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The dynamic effect of necking in Hopkinson bar tension tests

Giuseppe Mirone ⇑Dipartimento di Ingegneria Industriale e Meccanica, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

a r t i c l e i n f o

Article history:Received 13 April 2012Received in revised form 14 September2012Available online 2 December 2012

Keywords:Hopkinson barNeckingTrue strain rateTrue strainHardening

a b s t r a c t

The determination of the stress–strain curves from static and dynamic tension tests isaffected by the necking which locally modifies the stress distributions and the stress state,so that uniformity and uniaxiality of the stress state cease to apply and the load-areareduction measurements do not allow anymore to calculate the equivalent plastic strainand the equivalent stress at any material point within the resisting cross-section.

In case of dynamic tension tests, the necking also influences the effective strain rate,causing it to substantially differ from the nominal applied strain rate.

The effects of the necking on the strain rate and on the related material response areinvestigated here, and it is also checked whether or not a material-independent functionpreviously developed for correcting the post-necking true curves in quasi static tests,can also be used for correcting the stress–strain curves from Hopkinson bar testing andtransforming them into equivalent stress vs. equivalent strain curves at a given strain rate.

Finite elements analyses simulating experimental tests are compared to experimentaldata from the literature so that, from the validated numerical results, stress and strain dis-tributions in the interiors of the specimens can be investigated in detail.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Material testing at strain rates in the range (102

� 104) s�1 is usually conducted by way of the Split Hopkin-son Pressure Bar (SHPB) (Gilat et al., 2009; Staab and Gilat,1991; Lee and Kim, 2003; Meng and Li, 2003), based on thereflection and the transmission of stress–strain wavestravelling along two slender bars, elastically loaded, andinto the specimen, placed between the two bars and loadedbeyond its elastic range and up to failure.

In the SHPB, the compressive strain wave on the inputbar is generated by the impact of a projectile launchedby a gas-gun or by a similar device (Fig. 1(a), but variousother system configurations as in Fig. 1(b) have beendeveloped for inducing tensile and shear dynamic stressstates (Gilat et al., 2009; Staab and Gilat, 1991; Lee andKim, 2003; Kobayashi et al., 2008; Suhadi et al., 2009).Here the attention is focused on the split Hopkinson

tension bar (SHTB) which facilitates the attainment ofhigher strains and the failure of ductile metals.

In typical SHTBs, a cylindrical specimen with threadedends is mounted between the input and the output bars,and a trapezoidal or approximately rectangular tensilestrain pulse is generated at the free end of the input barby adopting various machine layouts ranging from tubularstrikers concentric to the bars (Verleysen and Degrieck,2004; Verleysen et al., 2005), to the input bar pre-tension-ing (Staab and Gilat, 1991), to compression setups with re-bound sleeves coaxial to the specimen (Sasso, 2005; Sassoet al., 2007).

The imposed tensile wave (incident wave) travelsthrough the input bar till the bar–specimen interface,where it is partially reflected and partially transmittedaccording to the impedances of specimen and bars. Theimposed tensile wave has an overall wavelength muchgreater than the specimen’s length, so that many succes-sive wave reflections dynamically load the specimen insmall discrete load steps, eventually up to failure. Thereflections of long waves within a short specimen ensure

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that deformation and strain rate within the specimen canbe assumed as almost uniform after a few reflectionsoccurred.

Input and output bars have a low D/L ratio for ensuringuniformity and uniaxiality of stress and, in turn, minimis-ing radial strains and wave dispersion. At the same time,the bars’ cross section cannot be too small for avoidingyielding.

The impulsive character of the loading causes typicalincident waves to have a large number of harmonics withdifferent frequencies, travelling into the bars at differentspeeds. This means that the wave distorts its shape as ittravels along the bars, and various ways are available forpreventing, reducing, correcting or compensating this effect(Verleysen et al., 2005; Meng and Li, 2003; Sasso, 2004;Ramirez and Rubio-Gonzalez, 2006; Sasso, et al., 2007).

A very well known phenomenon introducing significantapproximations in SHTB experiments is the necking, whoseeffect, investigated here, is usually neglected or only par-tially accounted for.

Experimental data available in literature (Johnson andCook, 1983; Noble et al., 1999; Ruggiero, 2005) are usedfor validating finite elements simulations of four SHTBtests (Balokhonov et al., 2009). The validated numericaldata are then used for calculating local distributions ofstresses and strains in meaningful zones within the speci-men, also during the post-necking phase.

The numerical simulations confirm that, in case of dy-namic straining, the necking introduces one more effectthan it does for quasi-static tests. In fact, while for statictests the only necking-induced modification of idealconditions is the gradually increasing triaxiality and non-uniformity of stress enforced all over the minimum crosssection, when it comes to high strain rate tests, the necklocalization also induces very sharp peaks of strain rate.

This result is in perfect agreement with recent experi-mental results (Gilat et al., 2009), and also constitutes aquantitative analysis of the strain rate amplification whichcan be expected in SHTB tests due to the necking.

The usual strain gauge recordings from Hopkinson Bartests allow to derive the current specimen load (averagedon the two specimen–bars interfaces), and the current totalspecimen elongation; these two variables are easily trans-formed in engineering stress and engineering strain,respectively, by simply referring to the dimensions of theundeformed specimen.

However, for identifying the elastoplastic response ofmetals, the hardening functions relating the equivalentvon Mises stress rEq to the equivalent plastic strain eEq

must be determined. Before the necking initiation, rEq

and eEq coincide with the true stress (rTrue, current ratioof load to cross section) and the true strain (eTrue

logarithmic strain based on the area reduction), both eas-ily obtainable from the engineering stress and straindata.

After the necking initiates, the true stress and the truestrain cannot be derived anymore from the engineeringstress–strain data, and also if other means are adoptedfor their derivation (e.g. neck diameter measurement byfast cameras and image analysis), rTrue and eTrue cannotaccurately represent the post-necking material curve; infact, the necking induces a gradually increasing triaxialityof the stress state and the consequent departure of the truestress from the Mises stress.

Ductile metals may be subjected to pronounced neckingduring more than 90% of their straining life so that, at fail-ure, the error in approximating the Mises stress with thetrue stress may be greater than 15% or 20%.

Then a correction is needed in the post necking strainrange for eliminating the effect of the necking-induced tri-axiality from the true stress (Bridgman, 1956; Alves andJones, 1999; Zhang et al., 1999; Ling, 1996; La Rosa et al.,2003; Mirone, 2004).

Such a correction was found to be material-indepen-dent for the quasi-static response of metals, and it allowsconverting the post-necking true stresses into accurateestimations of the von Mises stress all over the post-neck-ing strain range (Mirone, 2004, 2007; Mirone and Corallo,2010).

The suitability of this corrective function for determin-ing the hardening response of metals at high strain rates(SHTB tests) is also evaluated in the next sections.

2. Stress and strain calculations for SHTB

In SHTB experiments, the engineering axial strain eZ,strain rate _eZ and stress rZ can be calculated as followsas long as their distributions are uniform within the spec-imen volume:

eZ ¼ �2 � c0=L0 �Z t

0erdt ð1Þ

_ (a)

+ +

(b)

Fig. 1. Hopkinson bar configurations for compression (a) and tension (b) test.

G. Mirone / Mechanics of Materials 58 (2013) 84–96 85


rZ ¼ EB � AB=AS0 � et ð2Þ

_eZ ¼ �2 � c0=L0 � er ð3Þ

where (ei), (er) and (et) are the incident, the reflected andthe transmitted waves respectively, measured by straingauges on the surface of the bars; AS0 and L0 are the initialcross section and the initial conventional length of thespecimen, AB and EB are the cross section and the Youngmodulus of the bars, and c0 is the nominal sound speedwithin the bars.

The uniaxiality and uniformity of stress in smooth ten-sile bars make the axial stress rZ and strain eZ coincident tothe equivalent von Mises stress rEq and the equivalentplastic strain eEq, respectively; so rZ and eZ are capable ofidentifying the hardening law.

Tensile specimens usually have connecting threadedends with larger cross section than the gauge length, thusin this case the strain is far from uniformity and Eq. (1)only gives a volume-averaged strain of the entire specimenrather than the uniform strain on gauge length. So, if L0 isassumed equal to the entire specimen length then Eq. (3)returns a large underestimation of strain; if instead L0 is as-sumed equal to the gauge length, then an overestimationof the real strain is obtained; its magnitude is quite vari-able depending on the specimen shape and on the strainlevel reached.

However, if the connection lengths of the tensilespecimens are short and stiff, then Eqs. (1) and (2) with L0 -= gauge length, may represent a reasonable approximationof the engineering strain and stress.

Beyond small strains, another cause of approximationappears in typical SHTB equations (1)–(3) due to finitedeformation taking place; then the true stress and the truestrain, accounting for the currently deformed configurationof the specimen, are required for describing the materialhardening and can be calculated from the engineering dataas in Eqs. (4)–(6):

e0True ¼Z L

L0

dLL¼ LnðL=L0Þ ¼ Lnð1þ eZÞ ¼ eEq ð4Þ

volume conservation ensures that:

L=L0 ¼ AS0=AS ¼ ða0=aÞ2 ð5Þ

r0True ¼ F=AS ¼ rZ � AS0=AS ¼ rZ � L=L0 ¼ rZ � ð1þ eSÞ¼ rEq ð6Þ

where AS is the currently deformed minimum cross sectionof the specimen.

The uniformity of strain and the uniformity – uniaxial-ity of stress in the gauge length area, ensuring the validityof Eqs. (4)–(6), ceases to apply when necking initiates; be-yond necking, these equations poorly approximate theMises stress and the equivalent plastic strain.

In fact, after necking initiates, the strain localisesaround the minimum cross section and cannot be relatedanymore to the gauge length. Then the calculation of thetrue stress and true strain in the post-necking phase is pos-sible only if the current radius of the cross section a is

available, and the formulation (4)–(6) must be substitutedby the following:

eTrue ¼ LnðAS0=ASÞ ¼ 2 � Lnða0=aÞ ð7Þ

rTrue ¼ F=p � a2 ð8Þ

where a0 and a are respectively the initial and the currentradii of the minimum cross section.

Given the conceptual difference between the true curvedefined in (4)–(6) applying only before necking and thatdefined in (7) and (8) applying also beyond necking, theformer is named ‘‘nominal true curve’’ while the secondis the ‘‘effective true curve’’ or also simply ‘‘true curve’’ inthe next sections.

The true curve is the most common way of describingthe hardening of ductile metals but, also when it is deter-mined through Eqs. (7) and (8) and measurements of theevolving cross section, rTrue differs from rEq.

In fact, Eqs. (7) and (8) can only return the effectivecross-section averaged current values of axial stress (rTrue)and the axial plastic strain (eTrue): before necking initiationthe stress state of smooth tensile specimens is perfectlyuniaxial and uniform so the true stress is perfectly coinci-dent to the von Mises stress rEq; but, after necking initi-ates, both uniformity and uniaxiality of the stress statecease to apply on the minimum cross section of the speci-men and then rTrue substantially departs from rEq.

Various methods are proposed in the literature (Bridg-man, 1956; Alves and Jones, 1999; Zhang et al., 1999; Ling,1996; La Rosa et al., 2003; Mirone, 2004) for transformingthe true curve into an estimation of the von Mises curvebut, in the following sections, the MLR method (Mirone,2004) is adopted for checking its suitability to the case ofhigh strain rate tests.

This method consists of a material-independent correc-tive polynomial function which can be multiplied by thebest-fitting function of the experimental true stress–truestrain curve: the resulting function is a good estimate ofrEq(eEq).

The polynomial MLR reported in Eq. (9) was originallyderived as the fitting of many experimentally-validatednumerical results; it allows to determine the von Misesstress up to strains of 1.2 and more, after neckinginitiation;

MLRðeEq � eNÞ ¼ 1� 0:6058 � ðeEq � eNÞ2 þ 0:6317 � ðeEq � eNÞ3

� 0:2107 � ðeEq � eNÞ4

rEqðeEqÞ ¼ rEqðeEqÞ �MLRðeEq � eNÞð9Þ

The different ways discussed here for determining amaterial curve are always approximate, but while the engi-neering curve rZ(eZ) defined in (1) and (2) and the nominaltrue curve r0Trueðe0TrueÞ in (4)–(6) are rather inadequate todescribe the hardening of ductile metals, the effective truecurve rTrue(eTrue) in (7) and (8) allows rough but acceptableengineering predictions (overestimation of the real hard-ening stress up to 15–20%,) and the MLR estimation ofthe Mises stress in Eq. (9), gives an accuracy within 5%for quasi-static elastoplasticity.

For the characterization of metals at high strain rates,the best possible approximation of rEq(eEq) must be deter-

86 G. Mirone / Mechanics of Materials 58 (2013) 84–96


mined from experiments at different strain rates, so that atwo-variables function rEqðeEq; _eEqÞ can be identified.

The effects of the necking in inducing stress nonunifor-mity and triaxiality are investigated here with special re-gards to their relationship with the strain rate; thepossibility of using the MLR function for determining thedynamic material curves is also verified in the next sections.

3. Numerical analyses

Tensile tests with a SHTB are simulated here by usingthe FE commercial implicit code MSC-MARC.

The input and output steel bars (210 GPa Young modu-lus and 0.31 Poisson coefficient) have diameter Db = 9 mmand length Lb = 1100 mm.

Two different specimens are modeled having the samegauge length and total length (Ls = 8 mm and LT = 16 mmrespectively) and diameters Ds1 = 3 mm and Ds2 = 2.25 mm.Two M5 threaded shoulders are modeled for each specimen,protruding 2 mm out from the input–output bars and con-nected to the gauge length by a 3 mm filleting radius.

Two loading conditions are simulated by applying trape-zoidal tension waves with amplitudes of 785 MPa (durationof 2.1 � 10�4 s) and 1200 MPa (duration of 1.25 � 10�4 s).

The combination of two geometries and two incidentwaves results in a total of four simulated tests at as manystrain rates, named Sim#1 (D = 3 mm, ri = 785 MPa),Sim#2 (D = 2.5 mm, ri = 785 MPa), Sim#3 (D = 3 mm,ri = 1200 MPa), Sim#4 (D = 2.5 mm, ri = 1200 MPa).

The duration of the wave at 785 MPa is the maximumpossible still avoiding the overlap of incident and reflectedwaves at the midsection of the input bar, while the dura-tion of the wave at 1200 MPa is such that the time integraland the final engineering strain achieved are close to thoseof the 785 MPa wave. The rise time and the decreasingtime are equal to 25% and 20% of the total wave durations.

In Fig. 2 is reported a detail of the specimen and barsinterfaces, all meshed with axisymmetric – Quad 4 fullintegration elements having minimum dimension of2 � 1 mm2 for the bars, and 0.025 � 0.1 mm2 for the spec-imens. The MSC-MARC implicit solver uses the updatedLagrangian formulation with additive decomposition ofthe strain rate, to better describe large displacement and fi-nite deformation, within the framework of the von Misesyield criterion and the associated flow rule.

The boundary conditions include an axial symmetryconstraint (no radial displacement allowed to the nodeson the axis of the bars and of the specimens), a fixed-endconstraint (no axial displacement allowed to the nodes of

Fig. 2. Detail of the deformed mesh of the specimen and specimen/bars interfaces.

Table 1REMCO Iron – mechanical properties and Johnson–Cook parameters.

Material Mechanical properties Johnson–Cook parameters

Density [kg/m3] Specific heat [J/kg K] Melting temp. [K] A (MPa) B (MPa) n C m

REMCO Iron 7890 452 1811 175 380 0.32 0.06 0.55



the end section of the output bar), and a time-dependenttensile load applied to the nodes of the first section ofthe input bar, constituting the incident wave.

Given the triaxiality and the non-proportionality of thestrain path in the post-necking range, and also consideredthe path-dependent response of ductile metals at highstrains, the implicit formulation of FE analyses is used be-cause it is more accurate, although more time-consuming,than the explicit formulation.

The analysed time of 6 � 10�4 s is fractioned into vari-able amplitude steps ranging from 2 � 10�9 to 6 � 10�8 s.The simulation Sim#1 corresponds to experimental testsreported in (Noble et al., 1999; Ruggiero, 2005), so it isused for validating the entire set of analyses. The materialof the specimen is a ductile metal modelled through theJohnson–Cook constitutive function.

rEq ¼ ðAþ b � enÞ � ð1þ C � ln _e�Þð1� T�mÞ ð10Þ

The constitutive parameters from literature are given inTable 1.

The above parameters, originally derived in Johnsonand Cook (1983) for an ARMCO iron, were also used by No-ble et al. (1997) for a REMCO iron.

Despite the term m is assigned, the stress–temperaturedependence is actually turned off by setting conductivity,specific heat and emissivity = 0. This might introduceapproximations because the temperature-induced metalsoftening is neglected, but given the comparative characterof this study and also considering the moderate rise of adi-abatic temperature reported in this class of experiments,together with the usual sensitivity of flow stress on tem-perature (Kapoor and Nemat-Nasser, 1998; Nemat-Nasserand Guo, 2005), this approximation can be easily accepted.Temperature measurements were also performed by Nobleet al. (1999) on the material simulated here at comparablestrain rates, evidencing that ‘‘no significant change in thespecimen temperature following fracture was apparent’’.

4. Validation and interpretation of numerical results

The FE results are firstly validated by comparing theevolving area reduction from SIM#1 with its experimental

counterpart, available from Noble et al., who obtained itthrough high speed cameras and image analysis.

Fig. 3 shows a good agreement between experimentsand simulations, indicating that also the local stressesand strains available from FE are reasonably accurate andcan be used for the successive phases of the investigation.

Then the typical procedure for calculating the materialcurve, from the strain gauges recordings on the input andoutput bars, is applied here to the numerical results ofthe validated FE simulations.

The time histories of axial strain shown in Fig. 4 areread on the FE meshes at the midsections of the two bars.Then, (ei), (er) and (et), incident, reflected and transmittedwaves respectively, are obtained by splitting and translat-ing the strain histories at the bars midsections, for refer-ring to strain values originated at the specimen interfacesat the same instant (Fig. 5).

The reflected waves of Fig. 5, entered in Eq. (3) togetherwith the gauge length of the specimens, result in the nom-inal strain rates in Table 2 and in the time-history of theengineering strain rate of Fig. 6:

5. Analysis of local stress–strain histories and neckingeffect

Now different approximations of the material curvesare calculated from the validated FE data and plotted inFig. 7, for the four simulated tests.

Area Reduction Percentage

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 25 50 75 100 125 150 175 200

time [ms]

ExperimentalNumerical

Fig. 3. Cross-section reduction from experiments and from finite ele-ments analyses.

Elastic strain on bars

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 0.0001 0.0002 0.0003 0.0004 0.0005

time [s]

stra

in

SIM#1 input SIM#1 output SIM#2 inputSIM#2 outputSIM#3 inputSIM#3 outputSIM#4 inputSIM#4 output

Fig. 4. Elastic strain waves at the midsections of bars.

Elastic strain on bars

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0 0.00005 0.0001 0.00015 0.0002 0.00025

time [s]

stra

in

SIM#1 reflectSIM#1 transmSIM#2 reflectSIM#2 transmSIM#3 reflectSIM#3 transmSIM#4 reflectSIM#4 transm

Fig. 5. Reflected and transmitted strain waves.



While the engineering curve rEng(eEng), the nominaltrue curve r0Trueðe0TrueÞ and the effective true curve rTrue

(eTrue) are calculated by introducing FE results (strains,loads and resisting areas from nodes displacements) inEqs. (1)–(9), the von Mises stress rEq(eEq) at each givenstrain rate is a known function of the plastic strain accord-ing to the Johnson–Cook parameters in Table 1.

The first three curves of each graph in Fig. 7 (engineer-ing stress, nominal true stress and effective true stress) arereferred to the current resisting area of the specimen; in-stead, rEq is a local data referring to a single material point,and in Fig. 7 it is evaluated at the nodes on the neck centreand on the neck outer surface.

After the necking initiation, the strain and the strainrate are not uniform within the specimen, so the equiva-lent stress, which is strain-rate-sensitive, is expected toevolve along different curves at different material points.Instead the two curves of rEq(eEq) at the neck centre andat the neck outer surface are perfectly overlapped eachother, as if the strain rate gradients on the neck sectionwas having a negligible effect on rEq. This apparentlydisagrees with expectations and is better analysed below.

No failure criteria is implemented in the simulations sothe stress–strain curves in Fig. 7 end with unloading rampsafter the loading wave travels away from the specimen orthe strain gauges.

The engineering curves are reliable only in the elasticrange up to first yielding.

The nominal true curves, obtained by Eqs. (4)–(6), allowa good reconstruction of the equivalent stress-equivalentstrain response only until the necking initiates, and arefar from giving reasonable predictions already at strainsexceeding 0.6.

The effective true curves (also simply named truecurves), calculated from the evolving diameter of the spec-imens and from Eqs. (7) and (8), are much more accuratethan the nominal true curves but, at later stages, also theeffective true stress moderately departs from the Misesstress. However a rough estimation of the material

SIM#1

0

200

400

600

800

1000

1200

1400

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

Strain

Stress [MPa]

Eng. Stress

Nominal True stress

Effective True stress

Mises Stress @ neck outerradiusMises Stress @ neck center

SIM#2

0

200

400

600

800

1000

1200

1400

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

Strain

Stress [MPa]

Eng. Stress

Nominal True stress


Mises Stress @ neckcenterMises Stress @ neck outerradius

SIM#3

0

200

400

600

800

1000

1200

1400

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

Strain

Stress [MPa]

Eng. Stress

Nominal True stress


Mises Stress @ neck center

Mises Stress @ neck outerradius

SIM#4

0

200

400

600

800

1000

1200

1400

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

Strain

Stress [MPa]

Eng. Stress

Nominal True stress


Mises Stress @ neck center

Mises Stress @ neck outerradius

Fig. 7. Stress–strain curves at three approximation levels.

Table 2Nominal strain rates related to the gauge length of specimens.

ri = 785 MPa ri = 1200 MPa

D = 3 mm D = 2.5 mm D = 3 mm D = 2.5 mm

SIM#1 SIM#2 SIM#3 SIM#44400 s�1 4600 s�1 7000 s�1 7200 s�1

0

1000

2000

3000

4000

5000

6000

7000

8000

0 0.2 0.4 0.6 0.8

Eng. strain

Strain rate (Eng.)

'SIM#1

'SIM#2

'SIM#3'SIM#4

Fig. 6. Time history of the engineering strain rate.



response is provided by rTrue and eTrue, as them approxi-mate rEq and eEq with an error of about 15–25% at largerstrains modelled.

This first result then confirms that, if large post-neckingstrains are expected in SHTB tests before failure, highspeed cameras or other means for monitoring the crosssection (Noble et al., 1997; Kajberg and Wikman, 2007;Gilat et al., 2009; Li and Ramesh, 2007) are essential forobtaining reasonable stress–strain characterization.

The true curves extracted from all the simulations arecollected together in Fig. 8, exhibiting negligible differ-ences all over the strain history, as if the strain rates ofFig. 6 are not playing any role, as it was also found abovefor the local values of rEq in Fig. 7.

The descending branches of the four curves in Fig. 8 dif-fer each other only due to the different strains attained be-fore unloading, but do not reflect any strain rate effect.

Also the nominal true curves, summarized in Fig. 9 andmeaningful only up to the onset of necking, indicate that amoderate strain rate effect barely occurs at strains within0.2.

This is in apparent contrast with the Johnson–Cook con-stant C = 0.06, which suggest that the Remco iron is moresensitive to the strain rate than many other alloys, as alsoconfirmed by the Johnson–Cook parameters compared inTable 3.

Then, for further investigating about the sensitivity ofthe material to the strain rate, _e is also calculated by refer-ring respectively to the time rate of the nominal true strain�eTrue (data of Fig. 4 put into Eq. (4)), to the time rate of theeffective true strain eTrue (time history of the area reductionfrom the FE mesh put into Eq. (7)), and to the time rate ofthe local strains eEq from FE readings at nodes on the neckcentre (r = 0) and at the neck outer surface (r = a).

These strain rates are compared in Fig. 10 together withthe engineering strain rate, each one plotted against thecorresponding strain.

Fig. 10 shows that the effective strain rate in the criticalareas of the specimens is much greater than the nominalstrain rate, because of the amplification effect of necking.

The engineering strain rate and the nominal true strainrate, both based on the reflected wave on the input bar andcalculated by usual SHTB formulae (3) and (4), represent alarge underestimation of the strain rate really experiencedby the material at the neck section.

The reason of this difference is that the engineeringstrain rate only represents the current volume-average ofthe strain rate distribution all over the specimen; this dis-tribution, initially uniform, exhibits an increasing non-uni-formity as the necking initiates; the peak of the strain ratedistribution is located at the neck centre and its magnitudeis 6–8 times greater than the volume-averaged value.

Similar considerations apply to the strains reported onthe horizontal axes of Fig. 10.

Also the strain rate distribution on the neck sectionexhibits a certain non-uniformity, as evidenced by the dif-ferent strain rate evolutions at the neck centre and at theouter neck surface.

The effective true strain rate (true SR from specimendiameter in Fig. 10) represents the average of the strainrate distribution over the current neck section. It stillunderestimates the strain rate peak of 15–25%, while theeffective true strain in the abscissa is lower than the max-imum local strain at the neck centre of about 10–15%.

These results are in agreement with the findings of Ericeet al. (2010), Gilat et al. (2009), and Verleysen et al. (2005).

In principle, the material hardening should be deter-mined at the neck centre where the values of strain, strainrate and stress reach greater levels than elsewhere withinthe specimen.

But, given that local stress and strains cannot be mea-sured by experiments at internal material points within

EFFECTIVE TRUE STRESS (strain from specimens)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

True strain

Stre

ss [M

Pa]

SIM#1SIM#2SIM#3SIM#4

Fig. 8. Comparison of true stress–true strain curves at different nominalstrain rates.

NOMINAL TRUE STRESS(strain from bars)

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5

True strain

Stre

ss [M

Pa]

SIM#1SIM#2SIM#3SIM#4

Fig. 9. Comparison of nominal true stress–true strain curves at differentstrain rates.

Table 3Strain rate sensitivity of different metals from literature.

Metal Value of coefficientC

Ref.

Steel AISI 1018 CR 0.045 Sasso et al. (2008)Pure Copper 0.025 Johnson and Cook

(1983)Aluminium 7039 0.010 Johnson and Cook

(1983)Steel 4340 0.014 Johnson and Cook

(1983)Steel Weldox 460

E0.012 Børvik et al. (2001)



the specimen, the neck-averaged values of the equivalentplastic strain, of its time rate, and of the Mises stress arethe best possible evaluation of the hardening variableswhich can be derived from experiments.

The true strain eTrue from Eq. (4) is a very good estima-tion of the current neck-averaged equivalent plastic strain,but the true stress rTrue largely differs from the currentneck-averaged Mises stress.

The most known model for converting the effective truestress into the Mises equivalent stress is due to Bridgman(Bridgman, 1956; Alves and Jones, 1999; La Rosa et al.,2003), but the suitability of the MLR model (Mirone,2004) to high strain rate histories is investigated in the fol-lowing sections.

6. Effect of the specimen size/shape

The true strain rate curves, summarised in Fig. 11 for allthe simulations, show how the combination of specimensslenderness and incident wave amplitude may affect thepeak of strain rate and the maximum strain achieved, pro-vided that no specimen failure occurs.

At the first stages of loading, the strain rate apparentlystabilizes at about 4500 and 7000 s�1, corresponding to thenominal engineering values predicted by the standardSHTB theory.

But after the necking initiates at strain levels of about0.18, the strain and the strain rate in the necked region un-dergo a progressive amplification and quickly diverge fromtheir nominal values.

As expected, the higher amplitude of the incident wave(1200 MPa) and the thinner specimen (SIM#4) cause thehighest peak of strain rate close to 60000 s�1, about ninetimes greater than the nominal value of 7000 s�1, and a fi-nal strain of 2.1, three times greater than the final engi-neering strain of 0.7.

The same incident wave passing through a thicker spec-imen (SIM#3) causes a lower strain rate peak (about 5times greater than the nominal strain rate) and a lower fi-nal strain (just more than twice the maximum nominalstrain).

According to classical SHTB theory and Eq. (1), the max-imum strain imposed to the specimen depends on theduration of the incident wave so it should be the same

d 3mm d 2.25mmW

ave

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MP

a SIM#1

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Eng. S.R. from barsTrue S.R. from barsLocal S.R. @ neck centerLocal S.R. @ neck outer radiusTrue S.R. from specimen

SIM#2

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Eng. SR from barsTrue SR from barsLocal SR @ neck centerLocal SR @ neck outer radiusTrue SR from specimen

Wav

e 12

00 M

Pa

SIM#3

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SIM#4

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20000

30000

40000

50000

60000

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Strain

Strain rate [s-1]


Fig. 10. Different approximations of strain rates.

0

10000

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0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

True Strain rate

SIM#1

SIM#2

SIM#3

SIM#4

True strain

Fig. 11. True strain rates calculated from the current diameters.



for SIM#3 and SIM#4 (see Fig. 6), but the occurrence ofnecking changes the ideal response by magnifying thestrain, and this magnification effect is more relevant forthe thinner specimen (SIM#4) than it is for the thickerone (SIM#3).

The above behaviour indicates that the strain rate reallyacting on the neck section of a round bar is the sum of twocontributions, _e ¼ _eL þ _eR; _eL is the elongation rate of theentire specimen, it corresponds to the nominal strain rateand accounts for the build up of the reflecting strain wavesat the ends of the specimen; _eR, is due to the radial contrac-tion rate which localises the strain around the thin slice ofmetal of the neck section, and it progressively amplifies thetotal strain in that area as the necking evolves.

These considerations explain the strain rates curves inFig. 11, which would remain unclear if a viewpoint onlybased on the engineering strain was maintained.

The very large strain rates occurring at the semi-localscale of the neck section (intermediate between the localscale of single material points and the global scale of theentire specimen), also explain why the stress–strain curvesof the four simulated tests (Figs. 8 and 9), exhibit so smalldifferences beyond necking, despite the strain rates are sodifferent each other.

In fact, at the large strain rates occurring over the necksection, the strain rate dependence modelled through thematerial parameters of Table 1 is almost completely satu-rated, and beyond _e ¼ 10000 s�1 no real strain rate sensi-tivity can be expected by the modelled material, asvisible in Fig. 12.

Fig. 12 shows the dependence of rEq on the strain rate,at six fixed levels of strain. The strain rate sensitivity of theREMCO iron is significant only up to strain rates of about5000 s�1, but becomes barely detectable for strain ratesbetween 5000 and 10000 s�1, and completely negligiblebeyond 10000 s�1.

Then it is perfectly reasonable that moderate differ-ences can be only found in the material response fromthe four simulations when the nominal true curves areconsidered (Fig. 9) at strains lower than 0.2–0.3. In fact,up to these strain levels the necking effect is just initiatedand the strain rates, still close to their nominal values of5000 s�1 and 7000 s�1, are below the saturation limit of10000 s�1.

As the necking localization takes place, the real strainrate progressively jumps beyond 10000 s�1 and the strainrate sensitivity is completely saturated, so the material re-sponses from the four simulations become identical eachother.

7. Stress and strain distributions over the neck section

The effect of necking has been analysed above with re-gard to the magnification it causes to the strain and to thestrain rate during dynamic SHTB tests, but also the neck-ing-induced perturbations of the stress state should beinvestigated for assessing how them affect the elastoplas-tic characterization of metals.

In fact, after the necking initiates, non-uniform distribu-tions of radial and hoop stresses, rr and rh, appear on theneck section, and also the axial stress rZ loses its unifor-mity within the specimen cross section.

So, the necking causes the loss of stress uniformity andthe occurrence of increasing stress triaxiality over the necksection; these, in turn, cause the most accurate experimen-tal stress, namely the effective true stress rTrue, to progres-sively differ from the equivalent stress rEq; the ratio rEq/rTrue starts from unity before necking and progressivelydecreases as the necking becomes more and morepronounced.

Given that the characterization of ductile materials istypically expressed by their hardening law rEq(eEq), amethod for transforming the experimental stress rTrue inan estimation of the flow stress rEq is necessary.

Frequently in the literature the flow stress is approxi-mated by rTrue, but this leads to a poorly accurate estima-tion of the hardening stress.

Various methods (Bridgman, 1956; Ling, 1996; Zhanget al., 1999; Mirone, 2004) are available in the literaturefor transforming the experimental rTrue into the estimatedrEq at quasi-static strain rates; so, the check of their suit-ability to high strain rate tests, not yet attempted in the lit-erature in the knowledge of the author, is of considerableinterest.

Only in Erice et al. (2010) evidence is reported that thenecking correction method by Mirone (2004) was used forsimulating experiments at ballistic strain rates with rea-sonable success.

In this work, the Johnson–Cook parameters identifyingthe function rEqðeEq; _eEqÞ for the REMCO iron are alreadyknown, so, here a reverse procedure involving the vali-dated FE simulations is carried out, for assessing the atti-tude of the MLR method at deriving rEq in case of highstrain rates.

The usual experimental data for obtaining rTrue andeTrue (current load F and neck area A) are read on thedeforming FE mesh, which also allows more local readingslike stress and strain on single nodes within the neckedcross section.

Fig. 13 reports the distributions of rEq, rZ, and rTrue un-der well developed necking conditions (true strain close to1.0), for all the simulations ran.

As for quasi-static necking, the current distributions ofthe Mises stress are almost uniform while the axial stressexhibits significant gradients over the neck sections. The

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Strain rate [s-1]

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Eq. strain 0.1Eq. strain 0.2Eq. strain 0.4Eq. strain 0.6Eq. strain 1Eq. strain 1.5

Fig. 12. Sensitivity of the REMCO iron at high strain rates.



Mises stress at the neck centre is about 10% lower than theeffective true stress, which is uniform by definition since itis the current neck-averaged axial stress.

The above differences are barely detectable just afternecking initiation, but become much more pronouncedthan those in Fig. 13 as the straining proceeds up to failure.

In principle, rEq and eEq identifying the material harden-ing at a given strain rate should be measured locally at theneck centre, which is impossible; but, in Mirone (2004,2007), it has been demonstrated that, thanks to the mildgradients usually occurring to rEq and eEq over the necksections, fairly simple and accurate material characteriza-tions can be achieved by associating each other the currentsection-averaged values of rEq and of eEq, the latter beingrepresented very well by eTrue.

Then the problem is how to estimate the current neck-averaged rEq starting from the usual measurements ofrTrue.

The simplest and most accurate tool actually availablefor a direct estimation of the Mises stress after necking isthe MLR polynomial (Mirone, 2004) which, independentlyfrom the material considered, expresses the ratio rEq_Avg/rTrue:

rEq Avg

rTrue¼MLRðeEq � eNÞ

¼ 1� 0:6058 � ðeEq � eNÞ2 þ 0:6317ðeEq

� eNÞ3 � 0:2107ðeEq � eNÞ4 ð11Þ

where rEq_Avg is the current Mises stress averaged over thecurrent cross section and eN is the Considère strain, atwhich the necking initiates.

Given the spatial variability of the stresses, their neck-averaged values are intended as their current integralsover the neck section AS, divided by the whole current necksection.

rEq Avg

rTrue¼

1AS

RAS

rEqðe; _eÞdA1

AS

RAS

rZðe; _eÞdA¼

1AS

RAS

rEqðe; _eÞdAF

AS

¼R

ASrEqðe; _eÞdA

F¼MLRðeEq � eNÞ ð12Þ

with F being the current load. The usual moderate gradi-ents of the equivalent stress and strain make the section-averaged hardening law rEq_Avg (eTrue) very close to the lo-cal function rEq (eEq); so the latter function (not experi-mentally measurable) can be approximated by justmultiplying the effective true stress from experiments,times the MLR polynomial:

rEqðeEq; eNÞ ¼ rTrueðeTrueÞ �MLRðeEq � eNÞ ð13Þ

The above formulation expresses a phenomenologicalmodel of the necking which works finely under quasi-sta-tic loading, but the suitability of this model to high strainrate loading is checked here, with reference to the fourcombinations of specimens and strain rates previouslysimulated.

In this case, the material hardening response is alreadyknown and is modelled through the Johnson–Cook param-eters; so, the check of the validity of Eqs. (11)–(13) at highstrain rates is done below by calculating the ratio rEqAvg/rTrue from the simulations and by comparing it to theMLR function.

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r/a


Axial stress

Mises stress

Fig. 13. Stress distributions along the radius of the necked cross section.



The stress distributions rEq and rZ are read at the nodesalong the neck radius, at different strain levels, from the fi-nite elements simulations; then their section-averagedvalues are calculated at each strain level according to thesecond term of Eq. (12), and the evolving ratio rEq_Avg/rTrue

is compared to the MLR function in Fig. 14.The ratio rEq_Avg/rTrue exhibits a significant discrepancy

from the MLR function, continuously increasing all overthe post-necking strain range. This means that the localstrain rate on the nodes where the stress readings aredone, affects the neck-averaged Mises stress and the truestress in different ways.

Then, instead of rEq_Avg/rTrue, a slightly different ratioequivalent stress/axial stress is investigated below, wherethe strain rate possibly affects both terms in the same way;

The equivalent stress identified as rEq_JC_nom is calcu-lated from the Johnson–Cook law, by referring to the cur-rent true strain and to the nominal, elongation-promotedstrain rate _eL ; in this way, the effect of the local, radial con-traction-promoted strain rate _eR on the flow stress is not in-cluded. The ratio rEq_JC_nom/rTrue is then calculated andcompared to the MLR polynomial in Fig. 15.

The ratio rEq_JC_nom/rTrue from the four dynamic testssimulated on Remco iron is approximated very well bythe MLR function, which was derived from static tensiontests of a wide variety of metals including aluminium al-loys 20XX and 70XX, pure copper, mild steels, low andmedium carbon steels, spheroidal steels, stainless steelsand high strength steels.

This means that the nominal, elongation-based strainrate _eL , amplifies the Mises stress and the true stress inthe same way, so that their ratio is identical to that ofthe quasi-static case and can be correctly described bythe MLR polynomial.

The explanation proposed for the above finding is thatrTrue includes the load F as a global variable dependingon the entire cylindrical length of the specimen, and thenecked cross section AS as a semi-local strain variable,referring to an intermediate scale between that of the sin-gle nodes and that of the overall cylindrical length of thespecimen.

Then, also the variable based on the equivalent stressshould be evaluated on a global/semi-local basis for beingmeaningfully related to the true stress.

So, rEq_Avg is not appropriate because it expresses thefully local flow stress rEq integrated over the semi-localneck section AS (see Fig. 14), while rEq_JC_nom is much moresuitable for our purpose as it is based on the semi-localtrue strain, Ln(AS0/AS), and on the global, elongation-based,nominal strain rate _eL (see Fig. 15).

If the current load was influenced by the total effectivestrain rate _e ¼ _eL þ _eR, similarly to rEq_Avg, then the ratiorEq_Avg/rTrue was expected to be reproducible by the MLRfunction; instead the latter function only applies to the ra-tio rEq_JC_nom/rTrue, indicating that the total load F only de-pends on the elongational part of the strain rate _eL and isnot sensitive to the contraction-induced strain rate _eR, asit is for rEq_JC_nom.

Furthermore, it is worth noting that the axial stress isthe sum of a deviatoric stress sZ and a hydrostatic stressrH:

rZ ¼ sZ þ rH ð14Þ

so also the axial load F pulling the specimen, in the post-necking phase, includes a hydrostatic component with anincreasing weight over the deviatoric component.

Given that the material response complies with theclassical von Mises plasticity, it is apparently unclearwhy the deviatoric component of the load seems unaf-fected by _eR and why its hydrostatic component seems af-fected by _eL.

The reason is probably that the necking acts like a nat-ural notch with an evolving shape; the severity of the neckand the related notch factor, expressing how the equiva-lent stress departs from the axial stress, depend on the en-tire neck shape extending from the minimum neck sectionup to the specimen zones unaffected by the necking.

So, the local contraction-related strain rate peak _eR actsover a too small volume of the specimen which cannot berepresentative of the complete necking shape nor of itsnotch effect, and then _eR cannot influence the overall ten-sile load nor its area-normalised value, rTrue.

Instead, the nominal elongation-based strain rate _eL in-cludes the contribution of the entire necked volume to thetangent stiffness of the specimen, then it reflects the com-plete evolving notch factor and can affect the load in bothits deviatoric and hydrostatic components.

σ σ

0

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Post-necking strain

σEq/σTrue

MLR POLYSIM1 - Avg Mises stress from FE SIM2 - Avg Mises stress from FE SIM3 - Avg Mises stress from FE SIM4 - Avg Mises stress from FE

Fig. 14. Effect of necking-induced triaxiality on dynamic tests.

σ σ

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σEq/σTrue

MLR POLYSIM1 - Mises from J-Cook @ nominal S.R.SIM2 - Mises from J-Cook @ nominal S.R.SIM3 - Mises from J-Cook @ nominal S.R.SIM4 - Mises from J-Cook @ nominal S.R.

Fig. 15. Effect of necking-induced triaxiality on dynamic tests.



Summarizing, the MLR function is capable of returningthe material hardening response at the nominal strain ratewithin an error level of about 5%, but it does not return thelocal instantaneous peak of equivalent stress on the mini-mum cross section, promoted by the local peak of thestrain rate _eR.

Then, it is possible to update Eq. (13) into the followingEq. (15) which formally includes the nominal strain rate _eL

within its variables.

rEqðeEq; eN; _eLÞ ¼ rTrueðeEq; _eLÞ �MLRðeEq � eNÞ ð15Þ

A thin slice of the specimen in the neck area undergoeslarger strain rates than the nominal one and, consequently,larger equivalent stresses than those calculated by Eq. (15).However, the material curve found by way of the MLRfunction is very accurate in expressing the dynamic hard-ening of material points on the minimum cross section,supposed to be subjected to the nominal strain rate, whichis the target of usual SHTB characterization.

8. Conclusions

The SHTB testing procedure for smooth round speci-mens has been analysed here, for identifying the approxi-mations intrinsic in the postprocessing of experimentaldata; the reasons of the approximations, their magnitude,and ways for improving the stress–strain–strain rate char-acterization are also discussed within the paper.

The strain rate locally occurring over the minimumcross section is known to diverge from its nominal valuedue to the necking occurrence, but here a quantitativeevaluation of the necking-induced strain rate amplificationis carried out; the magnifying factor is found to be in therange 5–9 for the tests simulated here.

The local effective strain rate has an elongation-basedcomponent and a necking-promoted radial contractilecomponent; the latter strain rate contribution may be seenas an amplification of the former one, depending on thepost-necking strain imposed.

While the elongation-based strain rate affects bothdeviatoric and hydrostatic components of the axial load,the necking-induced amplification of the strain rate onlyinfluences locally the deviatoric stresses and the Misesstress, leaving almost uninfluenced the current load andthe section-averaged axial stress;

The MLR function, capable of translating the experi-mental true stress into an estimation of the necking-af-fected equivalent stress during quasi-static tension tests,is also found to apply to dynamic SHTB tests. Althoughthe strain rate during SHTB tests locally goes beyond itsnominal magnitude, the hardening characterization basedon the MLR function only extends up to the nominal strainrate, with an accuracy within 5%.

Further investigation is deserved by the subject, mainlybased on SHTB experiments with different materials andfast image analysis devices, but the MLR function is foundto be a promising option for stress–strain characterizationof ductile materials also at high strain rates.

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author's personal copy the dynamic effect of necking in hopkinson bar tension tests

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