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Numerical simulation of the high strain-rate behavior of quenchedand self-tempered reinforcing steel in tensionGianmario Riganti, Ezio CadoniUniversity of Applied Sciences of Southern Switzerland, CH-6952 Canobbio, Switzerlanda r t i c l ei n f oArticle history:Received 27 September 2013Accepted 19 December 2013Available online 28 December 2013Keywords:Reinforcing steel barSimulationHigh strain-rateUpper yield dynamic stressTime dependent plasticitySplit Hopkinson Tension Bara b s t r a c tThis paper presents the numerical analysis of the high strain-rate behavior of quenched and self-tem-pered reinforcing steel in tension. The investigation has been performed properly simulating the exper-imental facility (SHTB-Split Hopkinson Tension Bar), highlighting criticism in the simulation andinterpretation of the experimental results. Finite element simulation has allowed a robust model valida-tion of the B450C reinforcing steel. Parametrical finite element model has been used to rebuild the inputand output signals of the SHTB. Physical influence of damping in input wave and modeling strategieshave been discussed. The elastic and damping dispersion fonts have been introduced into the model toexplain the real case variability in SHTB signals. Strain-rate dependent plasticity model has been usedby LsDyna code features. Time dependent plasticity has been developed to explain upper and lower yieldvalues of the material resulting into a loading rate sensitivity. Finally, the material model has been usedto reconstruct a virtual test over a rebar of 32 mm diameter, as an example of general procedure to cal-culate the global material response.? 2013 Elsevier Ltd. All rights reserved.1. IntroductionThe understanding of the dynamic behavior of concrete andreinforcing steels is essential for the precise assessment of existingreinforced concrete structures when they are subjected to a highloading rate. These assessment studies are usually conducted bymeans of finite element codes and the material models have tobe properly based on correct experimental data. The difficultiesconnected to the complexity of the experimental tests can beappropriately understood and solved by numerical simulation. Tobetter comprehend the experimental results it is essential to per-form the simulation of the testing machine 15 in order to obtainmutual verification.In the analysis of the experimental results often it is possible toface difficulties in interpreting the results due to the presence ofinstabilities (i.e. presence of the first peak), which are not consid-ered in the usual material constitutive laws as JohnsonCook 6.These instabilities are due to the upper and lower yield stress ofthe material and have been investigated by several authors. Theupper yield stress has been explained with metallic structureparameters such as the dislocation density and velocity 7. Inany case, material models involving microstructure parametersare not suitable for engineering purposes. Structural assessmentrequires relations between the upper and the lower yield valuewith the engineering variables associated to the loading pulse,structure geometry, stress and strain tensor. Models that requirethe definition of material variables in terms of structure and dislo-cation density/velocity can be considered a phenomenologicalexplanation of upper yield lacking of the complete parameteriza-tion of the stress strain curve including upper, lower yield and itstime dependencies.Engineering investigations of upper yield were made by Camp-ell and Harding 810. Campbell introduced the delay time andthermal activation theory by which the upper yield occurs after acharacteristic time after the start of the loading stress due to theshear band thermal activation 11.The value of the upper yield stress was further investigated byHarding 12, who introduced a linear relation between dynamicupper yield stress enhancement and loading rate. Hardings ap-proach is the most suitable engineering formulation for upper yieldfound in the literature.The experimental study of the dynamic tensile behavior of full-scale quenched and self-tempered rebar (1640 mm in diameter)is practically impossible, except maybe in the case of very largefacilities (i.e. the large facility of the Joint Research Centre, Ispra).The unfeasibility of this study has led us to proceed to the charac-terization of the material 13 and the numerical analysis of thedynamic behavior of the material with the present paper. Theimportance of the numerical simulation is definitely based on thepossibility of studying real scale structural elements by means ofnumericalsimulationoftestsotherwisenotfeasiblefor0261-3069/$ - see front matter ? 2013 Elsevier Ltd. All rights reserved./10.1016/j.matdes.2013.12.049Corresponding author. Tel.: +41 58 6666 377; fax: +41 58 6666 359.E-mail address: ezio.cadonisupsi.ch (E. Cadoni).Materials and Design 57 (2014) 156167Contents lists available at ScienceDirectMaterials and Designjournal homepage: /locate/matdestechnical or economic reasons. The present work completes, from anumerical point of view, what was started 13 with the experi-mental one, analyzing the various critical aspects regarding bothexperimental technique used and numerical simulation.The experimental technique used for the high strain ratemechanical characterization of B450C rebar was the Split Hopkin-son Tension bar (SHTB) and was described in 1315. In this par-ticular set-up the input pulse is not generated by a striker who hitsthe input bar, as in the traditional Split Hopkinson Pressure bar,but using the energy stored in a pre-stressed bar directly con-nected to the input bar 16.This set-up offers several advantages compared to the tradi-tional one, avoiding problems connected to the planar impact be-tween striker and input bar, to the pulse length, etc.The numerical analysis has been performed properly simulatingthe SHTB, highlighting criticism in the simulation and interpreta-tion of the experimental results.This paper is organized as follows. Section 2 presents the criti-cisms of the SHTB. Section 3 reports the numerical model of theexperimental set-up. The numerical model results are presentedin Section 4 both in terms of FEM and numerical analysis. These re-sults are discussed in Section 5. The model of the real size rebar ispresented in Section 6. Finally, Section 7 summarizes the wholework.2. Critical aspects of the Split Hopkinson Tension Bar2.1. Signals analysisSignal analysis is usually adopted in the traditional theory of theSplit Hopkinson Bar (SHB) to calculate stress, strain and strain-rate17. Another methods consists in the combined use of the simula-tion and experimental test data. The validation of material model isthenmadebynumericalandexperimentalgaugesignalcomparison.The advantages in combined use of simulation and experimen-tal data are: (i) accurate final material model verification; (ii) spec-imen geometrical non linearity is included; (iii) the hypothesis ofuniformity of stress/strain through the specimen is overcome;(iv) inertial effects are included; (v) multi material and small struc-ture specimen can be investigated; (vi) possible use of simulationfor experimental facilities accuracy enhancement; and (vii) optimi-zation techniques and sensitivity analysis can be applied.2.2. Effect of perturbations into the signalSHB relations contain several idealizations as the one-dimen-sional wave propagation through bars and specimen, the unifor-mity stress in the specimen, the absence of perturbations andinertial effects. It is well-known as a real input signal of SHTB dif-fers from the ideal trapezoidal pulse due to local perturbationswhen the real signals are used to obtain the material model param-eters, a series of errors are included due to simplified hypothesisand signal perturbations. The study of perturbed real signal effectsto material model response is suitable to enhance the materialmodel correctness. The influence of these factors on the materialmodel response can be checked by means of finite elementsimulation.The input signal is mainly characterized by amplitude, duration,and rising time. These main characteristics can be adapted to gen-erate the wanted dynamic loading conditions into the specimenreaching the wanted rate during the experiment.The wanted input amplitude and duration are generated by tun-ing the physical parameters of the input pulse generation method(striker or pre-stressed bar).The input stress rising time is another significant characteristicfor the material response and it is conditioned by the SHB set up(striker or pre-stressed bar), by the use of pulse shaper, and byother physical parameters out of direct control such as the facilitiesdamping.The pulse shaper technique 17,18 is generally applied tosmooth the input signal, in case of stress oscillations typical of stri-ker impact in SHB. By interposition of an intermediate deformableelement between striker and input bar, a higher repeatability and asmooth input pulse is obtained. If short rising time is wanted, theinput signal will be also affected by high frequency perturbations,especially in SHB configurations. High frequency perturbationswiden the repeatability of signals and are subject to the elasticand damping dispersion phenomena. Usually in SHB a ratiolength/diameter is adopted, which is always suitable to elasticand damping dispersion 17. The damping influences the inputdispersion and its effect should be evaluated such as the elasticdispersion.Damping is not directly controlled in SHB. Different facilitiescould generate pulses with significant differences in rising timeand perturbations.Referring to Fig. 1, three typologies of input signal could begenerated:1. Low loading rate, high rising time, no apparent wave dis-persion (curve a).2. High loading rate, dispersion with hypercritical damping(curve b).3. High loading rate, dispersion with sub critical damping(curve c).When a high loading rate is wanted to study loading ratedependent materials, input signal (b) or (c) has to be generated.The phenomena which generate the pulse perturbations can begrouped as:? Unlocking(SHTB)/contact(SHB) perturbations/combined useof shaping technique.? PochhammerChree wave dispersion 19,20.? Damping effects/damping dispersion.input pulsetime bcaFig. 1. SHB input pulse in the case of: (a) pulse shape technique is used; (b)dispersion and over critical damping; and (c) dispersion and sub critical damping.G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167157? Boundary conditions (friction and contact on holders,clamping, etc.).? Geometry/alignment errors.? Other unknown effects (bar homogeneity and isotropy).During the experiment, all the listed causes act simultaneously.The global effect on the input loading could be easily measured byinput/output signal recording.Elastic dispersion occurs by a frequency dependent wave speedpropagation. In SHB, the short distance between input/outputgauge and specimen is suitable to affect elastic wave to dispersion.Gauge signal correction techniques can be applied to obtain data atspecimen location. Those techniques are energy conservative anddoes not represent the dispersion due to damping. This hypothesisis usually correct because of distance between gauge and specimenis short. Analytical technique cannot be applied for signal correc-tion affected by dispersion damping.A long length of the input bar is suitable to stabilize signal per-turbations, but the input length increases the elastic dispersion ef-fects resulting in smaller loading rate.The influence of damping, dispersion and rising time will benumerically investigated before applying the simulation to theexperimental data.3. Numerical model of the experimental set-upExplicit time integration has been applied to simulate the dy-namic test with rate-dependent material modeling using LsDynacode.The SHTB geometry 1315 is basically axial-symmetric andthe non-symmetry is a result of the small geometrical and align-ment imperfections. The axial length of the whole facilities was15 m consisting of pre-loading bar (6 m), input bar (3 m), and out-put bar (6 m). The bar diameter was 10 mm and the estimatedalignment error was 0.1 mm. Bars were horizontally placed andvertical holders consist of Teflon bushing supporting the bars each500 mm.The non-symmetric static stress due to gravity were several or-ders of magnitude lower than the average stress during the test,with deformations lower than the geometrical imperfections. Thegravity was not modeled but the static shear component at Teflonbearings correspondent to slipping condition is applied as concen-trated loads at bearings location. Static pre-loading acted along theaxis direction.The axial-symmetric model was suitable to study the SHTBcause of geometrical and loading conditions. The axial-symmetricmodel allowed the inclusion of dispersion, damping, pre-loadingandaxial-stresswavepropagation.Axial-symmetricvolumeweighted elements have been used due to efficiency advantagesin computation while ensuring correct solution interpolation withthe adequate mesh size. The numerical efficiency of the model wasrequired for multiple runs in parametrical analysis.The element size in radial and axis direction was 2.5 mm. Thesize of the element was equal to the experimental gauge lengthto average the stress time history as the real test case.The variation of stresses in radial direction was of the secondorder influence with respect to the solution of wave propagationin axis direction for SHTB experimental purposes 20. Two ele-ments in radial direction allowed a correct interpolation of thesolution.Thespecimenmeshsizewas0.2 mminaxisdirection,0.275 mm in radial direction. The specimen was modeled usingcoincident nodes with bar at outer tread diameter.The axial-symmetric solution excluded non symmetrical geo-metrical perturbations. A full 3D analysis could include the align-mentperturbationandcontactsinSHTBholders,withacomputation cost increase of two orders of magnitude.Pre-loading was represented by initial stress conditions of pre-loading bar elements. A uniform axis direction stress was assigned.This solution is highly efficient and neglects the pre-loading energystored closer to the jack joint, which is too far from the specimenside to afflict the input wave shape.Fixed boundary conditions in axis direction were assigned tothe jack location. Nodes on axis were automatically constrainedin radial direction.Locking was modeled with an instantaneous release free of per-turbation. At the start of the analysis, the pre-stressed elements ofthe tension bar were free to deform and explicit calculation starts.The absence of unlocking perturbation allowed to focus the influ-ence of material model and dispersion.A full restart technique was applied to increase calculation effi-ciency, adding specimen elements and out bar elements before thearrival of the input wave. Total number of nodes/elements was19,285/28,955. Calculation time is 15 min at strain-rate 250 s?1.3.1. Damping and numerical modelDamping modifies propagation of waves with a frequencydependent function. Damping study is necessary to the followingmaterial response verification.The SHTB damping sources were grouped in four physicalsources:(i) Material damping: constitutive material of SHTB bar had itsown damping parameter. The damping parameter for barswas low compared to the damping induced by other SHTBphysical sources, as confirmed by simulation results.(ii) Friction: the static bar weight was distributed along theholders and acted in radial direction. Once the input wavewas released, the moving in axial direction through theholder was possible because the axial pre-loading force isgreater than the weight multiplied by the static friction coef-ficient. (pre-loading 104N, input and pre-loading bar 50 Nweight each, estimated static friction force 5 N). During thewave propagation, the bar hits the holder moving throughthe Teflon gasket gap several times. The resultant dynamicfriction forces are highly dependent on the experimentalset up by alignment and bars pre-deformation.(iii) Viscous interface: The bar was in atmospheric air and thehigh frequency vibration of the bar release energy was inradial direction.(iv) Dynamiccontacts:Thepreviouslydescribedbar/holderimpacted release energy at holder location with a phenom-ena dependent on gap distance, materials, pre-deformationand imperfections. The wave propagation through holdersdissipate energy.Damping must be introduced into the numerical model for cor-rect input signal generation. There are two different approaches tomodel damping in SHTB simulation:(a) The phenomenological approach consists in introducing thesingle physical effect by modeling the interaction rules withtheir driven parameters, e.g. contact, vibration, imperfection,influence. This method requires the maximum effort in mod-eling, and is time consuming with regards to the operatorand calculator.(b) Model the global effect of damping by assigning a dampingcoefficient which converges the numerical results to theexperimental ones. A parametric analysis is necessary toidentify the correct damping coefficient. As the global result158G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167of damping causes is easily detectable by input gauge signal,this method offers the best efficiency in modeling andresults.In the present work, the (b) method has been applied. Dampingwas modeled by using LsDyna keyword?damping_global. Dampingvalue was anisotropic in the axial and radial directions, in accor-dance with the two different damping sources 21. Iterative solu-tion of the numerical model compared to the experimental inputwave will allow to identify the optimal numerical values.To model the axial damping due to friction of bar over Teflongaskets, a series of damper elements with damping coefficient pro-portional to the estimated axial friction forces has been defined.These elements act in axial direction.4. Numerical model results4.1. FEM analysisThe FEM analysis has been performed to verify the dependencyon experimental facility and the capability of a standard materialmodel.4.1.1. Elastic/damping dispersion influence in input signalsUniform damping affects uniformly the rising time and disper-sion peak. This implementation does not allow to model input sig-nal inFig. 1 (curve b). The anisotropic damping correctlyrepresents the physical causes of damping in SHTB. Fig. 2 showshow the input signal changes with the global damping coefficientvariation. Damping affects principally the rising time, while thepulse amplitude is less influenced. In Fig. 3, two input signals attwo different pre-loading conditions are represented and it canbe noted how loading rate is also affected to the pulse amplitudebut the rising time remains unaffected. The three typologies(Fig. 1) of input waves can be consequently modeled varyingdamping parameters. In Fig. 4, the comparison between the exper-imental input pulse and the numerical one has been depicted,usingthedefinitivedampingparametersshowingagoodcorrelation.4.1.2. Dispersion and material model verificationThe input/output gauge signals were generated by interactionbetween SHTB and specimen. The material model has been testedby fitting the numerical output to the experimental one. If differ-ences are introduced in the numerical stress wave, the fitting ofnumerical output gauge to real test case by material model param-eters identification will include errors in parameters to compen-sate for input differences. The error propagation is numericallyinvestigated.Material verification was performed by studying a strain-ratedependent material subject to a damped and un-damped inputwave. The test material was the B450C type C 13, modeled as ex-plained in the next section. The input wave represents the maximaldifferences in input stress caused by dispersion error generationusing non-damped finite element model.The dispersion oscillations according to the acoustic impedancematching at specimen interface loads the tested material withdamped oscillations around the average plastic strain rate. Dueto the nonlinear strain-rate dependent plasticity, the response of02 10-44 10-46 10-48 10-41 10-31,2 10-313801390140014101420143014401450optimized-damphigh_dampno_dampexperimentalstrain -time sFig. 2. SHB input pulse variations due to different damping coefficients.05 10-41 10-31,5 10-32 10-31380139014001410142014301440pre-load 1pre-load 2strain -time sFig. 3. Input pulse amplitude and rising time for two pre-loading conditions.0200 10-6400 10-6600 10-6800 10-61 10-3130014001500160017001800optimized dampexperimentalstrain -time sFig. 4. Comparison between experimental and numerical input pulse.G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167159the material was modified in terms of time to breakage and stressamplitude. The calculated numerical differences were small due tothe greater influence of average strain-rate response compared tooscillations variations. Plastic rate-dependent response was less af-fected by dispersion oscillations. Damping affects rising time and itshould be considered for reliable material verification in case ofloading rate dependent material models.4.1.3. Strain-rate dependent plasticity responseOnce the input wave was tuned to the real one, the plasticmaterial response was studied.Strain-rate dependent plasticity material parameters can be cal-culated from the test data using the Hopkinson bar formulae.The SHTB finite element model was used to study the correla-tions between upper/lower yield stress to uniform plasticity mate-rial models and to wave reflection and inertia.The component materials of B450C steel has been numericallymodeled by the LsDyna keyword?material_piecewise_linear_plas-ticity 21. Strain-rate sensitive plasticity was defined by truestress/equivalent plastic strain curve definition for each strain-rateof interest. A total number of 5 curves were used to define thestrain-rate behavior from static to dynamic loading range. Eachcurve was defined by 5 points as shown in Table 1. The softwareautomatically interpolates the curve in order to obtain the elementresponse using element strain-rate. The static curve was insertedas first curve into the input file. The use of a greater number of in-put data in material plasticity definition is possible, but in this casethe use of automatic optimization procedure is suggested due tothe high number of material parameters to be characterized.The failure criteria were modeled by constant plastic strain tofailure. The value was set up using the specimen area reductionat fracture.Strain-rate dependent plasticity parameters were obtained byfitting the numerical and experimental gauge signals.The same final material model has been used for both SHTB dy-namic simulation of 250 and 1000 s?1. Fig. 5 shows a comparisonbetween numerical and analytical input signals where it is possibleto observe a good agreement. The differences between experimen-tal and numerical signals were of the same magnitude of experi-mental variability due to specimen differences. The failure strainwell represents the failure mechanism and fracture time. The frac-ture starts from the center of the specimen and propagates toexternal layers, in accordance with stress intensification along axisduring necking.The Lagrangian mesh has been used in the model. Neckingdeformation has been interpolated by 12 linear elements. The cal-culation stability to mesh size has been verified; differences inbreakage time due to mesh were negligible. The use of automaticre-meshing techniques was excluded in order to be critical for bet-ter result interpolation.This result has been reached by the explicit dynamic solutionincluding geometrical non linearity such as the specimen necking,and including the wave reflection, transient wave propagation andstress equilibrium.For material which exhibits necking and plastic deformability,as B450C, the numerical fitting, including the nonlinear sourcesshould be considered. This method is more accurate than analyticalsolution.Differences are visible between calculated and test signals atfirst phase of yielding. Experimental test record upper and loweryield significantly different to the numerical response usingstrain-rate dependent plasticity. A small upper yield value was vis-ible in numerical data and can be addressed to the higher plasticflow introduced into the specimen due to the input wave shape.The input wave shape results from a first rising phase, a loadingpeak due to wave dispersion, and a steady plateau before decay.The amplitude of the input stress peak was related to the plasticflow into the specimen. The specimen response was calculatedby using rate sensitivity scaling the plastic flow. A small peak inyield response results in a first yielding phase due to the dispersionpeak in input signal.The resultant value of yield peak was unable to predict theexperimental recorded upper yield value.4.1.4. Upper yield stress and its dependenciesDuring the experimental campaign 13 the materials responseshowed an upper yield and a lower yield values.The dynamic upper yield was higher than the static yield stressand increases with the strain-rate.The upper and lower yield appeared different when measuredto input or output gauge. This difference increases for test at highstrain-rates. Lower yield was significantly greater in input gauge,while it is coincident with the yield at uniform strain-rate re-sponse. Upper yield stress could be influenced by the variation be-tween numerical and real input waves, the inertia effects, the wavepropagation through specimen, and the material response.The FEM model has been used in order to analyze the influenceof inertia and wave propagation on upper yield stress. To this pur-pose, isotropic-elastic and ideal elasticplastic materials have beenused to maximize the effects of inertia and wave reflections whichpotentially affect the first peak. No correlations have been found.Also dispersion was excluded by using finite element method.Specimen and output bar density elastic properties were similar,ensuring by the acoustic transmittance an efficient wave propaga-tion in output direction. The reflection at output interface was min-imal, and the specimen loading proceeded almost monotonicallyuntil the yield stress value. The characteristic propagation timeof stress wave was about 2ls. The upper yield value was experi-mentally recorded with a characteristic time about 25ls, whichensured stress equilibrium into the specimen while the phenome-non appears.All previous effects did not influence the upper yield value. Thestrain-rate dependent material model was not adequate to fitupper yield data.4.2. Numerical analysis for a material model developmentThe aim of this part was to verify non-standard material modelto predict the real high strain-rate behavior of the material includ-ing the upper yield stress.4.2.1. Numerical solution based on yield propagation velocitydependent plasticityThis simulation, using Matlab, permitted to verify an arbitrarymaterial model able to interpolate the upper yield value, to verifymaterial model robustness, to simulate numerically SHTB and realsize reinforcing bar response subject to dynamic loading.The procedure, which assesses the material model defined asyield propagation velocity dependent plasticity, has been subdi-vided in the following three different calculation phases:(1) Determination of the material response imposing constantloading rate material response.(2) Determination of the material response simulating the SHTBtest by using the experimental input wave and the acousticalimpedances at the specimenbars interfaces. The transmit-ted and reflected gauge are calculated and compared withthe experimental ones.(3) DeterminationoftherealsizeB450Creinforcingbarresponsebasedonthecomponentmaterialdynamicresponses (type A, B, and C 13).160G. Riganti, E. Cadoni/Materials and Design 57 (2014) 1561674.2.2. Shear band velocity (or yield propagation velocity) dependentplasticityAs already known, the typical engineering approach consists ofstrain decomposition in elastic and plastic components, and in theyield definition as threshold limit between elastic and plastic do-main. The yield stress can be considered constant or dependentfrom strain-rate or from other variables e.g. temperature. Thehypothesis of uncoupling between variables and multiplicativefunctions of single variables is usually made to simplify data fittingand the model use. The material instantaneously switches to plas-tic behavior when the local equivalent stress reaches the yieldstress. A plastic stress function was used for stress calculation. Thisapproach has been extensively used by several authors 22 tomodel strain-rate dependent plasticity. One of those models hasbeen previously used to demonstrate the non suitability of thosemodels to explain the upper yield response.A material model with time dependent yield mechanism wasproposed to explain upper yield. Time dependency in yielding re-sults from the propagation of yield domains starting from the ini-tial activated dislocations. Microscopic heterogeneity of materialsand shear band propagation phenomenon 23 can be consideredas experimental evidence of velocity of propagation of yieldingcaused by combination of microstructure and external stresses.Time dependency in yielding is physically consistent to the energyexchanges generated during yielding. The energy release/absorp-tion should act in a finite scale of time.The specimen resultant response is calculated by two contribu-tions the elastic domain response and the plastic domain response.The transition from elastic to plastic domain was driven to thepropagation of sheared area front, which moves with finite velocityfrom the activation point. The shear band initialization was con-trolled by the material seed density. The material can express plas-tic response only if the sheared domain or band reaches the localmaterial coordinate.At elastic and plastic response are assigned simple elastic andplastic material models.Yield process starts when the static value is reached in one ini-tial point of the specimen section equation (1). Since that conditionequation (2), yield starts to propagate with constant speed equa-tion (6). The propagation speed allows to define a time dependentyielded area Ayand its plastic response according to the chosenplastic formulation equation (4). The definition of the area subjectTable 1Stressstrainstrain rate data used in strain rate dependent plasticity material model.True_strainStrain rate10?3s?1250 s?1800 s?12500 s?14000 s?1True_stress (MPa)True_stress (MPa)True_stress (MPa)True_stress (MPa)True_stress (MPa)Material A078088090098010000.12588011151140117012000.37104011151140117012000.8512001120114090010001.051200110011409001000213001300130013001300Material B05405856307007000.1257808559359309550.37780900990100011000.857809001000100011001.05800900100010001100211001100110011001100Material C03854856107007000.1256607758908909200.377357908978979000.857507958708709001.0580082087087090021000100010001000100005 10-71 10-61,5 10-62 10-62,5 10-63 10-63,5 10-61000150020002500300035004000experimentalnumericalstrain -time stime s05 10-71 10-61,5 10-62 10-62,5 10-63 10-63,5 10-6120014001600180020002200240026002800experimentalnumericalstrain -(a)(b)Fig. 5. Comparison between experimental versus numerical MHB input signal for:(a) 250 s?1; and (b) 1000 s?1.G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167161to elastic response equation (3) is consequent by the initial crosssection value A0. The resulting specimen is composed by theyielded and elastic domains contributes Eq. (5).ryry01ifrry02relreleel;mpar1for A0 Ay3rprp_e;ep;ry;mpar2for Ayandrry4rspcrp? Ay rel? A0? AyA05Vyield const:6whereryis the yield stress;ry0is a constant;ris real stress; A0isthe initial cross-section area;relis the elastic stress;rpis the plasticstress; Ayis the cross-section area during the plastic field; mpar1andmpar2are material parameters;rspcis the stress in the specimen;Vyieldis the shear band velocity.4.2.3. Model application: constant loading rate responseFrom the previous equation set, the material model constitutiveresponse has been developed under the following hypothesis: ini-tial single yield seed on specimen axis, round propagation of yielddomains, constant propagation of yield, constant loading rate.By hypothesis the material is loaded with linear stress throughtime as the following equation:rt _p ? t7where_p is the loading rate (MPa/s), t is the time (s).The initialization of shear band was considered process stressdependent. An initial hypothesis is made to fit initialization tothe static case, with single activated dislocation Ni. From the avail-able dislocations, the first activated is the one which exhibits thelowest activation stress. The initial seed from which the shear bandpropagates is initially corresponding to unity corresponding to theweakest dislocation into the section. While the shear propagates,the elastic domains are loaded with incremental stresses coher-ently with the loading wave rule. For incremental stresses, furtheryield initializations could occur when the elastic stress reaches thenew initialization value. This mechanism is formally resumed byEqs. (8) and (9), a function representing the relation between theresultant activated dislocation (Ni) and the activation stress level(rt) for each dislocation available into the cross-section.Nit Nirt;const:?8If the cross-section is small, a single activation could result,depending on material intrinsic characteristics. If loading rate issmall, a single activation will result too. Increasing cross-sectionand loading rate effect will be increasing the activated dislocations.Multiple activations cannot be excluded in case of large specimencross-section and due to the rising of the elastic stress over theactivation stress in peripheral section regions. Multiple seed pres-ence will be verified by interpolating the material model with theexperimental results.In the simplest case the material model presents one activation:Ni0 0;Nit ? ty 19The cross-section average material stress is implemented by theaddiction of the two domains contribute:rdt ry0_pt ?_ptAy=A010The apparent dynamic yield depends on time and on loadingrate. The region excluded from the shear band is elastically loadedwith the loading rate law. The ratio between specimen cross-sec-tion and yielded cross-section evolves with time, conditioningthe dynamic resultant yield stress. Time zero corresponds to yieldstress value. The yielded area projected to the section perpendicu-lar to the tension direction influences the specimen resultantcalculation.Ayt t2NipV2y11A combination of the previous formula leads to dynamic stressfunction of time and material/specimen constants:rdt ry0_pt ?_pt3NipV2y=A012The maximum of the upper yield stress is reached at the time tmafter reaching the static yield:tmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA03pNiV2ys13To calculate the total time to maximum stress the time to staticyieldry0_pshould be added.tm;totry0_pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA03pNiV2ys14Inserting tminto the time into the dynamic yield stress equation(12), the value of the upper yield is found.rdmry023_ptm15There is a linear dependence of maximum upper yield stress onthe stress rate result. For small loading rate, the upper yield is coin-cident with static value.The upper yield stress is conditioned by combined specimenand material parameters, by the characteristic time equation(14). The specimen cross-section raises the upper yield value,while an increase in shear band speed reduces it.4.2.4. Upper yield visibility in dynamic testingThe upper yield stress is observable when it exceeds the valueof the uniform yield stress inclusive of the plastic strain-rateenhancement. This condition is reached for defined relations be-tween the plastic strain-rate and the test loading rate. The plasticstrain-rate at regime depends on the input signal amplitude. Com-paring the upper yield stress with the uniform plastic strain-ratestress enhancement, the upper yield stress results visible in testsignal for loading rate above the critical loading rate are definedas follows:_pcr Dse32 ? tmry016Dseryd_ep ?ry0ry017Previous relations can be transformed into a loading rate toamplitude input wave condition.For the material of the test,_pcrcorresponds to 5 TPa/s with aspecimen of 3 mm diameter as recently hypothesized in 7.Comparing numerical and experimental solutions, some differ-ences occur in upper yield value and in loading rate at yielding asshown in Fig. 6. For 250 s?1test (Fig. 6a) differences include notuniformities in loading rate history. For both analyses, the realloading rate is higher than the numerical one, with correspondentincreasing in calculated upper yield. Once constant loading rate isimposed interpolating the test loading rates, good agreement inupper yield value is found (see Fig. 7).Test results suggest the inconstancy of the experimental acous-tic impedance, which lead to non-linearity in specimen loading162G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167rate due to coupling between acoustic transmittance and loadingrate.5. Discussion5.1. Lower yield stress valueLower yield stress measured at input gauge has been explainedby wave reflection consistent with yield domain propagation.During yielding the specimen can be considered composed bymaterials with different acoustic properties. The interfaces be-tween plastic and elastic material response influences the wavepropagation. The first interface is in the input side, from the mate-rial with elastic response to the plastic domain interface. The sec-ond one is at the output side, from the plastic domain to the elasticmaterial response. Both interfaces have area equal to the yieldeddomain time dependent. The remaining portion of the specimencross-section extends to the new interfaces. The elastic wave trav-els with elastic sound speed, related to elastic modulus and den-sity. The plastic wave travels with plastic wave speed, related tohardening modulus and density. Differences in elastic to plasticwave speed are significant. Plastic wave speed travels at about1500 m/s while elastic wave propagates at 5100 m/s.At the interface between elastic specimen portion and itsyielded one, an acoustic impedance lower than unit result. Withmaterial hardening and elastic values, an acoustic coefficient of0.47 is calculated. At output interface, the acoustic coefficient ishigher than the units result. This means that the stress is transmit-ted changing amplitude and frequency. As hypothesis, we assumeda unitary transmitting coefficient at the output because modifica-tion in frequency and amplitude requires further implementationafter the actual verification phases.Using specimen cross-section averaged stress resultant as in Eq.(5), the input gauge resultant differs to the output gauge resultantdue to the contribution of the reflection coefficient applied to theinput signal. Input signal is increasing with time, an input speci-men resultant decay appears as a contribution of wave reflections.Left resultant appears to be lower than right resultant. On outputinterface, the dynamic yield stress will be simply applied as previ-ously specified in Eq. (12).The output gauge response can be considered more appropriateto characterize material properties because it is extent to inputwave influences.Input specimen side average stress is calculated as follows:rd inputt A0? Asb ?rinpt Asb?rinpt ? 1 ? R2sbA018where R2sbis a reflected coefficient considered as positive.The previous formula can be combined with linear loading ratehypothesis or with yield area growing rules. In Fig. 8, the compar-ison of the experimental and numerical curves (using Eq. (18) ofthe lower yield stress at input gauge is shown.5.2. Material properties interpolationMaterial model identification consists of one material constantsdefinition in the case of the specimen here used. By using geomet-rical properties of specimen into Eqs. (13) and (15), upper yield va-lue was used to tune the velocity of yield propagation. Consequenttime history of upper yield was then calculated. Further verifica-tion of upper yield response of material subject to real input wavetime history was then implemented.The single yield seed was used in simulations. Multiple yieldseed were verified by numerical sensitivity application.The material parameter identification was obtained from a sin-gle SHTB test data. Multiple SHTB data at different rates were usedfor material verification. The verification consists in further upperyield and decay calculation.Lower yield interpolation was also used as further verificationof material model response.5.3. Fit of upper yield stressThe average loading rate values were extracted from the exper-imental input signals of 250 s?1and 1000 s?1rate test. Loadingrates which interpolate the yield loading rate were assigned toeach simulation. The fit was made by tuning the material shearband speed and using single activation in yield. Multiple activa-tions were not suitable to fit the test data for the specimen3 mm in diameter.Specimen cross-section influence was linear. When upper yieldstress appears in a 3 mm diameter specimen, higher upper yieldstress was expected for higher structure size. The increasing ofupper yield stress was expected until a characteristic size ofcross-section in which two or more yielding points appears. Thedefinition of those limits have critical influence in practical appli-cations. Under those limit, the upper yield peak is expected to be0100020003000400050006000050 10-6100 10-6150 10-6200 10-6numericalexperimentaloutput pulse Ntime s0100020003000400050006000020 10-640 10-660 10-680 10-6100 10-6numericalexperimentaloutput pulse Ntime s(a)(b)Fig. 6. Comparison between experimental and numerical curves of output signalsof B450C type C material at (a) 250 s?1and (b) 1000 s?1.G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167163sensitive to specimen scale effect. Around a critical size, theappearance of multiple yielding seed generates bifurcations inthe material response. Over the critical size, materials stabilizewith multiple yielding response. The decay of the upper yield re-sponse in multiple activation is driven to the saturation of thespecimen cross-section. In this case, the geometrical size and shapeof specimen plays an important role in upper yield response. Thisissues lead to applications in structural response to impulsiveloading.The initial raising of upper yield stress and its decay can beinterpolated with a combination of possible set of seed numberand decreasing shear band speed.In Fig. 9, the comparison between experimental and numericaloutput signals curves for material A (a) and material B (b) areshown, respectively.The first method is suitable to present the lower yield, becauseit is influenced by the real loading rate vale after the yield point.The calculation of lower loading rate is in good agreement withexperimental data, for output and input gauge. The calculation ofupper yield stress using input wave time history and uncoupled0100020003000400050006000-50 10-6050 10-6100 10-6150 10-6200 10-6numericalexperimentaloutput pulse Ntime s0100020003000400050006000-40 10-6-20 10-6020 10-640 10-660 10-680 10-6numericalexperimentaloutput pulse Ntime s(a)(b)Fig. 7. Comparison between experimental and numerical curves of output signalsof B450C type C material at (a) 250 s?1and (b) 1000 s?1imposing loading rateconstancy.050 10-6100 10-6150 10-6200 10-6250 10-6050 10-6100 10-6150 10-6200 10-6250 10-6experimentalnumericalstrain -time sFig. 8. Comparison between the experimental and numerical curves of the loweryield stress at input gauge.010002000300040005000600070008000-50 10-6050 10-6100 10-6150 10-6experimentalnumericaloutput pulse Ntime s010002000300040005000600070008000-50 10-6050 10-6100 10-6150 10-6200 10-6experimentalnumericaloutput pulse Ntime s(a)(b)Fig. 9. Comparison between experimental and numerical output signals curves formaterial A (a) and material B (b).164G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167acoustic impedance underestimates yield loading rate. The con-stant loading rate calculation leads to an accurate estimation ofupper yield stress but it is not suitable to interpolate the loweryield stress. This fact enforces the origin of the lower yield stressin reflection mechanism of the input wave at input interface dueto variation of acoustic impedance of sheared region.The shear band propagation seed found by the model was con-sistent with literature experiment and suggests an adiabatic shearband speed of about 500 m/s 15. This value corresponds to thesingle seed and the interpolated value of speed equal to 120 m/s.Material type B and C have the same propagation speed.Pulse shaper techniques are generally not suitable to investigateupper yield response smoothing the input pulse and resulting inlower specimen loading rate hiding the upper yield appearanceas in Eqs. (16) and (17). Real impacts could often generate a loadingrate involving an upper yield response. Experimental investigationshould be performed, planning a test in the same range of realevent loading rate and plastic flow. Structural response influencedby the upper yield criticism are expected for real applicationswhich stress the material with sufficient loading rate. In thosecases, a material investigation by means of SHTB apparatus ishighly recommended.6. Model of the real size barUpper yield value parameterization is not performed by the de-lay time theory. Delay time observations is in accordance with theEq. (13), which can be considered a generalization of the delay timefor constant loading rate and structure size. With the proposed ap-proach, it is possible to calculate the upper yield time as resultantof shear band propagation velocity, specimen and loading wavecharacteristics. The delay time is not considered a fundamentalconstant, but it appears as related when constant specimen sizeare used in tests and the input pulse is approximated with constantloading rate.Eqs. (14) and (15) includes the Harding approach 12 general-izing the dependencies of the upper yield by other physical param-eter of the problem. In case of arbitrary loading wave, and forgeometrical size variation, this approach cannot be considered va-lid because the proportionality of the upper yield stress to theaverage loading rate contains loading pulse and geometric charac-teristics and it is not a fundamental material constant.The introduction of the specimen size dependency is a funda-mental task to compare test data of different authors and to trans-late to the engineering structures the result of the dynamic tests.Without this generalization, relevant errors are potentially intro-duced at the structural assessment phase under impulsive loadingconditions.The material model of the constituent materials of B450C rein-forcing bar (type A, B, C) allows to reconstruct the dynamic behav-ior of real size rebar. For example, the response rebuilding of a32 mm diameter bar subject to impulsive loading is presentedhere. The measure of hardening through real bar cross-section al-lows to correlate the material typology thickness. When increasingthe diameter, the temper process modifies the ratio between mate-rial typologies by heat exchange coefficient and thermal inertia.The hardening measure can be applied to generic complex struc-tures to investigate material typologies percentages.In case of tension, the result on the rebar is obtained by the con-tributions of each constituent material at calculated strain/loading-rate. This simple rule can be used for material equivalence re-sponse calculation. A step by step explicit procedure is requiredto perform the calculation imposing arbitrary loading wave pulse.Rebar response is divided in two time calculation domains usingthe loading rate material model for the first one, and the strain ratedependent plasticity rules by the FEM approach. Yield propagationvelocity parameters previously identified (260 m/s for material Aand 160 m/s for material B and C) are applied to the Matlab numer-ical model dedicated to the loading rate response calculation. Fur-ther development of user defined material models into the FEMwill allow to simultaneously calculate the upper yield and the plas-tic response in conjunction with the numerical capability of the ex-plicit codes.The size and the geometry of the structure play a significant rolein the sensitivity definition to loading rate effect. Thickness andcross-section of material typologies should be studied in order tounderstand the characteristic size at which multiple yielding acti-vation occurs. This parameter influence is studied into Fig. 10 inwhich the rebar upper yield stress response is compared using sin-gle to four activation seed for the core material, in combinationwith 510 seed for the external one. Differences in amplitudeand duration are calculated.050010001500020 10-640 10-660 10-680 10-6100 10-6120 10-6rebar_stress_activation_core1_external5rebar_stress_activation_core2_external5rebar_stress_activation_core4_external5rebar_stress_activation_core4_external10stress MPatime sFig. 10. Stress vs. time curves of the rebar obtained with several activationconditions.0200400600800100012001400160000,00050,0010,0015strain-rate 750/sstrain-rate 1000/sstrain-rate 1500/sstrain-rate 2500/sloading-rate 106 MPa/sloading-rate 206 MPa/sloading-rate 406 MPa/sloading-rate 606 MPa/sstress MPatime sFig. 11. Rebar response to arbitrary combinations of loading rate and strain rate.G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167165The rebar dynamic response to arbitrary combination of loadingrate and strain rate is presented in Fig. 11. The response of theloading pulse can be chosen selecting the correspondent loadingrate and strain rate. In dynamic test facilities, higher loading ratescorrespond to higher strain rate. Fig. 12 shows two loading pulseresponses dedicated to test the strain rate at 750 and 1500 s?1withloading rate from 10 to 40 TPa/s.In combination of loading wave, upper yield stress can lower orhigher the ultimate stress. Fig. 13 synthesizes upper yield and ulti-mate stress amplitude in dependence with the loading rate andstrain rate.The upper yield response appears to be significant in real struc-ture and should be investigated and considered in blastingstructural assessment. Upper yield also influences the high speeddeformation phenomena such as perforation and impacts, result-ing in higher contact forces, reduced deformability and increasedcapability to transfer energy into the target material.Other dynamic loading conditions could be investigated, withdifferent cross-section, in any other rate conforming to the pro-posed material models. The material data is suitable to investigatedynamic composite models for concrete bar reinforcement.7. ConclusionsExperimental tests by means of SHTB involve non line
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