materials with (partial and full) instantaneous strain-rate sensitivity (which are mo re stable), in agreement with the predictions of the stability analysis (see Fig. 11 ). Notice also that the differences in the results obtained for . = 0 , 0.5 and 1 are reduced as the expansion velocity incr eases. Specifically, for P = 300 m/s , the maximum value of ̅« for α = 0 .5 and α =0 is 1.23 and 1.1, respectively, while in the case of P = 70 m/s the normalised plastic strain rate does not exceed 1.07 and 1.03 for these two values of α. This is also consistent with the predictions of t he stability analysis (see Fig. 11 ), which showed that, for a given effective plastic strain (i.e. for a given loading time), the differences in the critical cumulative instabil ity index (i.e. the differences in the stability of the material) for . = 0 , 0.5 and 1 decrease with the expansion velocity. The reason of this phenomenon is that at very high expansion velocity, the neck retardation effect caused by inertia is enhanced and, therefore, the stabilising effect of the instantaneous strain-rate sensitivity becomes comparatively less important. (a) (b) 0.2 0.4 0.6 0.8 11.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 Normalised circumferential coordinate α=1 α=0.5 α=0 FEM -Ve= 70 m/s - t=130 µs Normalised circumferential coordinate, ·®Normalised plastic strain rate, N%© 27 Fig. 12. Finite element results: normalised plastic strain rate ̅« versus normalised circumferential coordinate ¬© for several values of ., and different expansion velocities and loading ti mes: (a) P = 70 m/s and U = 130 µs , (b) P = 300 m/s and U = 53.7 µs . The normalised plastic strain rate is defined as ̅«= ̅̅¸⁄, where ̅¸ is the background strain rate (average value of ̅ along the ring circumference). The ̅«− ¬© curves show a series of peaks and valleys, so that the excursions of strain rate indicate multiple plastic localisations, and t he valleys correspond to the unloading portions of the ring surrounding the localising sections. The loadi ng times considered in Figs. 12-a and 12-b have been chosen in order to have nearly the same maximu m value of the normalised plastic strain rate. The corresponding values the background effective p lastic strain are (a) 0.45 and (b) 0.7. Other material parameters are given in table 1. The interplay between inertia and strain-rate histo ry effects is further illustrated in Fig. 13 , which shows the evolution of the necking strain ̅B c¹ with the expansion velocity P , for . = 0, 0.5 and 1. The finite element results are compared w ith the stability analysis predictions. In the numerical calculations, the necking strain corr esponds to the minimum plastic strain in the inner perimeter of the ring when elastic unloading first occurs, i.e. when the plastic strain rate vanishes at any point of the internal surface of th e specimen. The loading time corresponding to the necking strain is referred to as the necking time UB c¹ . Moreover, to determine the necking strain with the stability analysis, we have relied on the concept of effective instability introduced by Dudzinski and Molinari (1991) , and later adopted by El Maï et al. (2014) and N’souglo et al. (2020, 2021) , which states that localised necking occurs when t he critical 0.2 0.4 0.6 0.8 11.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 Normalised circumferential coordinate α=1 α=0.5 α=0 FEM -Ve= 300 m/s - t=53.7 µs Normalised circumferential coordinate, ·®Normalised plastic strain rate, N%© 28 cumulative instability index reaches a reference va lue °B c¹ . In the present work, °B c¹ is taken to be linearly dependent on the initial strain rate3 (i.e. on the expansion velocity). Identification of the parameters } and  of the function °B c¹ = } +  ∙ ̅  has been performed using the finite element results for the material with α = 0 (standard viscoplastic material with no strain-rat e history effects). Specifically, for the extrema of the range of strain rates considered, 312.5 s and 25000 s, we assume that °B c¹ corresponds to the value of °c calculated with the fundamental solution of the problem corresponding t o the necking strain. Thus, we have obtained the values } = 1.0025 and  = 1.518 ∙ 10 ¦ s, which are assumed to be valid also for the materials with α = 0.5 and α = 1 . The necking strain increases monotonically with the expansion velocity, due to the stabilizing effect of inertia, in agreement with the experiment al observations and the numerical calculations reported by Altynova et al. (1996) , Hu and Daehn (1996) , and Guduru et al. (2002) , among others. The linear stability analysis results are in qualitative agreement with the finite elements, with greater necking strains as the param eter α decreases (i.e. as the contribution of strain-rate history effects to material viscosity d ecreases). It becomes apparent that the analytical model captures the underlying mechanisms which determine the respective influence of instantaneous strain-rate sensitivity and work-h ardening strain-rate sensitivity on necking localisation. On the other hand, the quantitative d ifferences between numerical calculations and analytical predictions increase with .. In this regard, recall that °B c¹ was calibrated using the numerical results obtained for . = 0 , which explains the perfect matching between finit e element calculations and stability analysis obtaine d for this material for the extrema of the range of expansion velocities considered, and consequentl y the small differences observed for intermediates values of P . For . = 0.5 and . = 1 , the stability analysis predictions underestimate the finite element results for most v elocities tested, with a gap that reaches 18 % in the case of the material showing pure work-harde ning strain-rate sensitivity ( . = 1 ). Moreover, notice that for α = 1 and the lower expansion velocity P = 5 </q , the necking strain is only slightly greater than the instabilit y strain associated to the Considère criterion. This shows that, as in the case of the bar under im pact tension, the work-hardening strain-rate sensitivity does not yield neck retardation effect. In contrast, for . = 0 and 0.5, and the same expansion velocity of 5 m/s (for which the contribu tion of inertia is negligible), the necking strain is significantly higher, ~0.45 and ~0.55 , respectively, showing the material stabilisation 3 Dudzinski and Molinari (1991) , El Maï et al. (2014) and N’souglo et al. (2020, 2021) assumed that °B c¹ is constant. 29 induced by instantaneous strain-rate sensitivity at low strain rates. Even if there is not a perfect quantitative agreement between the finite element c omputations and the linear stability analysis, the important observation is that both approaches p redict a significant influence of strain-rate history effects on ductility. When the relative con tribution of the work-hardening strain-rate sensitivity increases, the necking strain decreases . Contrary to the case of the bar under impact tension investigated in section 3, there is no crit ical velocity for the ring expansion problem (Hu and Daehn, 1996) . Therefore, the influence of strain-rate history e ffects remains significant in the whole range of expansion velocities consider ed. However, the differences between the necking strains corresponding to the three values o f . tend to decrease for very high expansion velocities, as the stabilising effect of inertia be comes more important. Fig. 13. Evolution of the necking strain ̅B c¹ as a function of the expansion velocity P for several values of the strain-rate history effect parameter .. Comparison between finite element results and linear stability analysis predictions. The instabil ity strain associated to the Considère criterion is also shown. Other material parameters are given in table 1. 00.2 0.4 0.6 0.8 1 0 100 200 300 400 Expansion velocity (m/s) FEM - α=0 FEM - α=0.5 FEM - α=1 LSA - α=0 LSA - α=0.5 LSA - α=1.0 Considère strain Expansion velocity, h¼(m/s) Necking strain, N%½¼p¾ 30 5. Concluding remarks We have investigated the development of necking ins tabilities in materials exhibiting strain- rate history effects. Two different problems have b een considered: a bar under impact tension and a ring subjected to rapid radial expansion. For these two problems, finite element computations using a constitutive model that takes strain-rate history effects into account have been carried out. Moreover, for the ring expansion problem, the one-dimensional linear stability analysis introduced by (Zhou et al., 2006) has been extended to consider strain-rate history effects. The results presented in this paper show t hat strain-rate history effects may have a significant influence on neck formation. For materi als presenting strain-rate history effects, two different strain-rate sensitivities are operative: the instantaneous strain-rate sensitivity and the strain-rate sensitivity of work hardening. We have analysed the influence of these two kinds of strain-rate sensitivities on neck formation. The ma in finding is that only the instantaneous strain-rate sensitivity leads to neck retardation e ffect, when the neck formation is related to a plastic flow instability. In the case of the bar under impact tension, we hav e observed that the influence of strain-rate history effects vanishes when the applied velocity reaches the critical impact velocity (CIV). For supercritical velocities, necking is no longer caused by a plastic flow instability, but is due to wave trapping. The strain-rate history effects d o not significantly influence the propagation of plastic waves so that necking formation under su percritical loading conditions is largely insensitive to strain-rate history effects. Note th at no critical velocity exists for the expanding ring configuration and therefore strain-rate histor y effects on necking remain significant even for very high expansion velocities. An implication of the results presented in this pap er is that the experimental characterisation of the constitutive behaviour of materials should not only rely on tests at constant strain rate if the analysis of plastic flow instabilities is of concer n. Tests with strain-rate jumps should also be performed to determine if the material exhibits str ain-rate history effects. If this is the case, specific constitutive models taking these effects i nto account should be employed to analyse the development of necking instabilities. 31 Acknowledgements J.A.R.-M acknowledges that the research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme. Project PURPOSE, grant agreem ent 758056. Appendix: stochastic variations in the finite eleme nt ring expansion simulations The ring expansion simulations described in section 4.1 involve a random material imperfection. For a given defect amplitude, it is p ossible to create several realisations of the imperfection (i.e. several distributions of initial yield stress). Table A.1 provides numerical results corresponding to 5 different realisations o f an imperfection of amplitude z = 0.02 (see Eq. (15)) for an expansion velocity P = 70 m/s and several values of the strain-rate history effect parameter .. The necking time (time at which elastic unloading first occurs), the necking strain and the number of necks (as defined in secti on 4.3) are presented. For these three quantities, the five realisations yield very close results. The relative standard deviation is lower than 2% for the necking time and the necking strain , and about 5% for the number of necks. Realisation # Necking time (ms) Necking strain Number of necks .=0 .=0.5 .=1 .=0 .=0.5 .=1 .=0 .=0.5 .=1 1 0.234 0.191 0.135 0.622 0.537 0.443 23 25 26 2 0.232 0.188 0.134 0.612 0.53 0.44 26 28 29 3 0.224 0.183 0.129 0.587 0.508 0.424 26 26 27 4 0.234 0.189 0.134 0.614 0.528 0.439 28 28 29 5 0.229 0.186 0.133 0.602 0.527 0.437 25 25 29 Mean value 0.231 0.188 0.133 0.609 0.526 0.437 25.6 26.4 28 Relative standard deviation (%) 1.58 1.52 1.42 1.93 1.825 1.47 6.35 5.14 4.52 Table A.1. Necking time, necking strain and number of necks obtained with ring expansion simulations for several realisations of a random imperfection o f amplitude z = 0.02 (see Eq. (15)) and several 32 values of the strain-rate history effect parameter .. The expansion velocity is P = 70 m/s (corresponding to an initial strain rate of 4375 s).