PVP2014
July 20-24, 2014, Anaheim, California, USA
PVP2014-28772
ADDITIONAL GUIDANCE FOR INELASTIC RATCHETING ANALYSIS USING THE
CHABOCHE MODEL
William F. Weitze, P.E.
Structural Integrity Associates, Inc.
San Jose, California, USA
ABSTRACT
This 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.
INTRODUCTION
Ratcheting 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.
Timothy D. Gilman
Structural Integrity Associates, Inc.
San Jose, California, USA
Paper PVP2013-98150 by Kalnins, Rudolph, and Willuweit [1] provided guidance for ratcheting analysis using the Chaboche NLK material model for stainless steel. This paper continues this line of work as described in the abstract. NOMENCLATURE
= material parameter for the Kth component CK
modulus of elasticity Ey =
K = 1 to N, a Chaboche model component N = number of Chaboche model components
proof stress Rp0.2 = 0.2%
= backstress
= backstress for the Kth component K
NLK = total backstress from NLK model
= material parameter for the Kth component K
= uniaxial engineering strain
= uniaxial plastic engineering strain p
true = uniaxial true strain = engineering stress 0 = initial yield stress at the elastic limit
stress true = true
= ultimate tensile strength uts
yield strength ys =
DEVELOPMENT OF CHABOCHE MODEL PARAMETERS
As in PVP2013-98150, the Chaboche model is based on the 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 hardening that 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.
1 Copyright © 2014 by ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/25/2014 Terms of Use: http://asme.org/termsTable 1: Selected Properties of SA-312 TP304 at 70°F, at 70°F, at 400°F, at 400°F, in ksi in MPa in ksi in MPa uts 75.0 517.10 64.0 441.26 ys 30.0 206.84 20.7 142.72 Ey 28300 195119 26400 182019 As 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]: (1) = exp(true) – 1 (2) = true/(1 + ) Figure 1 shows the two stress-strain curves after conversion, and Figure 2 shows the portion of the curves used in the current analysis. As before, the initial yield stress at the elastic limit, 0, is estimated as equal to 0.55 times the 0.2% proof stress, Rp0.2 [1, Section 3.3]. Table 2 shows Rp0.2 and the elastic limit for the two curves. Table 2: Determination of Elastic Limit T, °F Rp0.2, ksi 0, ksi 70 20.1 11.055 400 29.23 16.0765 The backstress is the translation of the yield surface in stress space, and is calculated as [1, Section 3.4]: (3) = – 0 Plastic strain is approximated as the total engineering strain minus the strain at the elastic limit. Figure 3 shows the backstress for the two stress-strain curves; the same scales are chosen as in Figure 2 for ease of comparison. Figure 1: Engineering Stress-Engineering Strain Curves Based on Section VIII Division 2 Figure 3: Backstress as a Function of Plastic Strain Figure 2: Curves Up to 5% Engineering Strain The Chaboche material model is used to perform a curve fit of the curves in Figure 3 so that ratcheting analyses can be performed. For a Chaboche model of N components, the NLK backstress is as follows [1, Section 3.4]: K = (CK/K)[1 – exp(-K p)] (4) NLK = ∑ (5) K There are several differences between the current analysis and the analysis in PVP2013-98150: The current analyses use N=2 instead of 4. In addition to the analysis using the 400°F material model, a second run is made using the 70°F model. A third run is made using temperature-dependent properties. 2 Copyright © 2014 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/25/2014 Terms of Use: http://asme.org/terms Additional runs are made by applying the “Abaqus-4” (N=4, developed in ABAQUS) model from PVP2013-98150 [1] in ANSYS, as well as an N=4 model based on properties at 70°F, provided by Prof. Kalnins [3]. These parameters are used for comparative purposes, and are shown in Table 3. Table 3: Chaboche Parameters from ABAQUS 21°C 200°C N C, MPa C, psi C, MPa C, psi 1 1305 189277 6.75 915.53 132788 0 2 10403 1508851 176.13 8480 1229939 175.36 3 38911 5643651 822.83 34288 4973132 823.81 4 90857 13177899 4272.00 66450 9637908 4551.1 The procedure for generating the N=2 models in ANSYS is as follows: For each curve, the initial values of C1, 1, C2, and 2 are chosen as a set of values that yields a curve that is reasonably close to the curves shown in Figure 3. An additional restriction is that the same values of 1 and 2 must be used for both the 70°F and the 400°F model when used in a temperature-dependent analysis. This is a limitation of the ANSYS 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, 1, C2, and 2, as well as a yield strength of zero (appropriate since these are backstress curves), are entered in the curve fitting module (under Plasticity, Plasticity, Kinematic Hardening, 2 Term Chaboche). For 1, 2, 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 the input backstress curve with the calculated curve. Table 4 shows the resulting parameters, and Figures 4 and 5 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, is backstress, 1 is the first Chaboche term, 2 is the second Chaboche term, and NLK is the sum of the Chaboche terms.) Table 4: Chaboche Parameters from ANSYS 70°F 400°F N C, ksi C, ksi 1 13024 950 9114 950 2 685.5 40 560 40 Figure 4: Comparison of Backstress and Chaboche Model, 70°F Figure 5: Comparison of Backstress and Chaboche Model, 400°F EXAMPLE ANALYSIS The 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 (peq) 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. 3 Copyright © 2014 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/25/2014 Terms of Use: http://asme.org/termsREFERENCES
Software, Software, Parameter 1. Kalnins, A.; Rudolph, J.; Willuweit, A., “Using the model finite element tempera-peq after Nonlinear Kinematic Hardening Material Model of
60 cycles development analysis N ture, °F Chaboche for Elastic-Plastic Ratcheting Analysis,”
ABAQUS ABAQUS 4 70 0.003156 Proceedings of the ASME 2013 Pressure Vessels and ABAQUS ANSYS 4 70 0.003035
Piping Conference, Paper No. PVP2013-98150.
ANSYS ANSYS 2 70 0.004965 ABAQUS ABAQUS 4 * 0.005186 2. ASME Boiler and Pressure Vessel Code, 2013 Edition.
3. E-mail from A. Kalnins (Lehigh) to W. Weitze (SI) and J. ABAQUS ANSYS 4 * **
Gregg (Westinghouse) dated 8/14/2013, Subject: “RE: The ANSYS ANSYS 2 * 0.008031
Final Presentation file” ABAQUS ABAQUS 4 400 0.02608
ABAQUS ANSYS 4 400 0.02589 4. E-mail from Rajanikanth Jayaseelan (ANSYS) to ANSYS ANSYS 2 400 0.02928 XANSYS forum dated 10/11/2013, Subject: “Re: [Xansys] * Temperature-dependent parameters Xansys Digest, Vol 120, Issue 14” ** Since the values were not constant, this model could not be used 5. E-mail from A. Kalnins (Lehigh) to W. Weitze (SI) dated in a temperature-dependent ANSYS analysis. 7/24/2013, Subject: “Re: Material Models for Ratchet
Analysis”
6. Mechanical APDL Technology Demonstration Guide,
DISCUSSION Release 14.5, ANSYS, Inc., 2013.
Table 5 shows that, when using the N=4 Chaboche model 7. Rezaiee-Pajand, M. and Sinaie, S. “On the Calibration of without temperature dependency, ANSYS results matched those the Chaboche Hardening Model and a Modified Hardening from ABAQUS quite well. One can conclude that a four-term Rule for Uniaxial Ratcheting Prediction.” International Chaboche model based on a bounding stress-strain curve (that Journal of Solids and Structures, Volume 46, Issue 16, pp. is, without material temperature dependency) is adequate for 3009-3017 (2009). 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, 1, C2, 2, and C3 to match the curve [6, Chapter 32]. However, in the recommended methodology, 3 cannot be determined from the hysteresis curve, and must be determined from ratcheting data [6, Section 32.3.4] [7]. An exploration of this 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.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Prof. Arturs Kalnins for his kind assistance in providing model parameters and analysis results.
Table 5: Comparison of Results
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Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 11/25/2014 Terms of Use: http://asme.org/termsANNEX A
STRESS-STRAIN CURVES USED IN ANALYSIS
Table 6: 70°F Curves
t, ksi t , ksi 0 0.00000 0.000 2 0.00007 2.000 4 0.00014 3.999 6 0.00021 5.999 8 0.00029 7.998 10 0.00036 9.996 12 0.00044 11.995 14 0.00054 13.992 16.0765 0.00066 16.066 18 0.00079 17.986 20 0.00097 19.981 22 0.00121 21.973 24 0.00153 23.963 26 0.00196 25.949 28 0.00256 27.929 29.23 0.00303 29.141 32 0.00459 31.853 34 0.00637 33.784 36 0.00907 35.675 38 0.01315 37.504 40 0.01908 39.244 42 0.02703 40.880 44 0.03651 42.423 46 0.04647 43.911
p , ksi
0.00000 --- --- 0.00007 --- --- 0.00014 --- --- 0.00021 --- --- 0.00029 --- --- 0.00036 --- --- 0.00044 --- --- 0.00054 --- --- 0.00066 0.00000 0.000 0.00079 0.00014 1.920 0.00097 0.00032 3.915 0.00121 0.00056 5.907 0.00153 0.00088 7.897 0.00196 0.00131 9.883 0.00256 0.00190 11.863 0.00304 0.00238 13.075 0.00460 0.00395 15.787 0.00639 0.00574 17.718 0.00911 0.00846 19.609 0.01323 0.01258 21.438 0.01926 0.01861 23.178 0.02740 0.02675 24.814 0.03718 0.03652 26.357 0.04756 0.04691 27.845
t, ksi 0 2 4 6 8.3 10 11.055 14 14.88 18 19.14 20.1 24 26 28 29.77 32 34.52
Table 7: 400°F Curves
t , ksi 0.00000 0.000 0.00008 2.000 0.00015 3.999 0.00024 5.999 0.00036 8.297 0.00048 9.995 0.00057 11.049 0.00094 13.987 0.00110 14.864 0.00189 17.966 0.00232 19.096 0.00276 20.045 0.00606 23.855 0.00964 25.751 0.01570 27.564 0.02364 29.074 0.03610 30.865 0.05013 32.832
p , ksi 0.00000 --- --- 0.00008 --- --- 0.00015 --- --- 0.00024 --- --- 0.00036 --- --- 0.00048 --- --- 0.00058 0.00000 0.000 0.00094 0.00037 2.938 0.00110 0.00052 3.815 0.00189 0.00132 6.917 0.00232 0.00175 8.047 0.00277 0.00219 8.996 0.00608 0.00551 12.806 0.00969 0.00911 14.702 0.01582 0.01525 16.515 0.02392 0.02335 18.026 0.03676 0.03618 19.817 0.05141 0.05083 21.784
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