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Numerical simulation of the high strain-rate behavior of quenched and self-tempered reinforcing steel in tension Gianmario Riganti, Ezio Cadoni University of Applied Sciences of Southern Switzerland, CH-6952 Canobbio, Switzerland a r t i c l ei n f o Article history: Received 27 September 2013 Accepted 19 December 2013 Available online 28 December 2013 Keywords: Reinforcing steel bar Simulation High strain-rate Upper yield dynamic stress Time dependent plasticity Split Hopkinson Tension Bar a b s t r a c t This 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 and interpretation of the experimental results. Finite element simulation has allowed a robust model valida- tion of the B450C reinforcing steel. Parametrical fi nite element model has been used to rebuild the input and output signals of the SHTB. Physical infl uence of damping in input wave and modeling strategies have been discussed. The elastic and damping dispersion fonts have been introduced into the model to explain the real case variability in SHTB signals. Strain-rate dependent plasticity model has been used by LsDyna code features. Time dependent plasticity has been developed to explain upper and lower yield values of the material resulting into a loading rate sensitivity. Finally, the material model has been used to 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. Introduction The understanding of the dynamic behavior of concrete and reinforcing steels is essential for the precise assessment of existing reinforced concrete structures when they are subjected to a high loading rate. These assessment studies are usually conducted by means of fi nite element codes and the material models have to be properly based on correct experimental data. The diffi culties connected to the complexity of the experimental tests can be appropriately understood and solved by numerical simulation. To better comprehend the experimental results it is essential to per- form the simulation of the testing machine 15 in order to obtain mutual verifi cation. In the analysis of the experimental results often it is possible to face diffi culties in interpreting the results due to the presence of instabilities (i.e. presence of the fi rst 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 of the material and have been investigated by several authors. The upper yield stress has been explained with metallic structure parameters such as the dislocation density and velocity 7. In any case, material models involving microstructure parameters are not suitable for engineering purposes. Structural assessment requires relations between the upper and the lower yield value with the engineering variables associated to the loading pulse, structure geometry, stress and strain tensor. Models that require the defi nition of material variables in terms of structure and dislo- cation density/velocity can be considered a phenomenological explanation of upper yield lacking of the complete parameteriza- tion of the stress strain curve including upper, lower yield and its time dependencies. Engineering investigations of upper yield were made by Camp- ell and Harding 810. Campbell introduced the delay time and thermal activation theory by which the upper yield occurs after a characteristic time after the start of the loading stress due to the shear band thermal activation 11. The value of the upper yield stress was further investigated by Harding 12, who introduced a linear relation between dynamic upper yield stress enhancement and loading rate. Hardings ap- proach is the most suitable engineering formulation for upper yield found 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 large facilities (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 the dynamic behavior of the material with the present paper. The importance of the numerical simulation is defi nitely based on the possibility of studying real scale structural elements by means of numericalsimulationoftestsotherwisenotfeasiblefor 0261-3069/$ - see front matter ? 2013 Elsevier Ltd. All rights reserved. /10.1016/j.matdes.2013.12.049 Corresponding author. Tel.: +41 58 6666 377; fax: +41 58 6666 359. E-mail address: ezio.cadonisupsi.ch (E. Cadoni). Materials and Design 57 (2014) 156167 Contents lists available at ScienceDirect Materials and Design journal homepage: technical or economic reasons. The present work completes, from a numerical point of view, what was started 13 with the experi- mental one, analyzing the various critical aspects regarding both experimental technique used and numerical simulation. The experimental technique used for the high strain rate mechanical 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 hits the 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 simulating the 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 the experimental set-up. The numerical model results are presented in 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 is presented in Section 6. Finally, Section 7 summarizes the whole work. 2. Critical aspects of the Split Hopkinson Tension Bar 2.1. Signals analysis Signal analysis is usually adopted in the traditional theory of the Split Hopkinson Bar (SHB) to calculate stress, strain and strain-rate 17. Another methods consists in the combined use of the simula- tion and experimental test data. The validation of material model is thenmadebynumericalandexperimentalgaugesignal comparison. The advantages in combined use of simulation and experimen- tal data are: (i) accurate fi nal material model verifi cation; (ii) spec- imen geometrical non linearity is included; (iii) the hypothesis of uniformity 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 simulation for experimental facilities accuracy enhancement; and (vii) optimi- zation techniques and sensitivity analysis can be applied. 2.2. Effect of perturbations into the signal SHB 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 and inertial effects. It is well-known as a real input signal of SHTB dif- fers from the ideal trapezoidal pulse due to local perturbations when the real signals are used to obtain the material model param- eters, a series of errors are included due to simplifi ed hypothesis and signal perturbations. The study of perturbed real signal effects to material model response is suitable to enhance the material model correctness. The infl uence of these factors on the material model response can be checked by means of fi nite element simulation. 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 specimen reaching 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 signifi cant characteristic for the material response and it is conditioned by the SHB set up (striker or pre-stressed bar), by the use of pulse shaper, and by other physical parameters out of direct control such as the facilities damping. The pulse shaper technique 17,18 is generally applied to smooth the input signal, in case of stress oscillations typical of stri- ker impact in SHB. By interposition of an intermediate deformable element between striker and input bar, a higher repeatability and a smooth input pulse is obtained. If short rising time is wanted, the input signal will be also affected by high frequency perturbations, especially in SHB confi gurations. High frequency perturbations widen the repeatability of signals and are subject to the elastic and damping dispersion phenomena. Usually in SHB a ratio length/diameter is adopted, which is always suitable to elastic and damping dispersion 17. The damping infl uences the input dispersion and its effect should be evaluated such as the elastic dispersion. Damping is not directly controlled in SHB. Different facilities could generate pulses with signifi cant differences in rising time and perturbations. Referring to Fig. 1, three typologies of input signal could be generated: 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 rate dependent materials, input signal (b) or (c) has to be generated. The phenomena which generate the pulse perturbations can be grouped as: ? Unlocking(SHTB)/contact(SHB) perturbations/combined use of shaping technique. ? PochhammerChree wave dispersion 19,20. ? Damping effects/damping dispersion. input pulse time b c a Fig. 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 by input/output signal recording. Elastic dispersion occurs by a frequency dependent wave speed propagation. In SHB, the short distance between input/output gauge and specimen is suitable to affect elastic wave to dispersion. Gauge signal correction techniques can be applied to obtain data at specimen location. Those techniques are energy conservative and does not represent the dispersion due to damping. This hypothesis is usually correct because of distance between gauge and specimen is 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 infl uence of damping, dispersion and rising time will be numerically investigated before applying the simulation to the experimental data. 3. Numerical model of the experimental set-up Explicit time integration has been applied to simulate the dy- namic test with rate-dependent material modeling using LsDyna code. The SHTB geometry 1315 is basically axial-symmetric and the non-symmetry is a result of the small geometrical and align- ment imperfections. The axial length of the whole facilities was 15 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 estimated alignment error was 0.1 mm. Bars were horizontally placed and vertical holders consist of Tefl on bushing supporting the bars each 500 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. The gravity was not modeled but the static shear component at Tefl on bearings correspondent to slipping condition is applied as concen- trated loads at bearings location. Static pre-loading acted along the axis direction. The axial-symmetric model was suitable to study the SHTB cause of geometrical and loading conditions. The axial-symmetric model allowed the inclusion of dispersion, damping, pre-loading andaxial-stresswavepropagation.Axial-symmetricvolume weighted elements have been used due to effi ciency advantages in computation while ensuring correct solution interpolation with the adequate mesh size. The numerical effi ciency of the model was required for multiple runs in parametrical analysis. The element size in radial and axis direction was 2.5 mm. The size of the element was equal to the experimental gauge length to average the stress time history as the real test case. The variation of stresses in radial direction was of the second order infl uence with respect to the solution of wave propagation in axis direction for SHTB experimental purposes 20. Two ele- ments in radial direction allowed a correct interpolation of the solution. Thespecimenmeshsizewas0.2 mminaxisdirection, 0.275 mm in radial direction. The specimen was modeled using coincident 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,witha computation 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 effi cient and neglects the pre-loading energy stored closer to the jack joint, which is too far from the specimen side to affl ict the input wave shape. Fixed boundary conditions in axis direction were assigned to the jack location. Nodes on axis were automatically constrained in radial direction. Locking was modeled with an instantaneous release free of per- turbation. At the start of the analysis, the pre-stressed elements of the tension bar were free to deform and explicit calculation starts. The absence of unlocking perturbation allowed to focus the infl u- 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 the arrival of the input wave. Total number of nodes/elements was 19,285/28,955. Calculation time is 15 min at strain-rate 250 s?1. 3.1. Damping and numerical model Damping modifi es propagation of waves with a frequency dependent function. Damping study is necessary to the following material response verifi cation. The SHTB damping sources were grouped in four physical sources: (i) Material damping: constitutive material of SHTB bar had its own damping parameter. The damping parameter for bars was low compared to the damping induced by other SHTB physical sources, as confi rmed by simulation results. (ii) Friction: the static bar weight was distributed along the holders and acted in radial direction. Once the input wave was released, the moving in axial direction through the holder was possible because the axial pre-loading force is greater than the weight multiplied by the static friction coef- fi cient. (pre-loading 104N, input and pre-loading bar 50 N weight each, estimated static friction force 5 N). During the wave propagation, the bar hits the holder moving through the Tefl on gasket gap several times. The resultant dynamic friction forces are highly dependent on the experimental set up by alignment and bars pre-deformation. (iii) Viscous interface: The bar was in atmospheric air and the high frequency vibration of the bar release energy was in radial direction. (iv) Dynamiccontacts:Thepreviouslydescribedbar/holder impacted release energy at holder location with a phenom- ena dependent on gap distance, materials, pre-deformation and imperfections. The wave propagation through holders dissipate energy. Damping must be introduced into the numerical model for cor- rect input signal generation. There are two different approaches to model damping in SHTB simulation: (a) The phenomenological approach consists in introducing the single physical effect by modeling the interaction rules with their driven parameters, e.g. contact, vibration, imperfection, infl uence. This method requires the maximum effort in mod- eling, and is time consuming with regards to the operator and calculator. (b) Model the global effect of damping by assigning a damping coeffi cient which converges the numerical results to the experimental ones. A parametric analysis is necessary to identify the correct damping coeffi cient. As the global result 158G. Riganti, E. Cadoni/Materials and Design 57 (2014) 156167 of damping causes is easily detectable by input gauge signal, this method offers the best effi ciency in modeling and results. In the present work, the (b) method has been applied. Damping was modeled by using LsDyna keyword ?damping_global. Damping value was anisotropic in the axial and radial directions, in acco