Proceedings of the ASME 2014 Pressure Vessels & Piping ConferencePVP2014July 20-24, 2014, Anaheim, California, USAPVP2014-28772 ADDITIONAL GUIDANCE FOR INELASTIC RATCHETING ANALYSIS USING THECHABOCHE MODELWilliam F. Weitze, P.E. Structural Integrity Associates, Inc.San Jose, California, USATimothy D. Gilman Structural Integrity Associates, Inc.San Jose, California, USAABSTRACTThis paper builds on PVP2013-98150 by Kalnins, Rudolph, and Willuweit [1], which documented two calibration processes for determining the parameters of the Chaboche nonlinear kinematic hardening (NLK) material model for stainless steel, and tested the material model using a pressurized cylindrical shell subjected to thermal cycling. The current paper examines (1) whether a Chaboche NLK model with only two terms (rather than four as in PVP-98150) is sufficiently accurate, (2) use of the ANSYS program for material model refinement and finite element analysis, and (3) analysis using temperature-dependent NLK model parameters, again using ANSYS.INTRODUCTIONRatcheting is progressive distortion of a component under cyclic duty. Taken to the extreme, it can lead to an unstable component geometry and subsequent collapse. Section III of the ASME Boiler and Pressure Vessel Code contains equations to prevent ratcheting in nuclear reactor components, such as Equations 10, 12, and 13 of NB-3650, for example [2]. Inelastic analysis is used to evaluate ratcheting when it is necessary to remove excess conservatism. When an inelastic analysis is performed, the design is considered acceptable if either shakedown occurs after a few cycles, or the maximum accumulated local strain does not exceed 5% (for certain materials only) [2, NB-3228.4(b)]. However, the ASME Code does not provide guidance as to how the inelastic analysis should be performed.A relatively simple inelastic analysis approach would be to assume elastic-perfectly plastic behavior. However, this approach is still significantly conservative compared to the actual behavior of ductile materials. Work is currently underway to develop more accurate inelastic analysis methodology. The Chaboche NLK material model is sufficiently sophisticated to model ratcheting behavior, but additional work is needed to further its application to real world problems.Paper PVP2013-98150 by Kalnins, Rudolph, and Willuweit [1] provided guidance for ratcheting analysis usingthe Chaboche NLK material model for stainless steel. This paper continues this line of work as described in the abstract. NOMENCLATUREC K= material parameter for the Kth componentE y = modulusofelasticityK = 1 to N, a Chaboche model componentN = number of Chaboche model componentsR p0.2 = 0.2%proofstressα = backstressαK= backstress for the Kth componentαNLK= total backstress from NLK modelγK= material parameter for the Kth componentε= uniaxial engineering strainεp= uniaxial plastic engineering strainεtrue= uniaxial true strainσ = engineeringstressσ0= initial yield stress at the elastic limitσtrue = truestressσuts= ultimate tensile strengthσys = yieldstrengthDEVELOPMENT OF CHABOCHE MODEL PARAMETERSAs in PVP2013-98150, the Chaboche model is based onthe monotonic stress-strain curve obtained from a tension specimen subjected to uniaxial loading [1, Section 3.1]. This is conservative because it neglects the beneficial cyclic hardeningthat occurs with stainless steels. Specific curves are taken from ASME Code Section VIII, Division 2, Annex 3-D, paragraph 3-D.3 [2], for SA-312 TP304 at 400°F as was previously done [1, Section 3.2], as well as the 70°F curve from the same source. Table 1 shows selected properties for these curves.Table 1: Selected Properties of SA-312 TP304at 70°F, in ksi at 70°F,in MPaat 400°F,in ksiat 400°F,in MPaσuts75.0 517.10 64.0 441.26 σys30.0 206.84 20.7 142.72E y28300 195119 26400 182019As before, the true stress-true strain curve from Section VIII-Division 2 is converted to engineering stress-engineering strain for the calibration as follows [1, Section 3.2]:ε = exp(εtrue) – 1 (1)σ = σtrue/(1 + ε) (2)Figure 1 shows the two stress-strain curves after conversion, and Figure 2 shows the portion of the curves used in the current analysis.Figure 1: Engineering Stress-Engineering StrainCurves Based on Section VIII Division 2Figure 2: Curves Up to 5% Engineering StrainAs before, the initial yield stress at the elastic limit, σ0, is estimated as equal to 0.55 times the 0.2% proof stress, R p0.2 [1,Section 3.3]. Table 2 shows R p0.2 and the elastic limit for thetwo curves.Table 2: Determination of Elastic LimitT, °F R p0.2, ksi σ0, ksi70 20.1 11.055400 29.23 16.0765 The backstress is the translation of the yield surface instress space, and is calculated as [1, Section 3.4]:α = σ – σ0(3)Plastic strain is approximated as the total engineering strainminus the strain at the elastic limit. Figure 3 shows the backstress for the two stress-strain curves; the same scales arechosen as in Figure 2 for ease of comparison.Figure 3: Backstress as a Function of Plastic StrainThe Chaboche material model is used to perform a curvefit of the curves in Figure 3 so that ratcheting analyses can be performed. For a Chaboche model of N components, the αNLK backstress is as follows [1, Section 3.4]:αK = (C K/γK)[1 – exp(-γKεp)] (4)αNLK = ∑K(5)There are several differences between the current analysisand the analysis in PVP2013-98150:∙The current analyses use N=2 instead of 4.∙In addition to the analysis using the 400°Fmaterial model, a second run is made using the70°F model.∙ A third run is made using temperature-dependentproperties.∙Additional runs are made by applying the“Abaqus-4” (N=4, developed in ABAQUS) modelfrom PVP2013-98150 [1] in ANSYS, as well asan N=4 model based on properties at 70°F,provided by Prof. Kalnins [3]. These parametersare used for comparative purposes, and are shownin Table 3.Table 3: Chaboche Parameters from ABAQUS21°C 200°C N C, MPa C, psi γC, MPa C, psi γ1 1305 189277 6.75915.53 132788 02 10403 1508851 176.13 8480 1229939175.363 38911 5643651 822.83 34288 4973132823.814 90857 13177899 4272.00 66450 96379084551.1 The procedure for generating the N=2 models in ANSYS isas follows:∙For each curve, the initial values of C1, γ1, C2, and γ2 are chosen as a set of values that yields a curvethat is reasonably close to the α curves shown inFigure 3.∙An additional restriction is that the same values of γ1 and γ2 must be used for both the 70°F and the400°F model when used in a temperature-dependent analysis. This is a limitation of theANSYS program [4].∙Each backstress curve is entered into the ANSYS curve-fitting module (under Preprocessor,Material Models, Structural, Nonlinear, Inelastic,Curve Fitting).∙For each curve, the initial values of C1, γ1, C2, and γ2, as well as a yield strength of zero (appropriatesince these are backstress curves), are entered inthe curve fitting module (under Plasticity,Plasticity, Kinematic Hardening, 2 TermChaboche).∙For γ1, γ2, and yield strength, the Fix box is checked to prevent these parameters from varying.∙The Solve button optimizes the C1 and C2 values, and the Plot button allows a comparison of theinput backstress curve with the calculated curve.Table 4 shows the resulting parameters, and Figures 4 and5 show the comparison of the backstress curves with the ANSYS-generated Chaboche models. These figures show that the two-term Chaboche models appear to be reasonably close to the backstress curves. (In Figures 4 and 5, α is backstress, α1is the first Chaboche term, α2 is the second Chaboche term, and αNLK is the sum of the Chaboche terms.)Table 4: Chaboche Parameters from ANSYS70°F 400°F N C,ksi γC, ksi γ1 13024 950 9114 9502 685.5 40 560 40Figure 4: Comparison of Backstress and ChabocheModel, 70°FFigure 5: Comparison of Backstress and ChabocheModel, 400°FEXAMPLE ANALYSISThe identical Bree cylinder problem from PVP2013-98150 [1, Section 5] is analyzed herein. Specifically, this is a cylindrical shell with a mean radius of 3.740 in and a thickness of 0.394 in under constant pressure of 1595 psi, with an inside surface temperature that varies between 21°C (69.8°F) and 200°C (392°F), and the outside kept at 21°C.The temperature distributions are determined in ANSYS using steady state analysis; that is, no transient effects are considered. A coefficient of thermal expansion of 1.84x10-5 1/°C (1.022x10-5 1/°F) is used [5].As in PVP2013-98150, equivalent plastic strain (εpeq) is used to measure locally accumulated strain [1, Section 5]. In ANSYS, this is the EPPLEQV parameter, which is the equivalent strain based on the plastic strain tensor. The appropriateness of this parameter is confirmed by comparison with results from ABAQUS.Table 5: Comparison of ResultsSoftware, model development Software,finite elementanalysis NParametertempera-ture, °Fεpeq after60 cyclesABAQUS ABAQUS 4 70 0.003156 ABAQUS ANSYS 4 70 0.003035 ANSYS ANSYS 270 0.004965 ABAQUS ABAQUS 4 * 0.005186 ABAQUS ANSYS 4 * ** ANSYS ANSYS 2* 0.008031 ABAQUS ABAQUS 4 400 0.02608 ABAQUS ANSYS 4 400 0.02589 ANSYS ANSYS 2400 0.02928 * Temperature-dependent parameters** Since the γ values were not constant, this model could not be usedin a temperature-dependent ANSYS analysis.DISCUSSIONTable 5 shows that, when using the N=4 Chaboche model without temperature dependency, ANSYS results matched those from ABAQUS quite well. One can conclude that a four-term Chaboche model based on a bounding stress-strain curve (thatis, without material temperature dependency) is adequate for modeling ratcheting behavior.The 400°F two-term Chaboche model produced fairly good results, exceeding the results from ABAQUS by about 12%.The 70°F two-term Chaboche model, as well as the temperature-dependent ANSYS run, did not match the corresponding results from ABAQUS, exceeding them by roughly 50%. Additional runs to determine the cause of this are recommended for future work. (Time was not available to perform this and other recommended work for this paper.) For ratcheting analysis, ANSYS documentation recommends using a three-term Chaboche model, which is calibrated by starting with a stable strain-controlled hysteresis curve, splitting it into three parts, choosing the three components of backstress to represent these three regions, and choosing C1, γ1, C2, γ2, and C3 to match the curve [6, Chapter 32]. However, in the recommended methodology, γ3 cannot be determined from the hysteresis curve, and must be determined from ratcheting data [6, Section 32.3.4] [7]. An exploration ofthis methodology is recommended for future work.The two-term Chaboche models produced conservative results as compared with the four-term models. Additional work involving two- and three-term Chaboche models and temperature dependency should be explored as a means to remove excess conservatism in the analyses, and to further validate these models.ACKNOWLEDGMENTSThe authors gratefully acknowledge Prof. Arturs Kalninsfor his kind assistance in providing model parameters and analysis results. REFERENCES1.Kalnins, A.; Rudolph, J.; Willuweit, A., “Using theNonlinear Kinematic Hardening Material Model of Chaboche for Elastic-Plastic Ratcheting Analysis,”Proceedings of the ASME 2013 Pressure Vessels and Piping Conference, Paper No. PVP2013-98150.2.ASME Boiler and Pressure Vessel Code, 2013 Edition.3.E-mail from A. Kalnins (Lehigh) to W. Weitze (SI) and J.Gregg (Westinghouse) dated 8/14/2013, Subject: “RE: The Final Presentation file”4.E-mail from Rajanikanth Jayaseelan (ANSYS) toXANSYS forum dated 10/11/2013, Subject: “Re: [Xansys] Xansys Digest, V ol 120, Issue 14”5.E-mail from A. Kalnins (Lehigh) to W. Weitze (SI) dated7/24/2013, Subject: “Re: Material Models for Ratchet Analysis”6.Mechanical APDL Technology Demonstration Guide,Release 14.5, ANSYS, Inc., 2013.7.Rezaiee-Pajand, M. and Sinaie, S. “On the Calibration ofthe Chaboche Hardening Model and a Modified Hardening Rule for Uniaxial Ratcheting Prediction.” InternationalJournal of Solids and Structures, V olume 46, Issue 16, pp.3009-3017 (2009).ANNEX ASTRESS-STRAIN CURVES USED IN ANALYSISTable 6: 70°F Curvesσt, ksi εtσ, ksi εεpα, ksi0 0.00000 0.000 0.00000 --- ---2 0.00007 2.000 0.00007 --- ---4 0.00014 3.999 0.00014 --- ---6 0.00021 5.999 0.00021 --- ---8 0.00029 7.998 0.00029 --- ---10 0.00036 9.996 0.00036 --- ---12 0.00044 11.995 0.00044 --- ---14 0.00054 13.992 0.00054 --- --- 16.0765 0.00066 16.066 0.00066 0.00000 0.000 18 0.00079 17.986 0.00079 0.00014 1.920 20 0.00097 19.981 0.00097 0.00032 3.915 22 0.00121 21.973 0.00121 0.00056 5.907 24 0.00153 23.963 0.00153 0.00088 7.897 26 0.00196 25.949 0.00196 0.00131 9.88328 0.00256 27.929 0.00256 0.00190 11.86329.23 0.00303 29.141 0.00304 0.00238 13.075 32 0.00459 31.853 0.00460 0.00395 15.787 34 0.00637 33.784 0.00639 0.00574 17.718 36 0.00907 35.675 0.00911 0.00846 19.609 38 0.01315 37.504 0.01323 0.01258 21.438 40 0.01908 39.244 0.01926 0.01861 23.178 42 0.02703 40.880 0.02740 0.02675 24.814 44 0.03651 42.423 0.03718 0.03652 26.357 46 0.04647 43.911 0.04756 0.04691 27.845Table 7: 400°F Curvesσt, ksi εtσ, ksi εεpα, ksi0 0.00000 0.000 0.00000 --- ---2 0.00008 2.000 0.00008 --- ---4 0.00015 3.999 0.00015 --- ---6 0.00024 5.999 0.00024 --- --- 8.3 0.00036 8.297 0.00036 --- ---10 0.00048 9.995 0.00048 --- ---11.055 0.00057 11.049 0.00058 0.00000 0.000 14 0.00094 13.987 0.00094 0.00037 2.938 14.88 0.00110 14.864 0.00110 0.00052 3.81518 0.00189 17.966 0.00189 0.00132 6.91719.14 0.00232 19.096 0.00232 0.00175 8.04720.1 0.00276 20.045 0.00277 0.00219 8.996 24 0.00606 23.855 0.00608 0.00551 12.806 26 0.00964 25.751 0.00969 0.00911 14.70228 0.01570 27.564 0.01582 0.01525 16.51529.77 0.02364 29.074 0.02392 0.02335 18.026 32 0.03610 30.865 0.03676 0.03618 19.817 34.52 0.05013 32.832 0.05141 0.05083 21.784。
