Calibration of a combined isotropic‐kinematic hardening material model for the simulation of thin electrical steel sheets subjected to cyclic loading

The combined isotropic-kinematic hardening model enables the description of cyclic transient elastic-plastic material behaviour steel. However, determination parameters and understanding their influence on response can be a challenging task. This study deals with individual steps calibration for simulation thin electrical steel sheets under loading. Specific recommendations are made kinematic isotropic parameters. In particular, evolution is described by Voce's exponential law simple multilinear approach. Based approach, which allows different slopes in yield surface size, an alternative component proposed. As result, presence plateau first half cycle accurately captured, while convergence issues definition numerical simulations avoided. comparison simulated load cycles experimental tests shows good agreement, indicates suitability proposed Die Berücksichtigung kombinierter isotroper-kinematischer Verfestigung ermöglicht die Beschreibung und Modellierung des transienten elastisch-plastischen Werkstoffverhaltens von Stahl. Ermittlung der zugrunde liegenden Materialparameter das Verständnis für deren Einfluss auf Materialantwort stellt jedoch häufig eine anspruchsvolle Aufgabe dar. vorliegenden Arbeit wird einzelnen Schritte bei Kalibrierung Materialmodells im Rahmen einer Simulation zyklisch beanspruchten dünnen Elektroblechen eingegangen. Dabei werden konkrete Empfehlungen kinematischen isotropen Verfestigungsparameter gegeben. Evolutionsgleichung sowohl durch einen Exponentialansatz nach Voce als auch einfachen multilinearen Ansatz beschrieben. Basierend dem Ansatz, unterschiedliche Steigungen Evolution Fließfläche ermöglicht, Komponente vorgeschlagen. Dadurch kann auftretende Fließplateau ersten Halbzyklus präzise erfasst werden, während Konvergenzprobleme vermieden werden. Der Vergleich simulierten Belastungszyklen mit experimentellen Daten zeigt gute Übereinstimmung, was Eignung vorgeschlagenen Kalibrierungsprozesses Elektroblech spricht. rotor electric drive, consists several sheets, exposed to high rotational speeds (∼20.000 min−1) loading due acceleration deceleration. With typical sheet thicknesses only 0.3 mm, locally exhibit mechanical stresses strains. context analytical fatigue life assessment, evaluated. For this purpose, suitable, calibrated required. recent years, constitutive models theories describing phenomenological mechanisms plasticity have been developed further improved. An overview single including respective equations viewed literature 1. choice industrial applications, however, task considering complexity, computation time data available 2. study, according Lemaitre Chaboche used describe 3. It well-established implemented most commercial finite-element applications. combination accurate representation complex phenomena, e. g., softening/hardening creep. To ensure satisfactory alignment between actual behaviour, carefully determined based data. present work supposed give modelling followed specific explanation Symmetric strain-controlled performed provide stress-strain hysteresis curves parameter determination. avoid buckling compressive forces, application anti-buckling device necessary more detailed paper. After that, presented. Numerical Abaqus comparing predicted tests. validated both strain- stress-control. characterised certain mechanisms, accounted plasticity. Typically, includes Bauschinger effect. Under uniaxial loading, strength specimen decreases, after direction reversed, Figure Considering symmetric test, it observed that elastic limit unloading corresponds double initial 2 . Influence effect strength. Bauschingereffektes Streckgrenze. stress or strain change microstructural effects, results decreasing increasing deformation resistance softening material-specific depends amplitude as well history. caused strain-control associated changing amplitudes, usually assumed intensity decreases stabilised state reached. Stress-strain a) hardening, b) softening. Spannungs-Dehnungsantworten symmetrischer dehnungskontrollierter zyklischer Belastung, zyklische Verfestigung, Entfestigung. unsymmetric stress-control, ratchetting (cyclic creep) has considered, 3a. case measured accumulation axial plastic rate magnitude mean stress. contrast, result relaxation, 3b. number cycles, tends approach zero. ratchetting, relaxation. Spannungs-Dehnungsverhalten unsymmetrischer Beanspruchung, zyklisches Kriechen, Mittelspannungsrelaxation. Besides mentioned (Lüders bands propagation) given particular consideration. idealised non-hardening region onset , 4 4. long not reached, remains less constant. reversal, disappears. Presence region, [4]. Vorhandensein Fließplateaus idealisiert nichtverfestigte Region, Darstellung three back components, [1]. Kinematische Verfestigungsentwicklung drei Rückspannungskomponenten, If considered simultaneously, we receive model, 6. Isotropic covers uniform expansion space development interpreted domain 1, for. Kinematic translation flow captures terms effect, Representation stress-space, one-dimensional [7]. kombinierten isotrop-kinematischen Verfestigungsmodells, Spannungsraum, eindimensional analysed fully-processed non-oriented nominal thickness 270 μm following composition (mass fraction): 3.32 % silicon, 1.10 aluminium, 0.16 manganese, 0.01 carbon, 0.002 sulfur, 0.010 phosphorus, balance iron. Therefore, exhibits ferritic microstructure small non-metallic inclusions. average grain size approximately 100 μm, electron backscatter diffraction (EBSD). Monotonic properties were ten specimens quasi-static tensile at room temperature constant 2.5 10−4 s−1, Table Due slight rolling texture, values Young's modulus lower than other orientations. Hence, investigated exclusively direction. [GPa] Poisson's ratio [−] Yield [MPa] Ultimate Elongation break [%] 187 0.28 447 540 15.3 GPa 0.28, respectively. MPa. limited amount resulting comparatively flat curve ultimate MPa elongation 15 %, 7. studied sheets. Monotone Spannungs-Dehnungs-Kurve untersuchten Elektrobleche unter Zugbeanspruchung. Experimental obtained from authors use stress-controlled without compression among reasons, specimens. preferably characterise therefore, perform model. Consequently, all amplitudes 5 10−3 s−1. Cyclic applied means dynamic test machine ElectroPuls E10000 INSTRON extensometer EXA 10-0,5 SANDNER-Messtechnik GmbH used. testing accordance DIN EN ISO 7500-1 23788, low proper mounting, 8a. Tests five 0.2 0.6 %. Despite using device, ( > %) possible extent, since increasingly influenced occurring waviness higher forces get. until = study. setup, view vertical guide rails, extensometer, tension springs specimen, latter highlighted yellow line, c) geometry. Versuchsaufbau, Versuchsmaschinenaufbau, Detailansicht Knickstütze vertikalen Führungsschienen, Extensometer, Zugfedern Probe, letztere gelbe Linie hervorgehoben, Probengeometrie. two halves fastened four screws sample, 8b. Two rails restrict unintended movement during test. A diamond-like carbon (DLC) coating reduces friction sliding contact anti-buckling-device specimen. Furthermore, polytetrafluoroethylene-foil device. gap must remain clamping jaws allow unrestricted extension positioning edges samples. By slippage prevented. area cutting edge lead local pressures. therefore ensured no crack initiation starting observed. smooth <1.0 μm), processed precision discharge machining (EDM). mm gauge length 10 width used, 8c. transition designed matching radii achieve homogeneous minimised notch factor. include representative 90 life, 9. Usually, defined 50 assumption failure. steel, condition reached Instead, initiation, 10. appropriately, selected life. chosen do yet crack, its influences shape. Experimentelle Spannungs-Dehnungs-Hysteresekurven Ermüdungslebensdauer. showing continuous saturation Wechselverformungsverhalten kontinuierliche ohne Stabilisierung bis zum Versagen. extend revealed accommodation general, division into responses depending behaviour: 0.4 %), substantial stabilisation fracture. rises slightly 0.25 beginning before rapid occurs. Regarding lowest changes No takes place, why these will on. On one hand, monotonic tests, conducted curve, 11a provides range Δ expected significantly differ shape varying sequence complete 11b. Quantities Größen Spannungs-Dehnungs-Hysterese-Kurven zur Bestimmung kinematischer isotroper set ) fitting equation (9) pairs internal routine nonlinear least-squares regression analysis. generally steep slope curved section leading flatter 14. points sloped significant residuals compared region. weighting process. At same time, parts where concentrated. Since Abaqus’ does sufficient options monitor quality fit, should external program. paper, toolbox Matlab components recognised fit considerably improved if 12. root squared error (RMSE) reduced value 7.2 0.84 indicating reliable result. Both captured. addition useful, minor adjustments expected. set, defined, graphical representation, plotted (4) consideration zero stress, 13. Results analysis components. Ergebnisse Kleinste-Quadrate-Regressionsanalyse Verfestigungskomponente zwei Rückspannungskomponenten. – 568467 4566 79017 817.9 11500 0 calculated strain. Entwicklung berechneten Rückspannungskomponenten Null-Anfangsrückspannung plastischer Dehnung. show imperfections 15. leads deviation 11. include, example, exceeding adjusted sharp corners curve. Moreover, relatively problems. Although deviations small, errors uncertainties identification becomes quite difficult task, Identification deviating 0.02 offset method [15]. Identifizierung elastischen Bereichs Spannungs-Dehnungs-Hysterese-Kurve abweichend idealisierten Form vertikale Linie, 0,02 Offset-Methode appropriate estimation nevertheless, apply line middle rounded corner 14a. intersections peak range. Alternatively, determine applying straight cycle, 14b. Starting upper right corner, end (maximum stress) when begins separate degree line. respect curves, suggested point amplitude, but negative methods cannot uncertainties, especially below separation portions difficult. represent better important note equivalent (accumulated) individually regardless plateau. disappears increases over occurs, represented ). dargestellt berechnete Datenpaare Thus, account (initial softening) subsequently slopes. Describing (10), sets provided. decrease expressed maximum equals difference speed then increase positive required basis, corresponding plotted, 16. Parameter 256 −202 49.56 6613 0.6645 law. 0,4 Voce‘sche Gesetz. difficulties. simultaneous implementation subsequent requires reversal. discontinuity relation consequently stability problems Abaqus. decline causes issues. prevent such problems, (11). makes flexible connected assuming linear them. experimentally conditions 17a. monotonously order point, smoothed calculations. (5), Prager's rule, 17b. experiment, rule. Spannungs-Dehnungs-Beziehung Halbzyklus, Versuch, beschrieben lineare Verfestigungsregel Prager. 18. addition, displayed. sum desired stress-strain-relation cycle. According calculation (18) decomposition complemented hardening. second last well. indicated dashed lines reversal pair before, Resulting isotropic, Resultierende isotropen, den einschließlich angedeuteten Beginns zyklischen Verfestigung. parameters, geometry two-dimensional static general-purpose shell elements type S4R (four-node shell, integration), 19. mesh sensitivity studies, element 1 area, density refined 0.7 mm. boundary correspond setup whereas fixed. effective force cross-section enable simulation, displacement updated Python script iteratively calculate specimen's cross-section. Two-dimensional Zweidimensionales Modell Elektroblechprobe numerischen Beanspruchung. definition, directly presented side steps. step, (18). provided tabular form 1.5 strain, assumes pairs. increment (e. 10−5) considered. calculating information, sequence. ), marks previously behaviour. Then, range, differs those component, available. case, estimate interpolation ranges lie known extrapolated extrapolation treated carefully, generalised smaller larger ranges. Instead extrapolating pairs, for, neglected. shown calibration, 20. each represented. seen basically agreement properly described. predicted. Minor differences explained cycles. Especially overcome affects irregular way. Comparison Versuchsdaten symmetrische dehnungskontrollierte Belastung Bezug Zyklen. already discussed imperfect seems related time-dependent relaxation 17. continues even though apparent deformation, material's exceeded, predicts could 500 MPa, 0.1) 21. contrast condition, Apart this, clearly exceeds (Δ %). estimated, before. unsymmetrische belastungskontrollierte zehn correctly represented, ratchet underestimated. may size. Another reason distinct again deformation. More precisely, creep process stress-control explicit -independent proves location lies curves. (time-dependent) smaller. Chaboche. Special considerations regarding 0.27 Using coated showed consisting capture component. appropriately noticeable underestimation results, works assess proportional part research project “Schwingfestes Elektroblech” funded Federal Ministry Education Research (BMBF) Audi AG. Technische Hochschule Ingolstadt cooperation Otto-von-Guericke-Universität Magdeburg. 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