Material characterization at high strain rate by Hopkinson bar tests and finite element optimization

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Materials Science and Engineering A 487 (2008) 289–300

Material characterization at high strain rate by Hopkinson bartests and finite element optimization

M. Sasso a,∗, G. Newaz b, D. Amodio a

a Mechanics Department, Polytechnic University of Marche, via Brecce Bianche, 60131 Ancona, Italyb Mechanical Engineering Department, Wayne State University, Engineering Bldg., 48202 Detroit, MI, USA

Received 16 May 2007; received in revised form 4 October 2007; accepted 10 October 2007

bstract

In the present work, dynamic tests have been performed on AISI 1018 CR steel specimens by means of a split Hopkinson pressure bar (SHPB). Thetandard SHPB arrangement has been modified in order to allow running tensile tests avoiding spurious and misleading effects due to wave dispersion,pecimen inertia and mechanical impedance mismatch in the clamping region. However, engineering stress–strain curves obtained from experimentalests are far from representing true material properties because of several phenomena that must be taken into account: the strain rate is not constanturing the test, the specimen undergoes remarkable necking, so stress and strain distributions are largely non-uniform, and the temperature increases

ecause of plastic work. Experimental data have been post-processed using a finite element-based optimization procedure where the specimenynamic deformation is reproduced. Optimal sets of material constants for different constitutive models (Johnson–Cook, Zerilli–Armstrong andthers) have been computed by fitting, in a least mean square sense, the numerical and experimental load–displacement curves.

2007 Elsevier B.V. All rights reserved.

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eywords: Hopkinson bar; High strain rate; Thermoplastic instability; Constitu

. Introduction

The interest in the mechanical behavior of materials at hightrain rates has increased in recent years, and by now it is wellnown that mechanical properties can be strongly influenced byhe speed of load application. The split Hopkinson pressure barSHPB) has become a widely used experimental method to deter-ine, for several classes of materials, the dynamic stress–strain

urves at strain rates that range from 102 to 104 s−1. Historically,he first who used long thin bars for measuring impact generatedressure waves, was Hopkinson [1]; later, this method was fur-her developed by Kolsky [2]. More recently, the Hopkinson barechnique, initially focused on compression tests, has been mod-fied and successfully applied to tensile [3,4] and torsion tests5]. Moreover, in addition to metallic materials that are now rou-inely studied, its use has been extended to many other kind of

aterials, such as polymers, ceramics, composites and foams.n order to study this wide variety of materials, including brittleaterials [6], a lot of studies have been conducted with the aim of

∗ Corresponding author. Tel.: +39 071 220 4520; fax: +39 071 220 4801.E-mail address: [email protected] (M. Sasso).

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921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2007.10.042

odeling; Finite element optimization

mproving the experimental methodology and the response qual-ty in the range of small deformations that typically is the weakpot of the SHPB technique. Anyway, also dealing with the largelastic deformations, distinctive of ductile metallic materials, aot of work is still needed to completely understand and model-ng the phenomena that are experimentally observed. In fact, as itill be shown in the next sections, the engineering stress–strain

urves obtained from experimental tests are far from represent-ng true material properties and, up to now, there is not anynified and universally accepted methodology to compute thectual material behavior from experimental data.

In this work, an experimental SHPB device has been used toerform tension tests on steel specimens. The collected data haseen post-processed using an optimization procedure based on anite element model in order to estimate the material constantsor some constitutive models that are generally adopted in impactpplications.

. Theoretical background

The compression Kolsky bar, or split Hopkinson pressure bar,onsists of two long metal bars. These bars sandwich a shortylindrical specimen, as schematically reproduced in Fig. 1.

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290 M. Sasso et al. / Materials Science and Engineering A 487 (2008) 289–300

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Fig. 1. Scheme

he free end of the first (input) bar is impacted by a projectilehat usually consists in a third bar (striker bar); the compressiveulse generated by the impact propagates down the input barnto the specimen. Several reverberations take place within thepecimen, causing the rise of its stress; in the meanwhile, a com-ressive pulse is transmitted to the second bar (output bar) andtensile pulse is reflected back to the input bar. Conventionally,

hese pulses are measured by strain gages placed on the input andutput bars. The bars must be designed to remain within theirlastic limit throughout the test while the specimen undergoesarge plastic deformation.

Let the incident, reflected and transmitted pulses be denotedespectively by εI(t), εR(t) and εT(t). All of them are function ofime and can be regarded also as displacement, strain and stresserturbations travelling along the bars with the sound speed ofhe material C0 and determining a particle velocity equal to C0ε.y applying the elementary one-dimensional elastic wave theorynd assuming that the specimen is in dynamic equilibrium, theverage nominal strain δ, strain rate δ̇ and stress S along theauge length of the specimen can be determined by applyinghe well known equations:

˙(t) = −2C0

LSεR(t), (1)

(t) = −2C0

LS

∫ t

0εR(t) dt, (2)

(t) = ABEB

ASεT(t), (3)

here AB and EB are the bar cross-sectional area and Youngodulus, and where LS and AS represent the specimen original

auge length and cross-sectional area, respectively.Thus the strain rate and the strain in the specimen can be

etermined from the reflected pulse only, and the stress can beetermined from the transmitted pulse only. Eliminating time t inqs. (1)–(3) by synchronizing the acquired signals, it is possible

o detect the stress–strain curve “followed” by the specimenaterial. It must be noted that unfortunately the strain rate is

ot constant during the test and also that the computed stressnd strain values are the engineering (or nominal) ones, becausehey are evaluated considering the initial specimen length andross-sectional area.

This treatment is based on some basic assumptions:

The waves propagating in the bars can be described by one-dimensional elastic waves propagation theory.

The specimen inertia effect is negligible and the specimen isin dynamic equilibrium.The stress and strain fields in the specimen are uniform.The friction effect in the compression test is negligible.

sdts

SHPB device.

These assumptions have been extensively studied in pastecades by several authors [7–13], with the result that numer-us experimental and numerical methods have been successfullystablished to satisfy these hypotheses or to overcome theirmperfect realization in practical application.

Theory and analysis of the tensile SHPB are basically identi-al to the compressive one [14], except for the change in shapend sign of the pulses. The principal drawback of tensile testsies in the non-uniaxial stress and non-uniform strain distribu-ions; in fact, the short dimension of the specimen accentuateshe necking instability much more than in standard tensile testserformed on comparatively much longer specimens. On thether hand, tensile tests permit to reach a higher effective strainate than in the compression tests carried out with the samequipment and under the same conditions of impacting massnd velocity.

. Experimental device

To perform tensile tests, the indirect tension method has beensed [15]; in Fig. 2, a picture of the experimental apparatuss given together with two details of the clamping region. Thehreaded-ends of the impact tension specimen can be screwedirectly into the Hopkinson bars or by means of two threaded col-ars. The first solution has been preferred because it reduces thenavoidable spurious reflections (due to mechanical impedanceariation) of the first wave when it transits across the collars.wisting the bars by hand, a small pre-tension load is applied on

he screwed joints and full contact is achieved between the barnds. In such a condition, the compressive strain pulse, initiallyenerated in the input bar by the impact with the striker bar, trav-ls through the contact surfaces from the input to the output barith no plastic deformation of the specimen [16]. As the wave

eaches the free end of the output bar, it is reflected back as aensile wave; from now on, the same conditions of the compres-ive test are reproduced except for the sign of the waves: whenhe tensile pulse reaches the specimen, it is partially transmittedo the input bar and partially reflected as a compressive wave tohe output bar.

The input and output bars, made in AISI 4140 steel, can slidento special supports with low friction; the striker bar is fired byn air gun at speed of approximately 10 m/s generating intensetress waves up to 200 MPa. The principal characteristics of thequipment are summarized in Table 1.

The bars have been chosen to have a small diameter and a high/D ratio in order to reduce the dispersion of the waves due to the

o-called Pochhammer–Chree effect. With the same aim, someiscs of soft material (e.g. sheet paper) have been interposed athe striker bar–incident bar interface, acting as “pulse shaper” touppress high frequency waves. Strain gauge rosettes, placed at

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M. Sasso et al. / Materials Science and Engineering A 487 (2008) 289–300 291

Fig. 2. Overview of the exp

Table 1Experimental rig data

Length (mm) Diameter(mm)

Material

Striker bar 121912.7

AISI 4140 steel, E = 207 GPa,density = 7800 kg/m3, soundInput bar 3073

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However, the general rule providing for flow stress increase athigher strain rates has been seen to remain valid also within thesmall variations occurred in each series, so that to slightly higher

speed = 5150 m/sutput bar 3073

pecimens 2, 4 and 8 4 AISI 1018 CR steel

alf-length of the bars, convert the strain into voltage signals thatre acquired by a 12-bit digital oscilloscope; a couple of opticalates allows to trigger the signal acquisition and to measure thetriker bar speed at the same time, that is very useful to verifyhe strain gages calibration. The adoption of different specimenengths permits to perform tests at different strain rate with theame experimental set-up; the small specimen diameter makeshe lateral and axial inertial stresses negligible (about 1 MPanly, computed by Davies and Hunter formula [11]) even if theptimal slenderness ratio is not fully respected.

. Experimental results

With the apparatus just described, dynamic tension tests haveeen performed on AISI 1018 CR steel specimens; the exper-mental results, in terms of engineering stress–strain curvesomputed by formulas (2) and (3), are shown in Fig. 3a, whilehe engineering strain rates are reported in Fig. 3b.

As outlined, different average strain rates have been achievedy using different specimen lengths, precisely 2, 4 and 8 mmith 4 mm constant diameter: accordingly to (6), the shortest

pecimen (referred D4L2) showed the highest average strainate of about 1500 s−1, the median length specimen (D4L4) had000 s−1, and the longest (D4L8) gave the lower strain rate ofbout 500 s−1.

Several tests for each specimen geometry have been car-ied out revealing an overall good repeatability of the testshemselves: considering five replications of each test, the stress

tandard deviations have been the 2.4, 2.0 and 5.1% of the aver-ge values for the D4L2, D4L4 and D4L8, respectively, whilehe strain rate standard deviations were 4.6, 3.5 and 3.1%. Theontinuous curves in Fig. 3 represent the experimental curves

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erimental equipment.

hat better approximate the average of the corresponding series;nstead, the dashed curves represent the band of ±1S.D. fromhe average curves.

It must be said that the striker bar speed was actually slightlyifferent among each test, varying from 9.7 to 10.3 m/s (this isainly due to the non-perfectly constant friction force between

he internal surface of the gun and the striker bar), so the aboveeported deviations actually refer to tests carried out with slightlyifferent conditions and strain rates.

ig. 3. (a) Tensile tests on AISI 1018 CR steel: stress–strain curves. (b) Tensileests on AISI 1018 CR steel: strain rate–strain curves.

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292 M. Sasso et al. / Materials Science and E

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•uniformly distributed into the specimen.

Fig. 4. Flow stress at 0.04 of strain vs. strain rate (engineering values).

train rate corresponded a slightly higher stress–strain curve;his means that the standard deviations here computed representsort of upper limit, or worst case, of the system repeatability,

nd if one could perform a perfect control on the impact speed,e would probably measure even smaller deviations.

Stress–strain curves are seen to move up as the strain ratencreases, making the strain rate effect quite evident for the steelested. Fig. 4 shows the flow stress at 0.04 strain at differenttrain rates, also including data coming from two quasi-staticests at 10−3 and 100 s−1.

From these dynamic results, a formidable strain rate sensitiv-ty of the material (for the reason that strain rate is only varyingrom 500 to 1500 s−1) may be deduced, but it is worth to antici-ate that the true strain rate levels experienced by the specimens,articularly in the necking region, are actually much higher thanhe approximate time averaged engineering strain rate reportedbove.

Moreover, the increase of flow stress is not only due to thetrain rate effect, but also to the triaxiality of the stress distri-ution: the nominal stress is proportional to the cross-sectionverage of axial stress component but, being the radial and hoopomponents not negligible, especially for the shorter specimens,he effective stress increment can be assumed to be less than thatbserved in nominal curves of Fig. 3a. This fact will be furtheriscussed in Section 7.

The main objective of this kind of tests is to describe the mate-ial behavior by means of constitutive models, whose constantsre determined by fitting the experimental data. The difficultiesf applying this approach moving from dynamic tensile tests, b

Fig. 5. Two tensile speci

ngineering A 487 (2008) 289–300

ies in the specimen geometry that is characterized by an abrupthange in diameter passing from the clamping to the gauge zone,nd by a very small length to diameter ratio; this determines aigh stress and strain non-uniformity along the entire specimenauge length, which is so short that there is not enough space toecover a certain homogeneity.

When a diffused necking occurs, as shown in Fig. 5, thetress and strain distributions become suddenly triaxial, and thetress–strain curves are characterized by an absolute maximumhat is reached at the very beginning of the test at quite lowtrain level (Fig. 3a), after which the stress has a large reductiongeometrical softening). Under these conditions, the engineer-ng measured data (S − δ) cannot be easily converted into truetress–logarithmic strain (σ–ε) relationship by means of thesual formulas:

= SA0

A, (4a)

= log

(A0

A

). (4b)

n fact, while (4b) can be considered applicable even if thetrain distribution is not constant across the section, Eq. (4a) pro-ides only the average axial true stress; to compute the averageffective stress the Bridgman formulation should be applied:

= σaxial

[(1 + 2R

a

)log

(1 + a

2R

)]−1

. (5)

owever, its theoretical hypothesis are not fully satisfied andqs. (4b) and (5) require the knowledge of the instantaneousross-section radius a and profile curvature radius R at the neckection that are very hard to be measured in such dynamic exper-ments.

Furthermore, two other aspects must be taken into consider-tion in order to assess the actual material properties:

The temperature rise is not negligible in this kind of tests,because the rapidity of the deformation makes the thermaldiffusion distance comparable with the specimen size; so highstrain rate processes can be considered adiabatic, and the largeplastic deformation work is transformed into heat with almostinstantaneous temperature increment [17].The strain rate is not constant during the tests, and not even

Because of these difficulties, fitting of experimental data haseen carried out by a finite element optimization procedure,

mens after the test.

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M. Sasso et al. / Materials Science

hat will be described in Section 6. In the next section, a briefescription of the constitutive models here adopted is presented.

. Constitutive models for the material

A variety of constitutive models exists for the representationf engineering materials behavior when the strain rate and theemperature effects cannot be neglected; their objective is to fithe experimental data such as shown in Fig. 4 with only onequation. A review of the most used models is given in [18]; inhis paper some of them have been implemented.

At low and constant strain rate, and at room temperature,etals are known to work harden along a path usually describedith the so-called “parabolic hardening” equations:

= (σ0 + kεn), (6a)

= σ0

(1 + ε

εy

)n

, (6b)

here σ0 is the yield stress, k and n are the work hardeningoefficient and exponential factors, respectively; often this rela-ionship is further simplified into the Hollomon equation:

= kεn. (7)

For mild steel it has been found that the effects of temperaturend strain rate are basically separated; Johnson and Cook [19]roposed the phenomenological constitutive relation:

= (A + Bεn)

(1 + C log

ε̇

ε̇0

)(1 − T ∗m), (8)

hich has been frequently used in impact and dynamic analysisue to its simplicity.

In Eq. (8), A, B and n are the three coefficients describing theuasi-static behavior of the material accordingly to (6a), while Cnd m account for the strain rate hardening and thermal softeningffects; ε̇0 is a user-defined reference strain rate, generally takens unity, and T* is the homologous temperature defined by

∗ = T − Troom

Tmelt − Troom, (9)

here T is the absolute temperature, and Troom and Tmelt repre-ent the room and melting absolute temperatures, respectively.

The main limitations with this model lie in the fact the strainnd strain rate hardening and the temperature softening phenom-na are uncoupled, and may be this is the principal reason thatakes the Johnson–Cook model to be considered to work well

t moderate strain rate level, not much over 102 s−1; anyway, its considered a “milestone” of constitutive modeling.

Many authors proposed additional empirical equation, amonghem Klopp–Clifton [20] and Litonski [21], whose models areespectively expressed by(

εp)n(

ε̇)m(

T)−τ

= σ0εy ε̇0 Troom

, (10)

= σ0

(1 + ε

εy

)n

(1 − aT )(1 + bε̇)m, (11)

Athc

ngineering A 487 (2008) 289–300 293

aking into account all the phenomena previously mentioned inmanner slightly different from the Johnson–Cook relationship.

While the Johnson–Cook constitutive relation is purelympirical, Zerilli and Armstrong [22] applied dislocationynamic concepts to develop a constitutive model able toccount for strain hardening, strain rate and temperature effectsn a coupled way; they expressed the flow stress of a material as

= σG + σ∗(T, ε̇), (12)

here σG represents the athermal barriers, determined by thetructure of the material, encountered by dislocations on theirourse, and σ* is due to the thermally activated barriers, that ishe barriers that can be overcome by thermal energy.

For bcc metals, the σ* is essentially determined by the forcepposing the dislocation movement at atomic level and theonsequent overall lattice potential; so the thermally activatedotion is not dependent on the strain, and the strain hardening

ecomes independent of strain rate and temperature, leading tohe well known formula:

σ = C0 + C1 exp(−C3T + C4T log ε̇) + C5εn

(for bcc metals). (13)

In literature, this model has been found to work well up toery high strain rate levels, even those reached in the Taylorylinder impact test.

These constitutive models have been implemented into thenite element optimization procedure that will be described in

he next section. All of them account for the thermal softeningo they are able to describe, even if in a macroscopic way, thetrain localisation in shear bands that could lead to the so-calledthermoplastic instability” if the softening rate turns out to beigher than the hardening rate.

For comparison, besides the aforementioned material mod-ls, finite element simulations have been performed also usinghe quasi-static material power law curve (6a) and the Perzynaodel, expressed by

= σ0(ε)

[1 +

(ε̇

C

)m], (14)

hat accounts for the strain rate, but not for the temperature effect.n Eq. (14) C and m are the coefficients governing the strain rateffect, and σ0(ε) represents the quasi-static material properties,odeled by (6a).

. Finite element optimization

Within the optimization module of the Ansys® finite elementommercial code [23], the experimental tests described in pre-ious sections have been numerically reproduced. Preliminarynite element simulations showed that the lateral extremitiesf the specimen remain within the elastic limit and no rele-ant error is committed if they are considered infinitely rigid.

ccordingly to this simplifying assumption and taking advan-

age of the symmetrical geometry, only the “useful” part of aalf-length specimen has been modeled in order to reduce theomputation time.

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294 M. Sasso et al. / Materials Science and Engineering A 487 (2008) 289–300

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Table 2Constrained and unconstrained optimization parameters

Totalparameters

Constrainedparameters

Designvariables

Johnson–Cook 5 3 2Zerilli–Armstrong 6 3 3Klopp–Clifton 5 3 2LP

e

S

wentTrb

das(

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pdtotrTaTable 3.

ig. 6. Virtual specimens geometry with three significant nodes highlighted.

Transient thermo-mechanical analysis have been performedn the three specimen geometries here considered; in Fig. 6, thenal deformed profiles of the specimens are compared with thendeformed ones (dotted lines). The virtual specimens, meshedith plane axial-symmetric elements, have been subjected to

he experimentally measured elongations imposing on the upperound nodes the vertical nodal displacements v(t) computed by

(t) = δ(t)LS, (15)

here δ(t) is the measured engineering strain (function of time)nd LS is the initial length of the specimen. The lower boundodes are constrained to slide along the horizontal x directionnly, representing the half-length symmetry plane, while the leftound nodes are constrained to slide along the vertical y directionnly, representing the axial-symmetry axis.

The temperature increment during the deformation has beenomputed considering that the entire plastic work is transformednto heat with attendant temperature increase; this leads to theollowing equation:

T = PLWK

ρCp

=∫ ε

0 σ dε

ρCp

, (16)

here PLWK denote the plastic work done (per volume unit), ρnd Cp are the material density and specific heat, respectively.

Clearly, this is a case of inverse procedure, where the actualnknown is not the structure behavior, but the material proper-ies, that still have to be computed. In this paper, such a problemas been solved with an optimization approach.

Generally speaking, the optimum design is the design thatinimizes an objective function; in the specific case, the

ptimum design has been defined as the set of material coef-cients that permits to minimize the standard deviation of theumerically obtained engineering stress–strain curve from the

tt

itonski 6 3 3erzyna 5 3 2

xperimental one:

.D. =√∑

ε[Snum(δ) − Sexp(δ)]2

N, (17)

here Snum(δ) and Sexp(δ) are the numerical and experimentalngineering stresses, δ is the engineering strain and N is theumber of simulation sub-steps that has been forced to be equalo the number of experimental sample points (1 �s time spaced).he numerical engineering stress is obtained by dividing the total

eaction force in the lower bound nodes of the virtual specimeny the initial cross-sectional area.

This is an unconstrained minimization problem, where theesign variable space is constituted by the material coefficients,nd the objective function is expressed by the average of thetandard deviation computed for the three different geometryD4L2, D4L4, D4L8):

BJ = S.D.D4L2 + S.D.D4L4 + S.D.D4L8

3. (18)

The only minimization constraints are represented by thepper and lower limit of the variability range of the material con-tants. Running the analysis for different material coefficients,ifferent OBJ values are obtained; the objective function is thenpproximated by a polynomial function of the design variables,s follows:

BJ(x) ∼= a0 +n∑

i=1

aixi +n∑

i,j=1

bijxixj, (19)

here n is the number of design variables x and the coefficient0, ai and bij are determined by a weighted least mean squaretting of the already computed OBJs. The polynomial function

n (19) is the function to be actually minimized.As already outlined, the constitutive models illustrated in the

revious section have been adopted, and the number of non-ependent parameters has been reduced imposing the constrainthat the equation should approximate the experimental databtained from quasi-static (ε̇ = 1 s−1, T = Troom = 273 K) test;hus, only the temperature and strain rate related coefficientsepresent the design variables xi of the optimization procedure.able 2 summarizes the situation about the optimization vari-bles, while the constrained parameter values are reported in

A brief note for the Zerilli–Armstrong model is required:he last term in Eq. (13) represents the strain hardening, whilehe first two terms have been considered to correspond to the

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M. Sasso et al. / Materials Science and Engineering A 487 (2008) 289–300 295

Table 3Optimization results

Quasi-static constraint Variable range Best set MinimumOBJ (MPa)

Perzyna A = 520, B = 269, n = 0.282 C: 6000–16,000, m: 0.1–1 C = 14,030 ± 279, m = 0.292 ± 0.021 47 ± 1.5Clifton σ0 = 525, εy = 0.0026, n = 0.0646 m: 0.01–0.08, τ: 0.2–1.5 m = 0.0392 ± 0.0017, τ = 0.9124 ± 0.0599 36 ± 2.0Johnson–Cook A = 520, B = 269, n = 0.282 C: 0.09–0.23, m: 0.3–0.9 C = 0.0476 ± 0.0021, m = 0.5530 ± 0.0250 36 ± 2.0Litonski σ0 = 520, εy = 0.0027, n = 0.0666 a: 0.0006–0.0025, b: 0.001–0.5,

m: 0.001–0.5a = 0.0021 ± 0.0003, b = 0.0687 ± 0.0116,m = 0.0676 ± 0.0037

39 ± 1.5

Z 10−5 −2

−3

ytC

anfrd

itraio

erilli–Armstrong C0 = 26, C5 = 269, n = 0.282 C1: 0–2000, C3:C4: 10−6 to 0

ield σ0 of Eq. (11); so from the static test it is not foundhe value of C0, but only its dependence on the other terms:

0 = 520 − C1 exp(−273C3).Of course, for a first evaluation of the coefficients in (19),

preliminarily investigation of the design variable space iseeded. To have a good estimation of the shape of the object

unction, some simulations have been performed with mate-ial constant sets representing equally spaced points inside theesign variable range; thus each variable range has been divided

orw

Fig. 7. Objective functions visualiz

to 10 , C1 = 1703 ± 149,C3 = 4.53 × 10−3 ± 0.0011,C4 = 1.83 × 10−4 ± 0.00004

35 ± 2.0

nto 5 intervals for the two-parameter models and into 3 forhe three-parameter models, requiring 52 and 33 simulations,espectively. Afterwards, the local minimum of the objectivepproximation function can be computed and the correspond-ng design variable set is reintroduced into the simulation codebtaining a new OBJ value, that is used to update the coefficients

f function (19), and to compute a new local minimum. This isepeated iteratively until the new solutions are not able to reduce,ith a relatively small tolerance (1%), the final OBJ value; gen-

ation for the applied models.

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a

96 M. Sasso et al. / Materials Science

rally only two or three additional simulations are required toerform this iterative procedure.

. Comparing results

An overview of the principal results of the FE optimizations given in the next pictures and in Table 3. In Fig. 7, the objec-ive functions are plotted as 3D meshed surfaces with someontour lines for the two-parameter models Johnson–Cook,lopp–Clifton and Perzyna; the local minimums at the optimalesign variable values, specified in Table 3, are clearly visible.he contour lines refer to isohypses of the 3D objective function

n geometrical progression starting from the minimum + 4 MPao minimum + 80 MPa.

The shaded surfaces instead, refer to the three param-ters Litonski and Zerilli–Armstrong models, and represent

Fig. 8. Numerical results for the optimum design set.

tmto

F

ngineering A 487 (2008) 289–300

sohypses of the 4D objective functions. The surfaces areoncentric encapsulated ellipsoids, grouping all the materialoefficient combinations that give the same OBJ value, withhe centre corresponding to best design set whose values areeported in Table 3. The black inner surface represents the min-mum + 1 MPa isohypse of the 4D OBJ approximation function,hile the others are 15 MPa equally spaced in increasing order.The optimization procedure seems to be stable with respect

o the experimental data scattering, without significant variationf the iterations number.

Further, the dispersion of the experimental data leads to a vari-bility range of the optimal parameters (also reported in Table 3)

hat is seen to be quite small, except for the Zerilli–Armstrong

odel; this is probably due to the exponential nature of the law,hat makes its parameters somewhat sensitive to the scatteringf the experimental data they are requested to fit, but this does

ig. 9. Stress state along the neck cross-section at 20% engineering strain.

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M. Sasso et al. / Materials Science

ot jeopardy the overall matching of the numerical model withhe real data, since the OBJ variation is observed to be in lineith other models.For the best sets of material constants, the numerical engi-

eering curves have been compared with the experimental onesFig. 8). It can be seen the good matching among the experi-ental stress–strain curve and all the simulated curves, except

hose obtained with the Perzyna and the quasi-static (denoted asstatic” in the legend) parabolic hardening models; in particularhe “static” curves remained constantly below the experimen-al curves for all the three geometries here considered, with aomprehensive standard deviation of 121 MPa, and showed aegative slope much lower than the experimental evidence. Thiss confirmed by the Perzyna curves that move up for increasingtrain rate but, as for the “static” ones, they cannot accuratelyeproduce the stress decay because they does not account forny material softening.

On the contrary, the models that account for both strain ratend thermal effects are able to provide a better matching ofhe experimental data, with the standard deviation indicatedn Table 3. It means that the main phenomena occurred inhe specimen have been correctly modeled; strong reductionf engineering stress after the initial peak cannot be explainednly by the geometric softening due to necking, but a certainffective softening, probably due to temperature rising, must bedmitted.

Furthermore, the curves of Fig. 8 give a numerical “a pos-eriori” confirmation of what has been mentioned in Section 4:ctually the increase of the experimental stress–strain curves isot only due to the strain rate effect. In fact, even in the simu-

erta

Fig. 10. Temperature

ngineering A 487 (2008) 289–300 297

ations performed with the parabolic hardening constitutive lawhere the strain rate effect is not really considered (Eq. (6a)), theominal curves moves up as the specimen length decreases, withmaximum stress of 600, 640 and 730 MPa for the D4L8, D4L4nd D4L2, respectively; this means that also the stress distribu-ion in the specimen contributes to the raising of the engineeringurves.

The situation is summarized in Fig. 9, where the stress com-onents and the triaxiality (computed as the hydrostatic to Vonises equivalent stress ratio) are plotted along the node A–nodenecking section, as computed by the Zerilli–Armstrong model

t 0.2 of nominal elongation.While Von Mises equivalent stresses are quite constant along

he section path, with average values of 815, 791 and 763 MPaor the D4L2, D4L4 and D4L8 specimens respectively, thexial, radial and circumferential components are not uniformlyistributed, especially in the shortest specimen which is charac-erized by a much higher triaxiality (>1.3 in the core).

It is to note that in general the equivalent stress is lower thanhe average axial component, which is particularly high in the4L2 specimen; this is another reason, beside the strain rate

ffect, why the shortest specimen follows the highest nominalow curve.

In Fig. 10 the temperature increase is shown as computed byhe Zerilli–Armstrong model (the other models give analogousesults) for the three specimen geometries at their maximum

longation; the D4L8 specimen, reaching lowest strain and strainate levels, undergoes the lowest temperature increase, while inhe other ones the temperature rises to relatively high values,bout 200 K for the D4L4 and more than 230 K for the D4L2

increase (K).

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2 and Engineering A 487 (2008) 289–300

stta

wss(icwte

co1

Fr

98 M. Sasso et al. / Materials Science

pecimen. These temperatures are enough to establish a sort ofhermoplastic instability into the material, that happens whenhe thermal softening rate exceed the strain hardening rate; forconstant strain rate, this is mathematically expressed by

dσ

dε=

(∂σ

∂ε

)T

+(

∂σ

∂T

)ε

(∂T

∂ε

)≤ 0, (20)

here dσ/dε represents the resulting slope of the truetress–strain curve, (∂σ/∂ε)T is the strain hardening rate at con-tant temperature, (∂σ/∂T)ε is the thermal softening rate and∂T/∂ε) represents the temperature increase relevant to a ∂ε strainncrement. The last term, as well summarized in [14], can beomputed in closed form if the Johnson–Cook model is adoptedith thermal softening exponent m set to unity; but generally

he dependence of temperature on the plastic strain must bevaluated by numerically integrating Eq. (16).

Fig. 11 shows the isothermal quasi-static true stress–strainurve compared with the adiabatic curves computed with theptimal constitutive models coefficients, and considering a03 s−1 constant strain rate. It can be noticed how the dynamic

ccld

Fig. 12. Logarithmic strain and strain

ig. 11. True stress–logarithmic strain curves at 103 constant logarithmic strainate.

urves, at relatively low strain, are much above the isothermalurve because of strain rate hardening, but after a critical strainevel is reached, the material starts to soften; at higher strain, theynamic curves go even below the static one. It must be said that

rate histories from simulation.

Page 11

and E

tifm

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ni

etc

vfmsbr

M. Sasso et al. / Materials Science

he critical strain, at which the thermoplastic instability occurs,s differently modeled by the constitutive laws used: it variesrom 0.075 for the Zerilli–Armstrong to 0.12 for the Litonskiodel.In Fig. 12 the strain and strain rate histories of the A, B and C

odes are reported, confirming that the true strain rate inside thepecimen is much higher than the overall average value deducedrom experimental engineering data (e.g. in the core of the D4L2pecimen the strain rate is almost permanently between 4000 and000 s−1). The curves also show how the strain rate distributions widely irregular, highlighting very different histories amonghe A, B, and C nodes of the specimens individually considered.he strain rate profiles vary from geometry to geometry as well;

n particular, while the intermediate and the longest specimenshow similar behaviors, in the D4L2 the corresponding nodesave very different histories with the node B, that lies on the

Fig. 13. True stress–logarithmic strain curves from simulation.

micmtsm

8

fddd

atbecmt

cin

crioh

btflm

R

ngineering A 487 (2008) 289–300 299

ecking cross-section, undergoing a lower strain rate than thenternal node C.

Because of all these issues, the numerical and experimentalngineering curves can be regarded as the “rough” response ofhe specimen, made by the overall summation of the differentontributes given by different parts of the specimen.

The fact that the sample nodes here considered reach strainalues very different among each other, experiencing very dif-erent strain rate levels even among a single geometry, actuallyeans that they are forced to follow different effective true

tress–strain curves during their deformation process. Moreover,ecause of the strain rate variation, the effective true stress–strainelationship followed by a certain elementary part of the speci-en is not even represented by the adiabatic curves of Fig. 11, but

t must be considered as the envelope of the different adiabaticurves corresponding to the different strain rate levels experi-ented by that elementary part of the specimen. The effective

rue stress–strain curves occurred at nodes A, B and C for eachpecimen, as numerically obtained with the Zerilli–Armstrongodel, are given in Fig. 13.

. Conclusions

In this work, high strain rate tension tests have been per-ormed on AISI 1018 CR steel specimens, by means of a SHPBevice; afterwards, the same tests have been numerically repro-uced with a finite element commercial code simulating the largeeformation process underwent by the specimens.

Analysis of the numerical results showed that the specimensre interested by widely non-uniform stress and strain distribu-ions; moreover, different internal points of the specimen haveeen demonstrated to experience largely different strain rate lev-ls at any given time, meaning that the engineering stress–strainurves represent only the “boundary conditions” of the speci-en behavior, and they are not quantitatively representative of

he material properties.An optimization procedure permitted to obtain the material

oefficients of some common constitutive models, provid-ng good correspondence between numerical and experimentalominal curves.

In this way it has been possible to take into account for theoupled effects of the material, characterized by strain and strainate hardening and thermal softening, and of the particular spec-men geometries, that lead to a “geometrical softening” becausef large reduction in area and simultaneously to an apparentardening because of high triaxiality.

These results encourage for future works aiming to cali-rate, with a similar approach, a damage evolution model ofhe material, giving the possibility not only to characterize theow stress–strain behavior but also to assess for the failureechanisms.

eferences

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[[[[[[[

[

[

[

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00 M. Sasso et al. / Materials Science

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