Chaboche Nonlinear Kinematic Hardening ModelSheldon Imaoka Memo Number: STI0805A ANSYS Release: 12.0.1 May 4, 2008 1 Introduction The Chaboche nonlinear kinematic hardening model was added in ANSYS 5.6 to complement the existing isotropic and kinematic hardening rules that users relied on. Despite its availability for nearly ten years as of the time of this writing, the Chaboche model has enjoyed limited popularity, in part because of the perceived complexity of calibrating the material parameters. This memo hopes to introduce the basics related to the Chaboche nonlinear kinematic model. Please note that many material parameters for the examples shown in this memo were taken from Reference [5]. The author highly recommends obtaining a copy of this book, as it is a useful reference not only for the nonlinear kinematic hardening model but also for plasticity and viscoplasticity in general. 2 2.1 Background on Material Behavior Linear vs. Nonlinear Kinematic Hardening The yield function for the nonlinear kinematic hardening model TB,CHABOCHE is shown below: r 3 F = ({s} − {α})T [M ] ({s} − {α}) − R = 0 (1) 2 In the above equation, which can be found in Section 4 of Reference [2], {s} is the deviatoric stress, {α} refers to the back stress, and R represents the yield stress. The back stress is related to the translation of the yield surface, and there is little in Equation (1) that distinguishes it from other kinematic hardening models at this point. 1 as covered in Subsection 2. where ntemp is the number of temperature sets and n is the number of kinematic models..net Tips Chaboche Nonlinear Kinematic Hardening The back stress {α} for the Chaboche model is calculated as follows: {α} = n X {αi } (2a) i=1 2 1 dCi {∆α}i = Ci {∆ǫpl } − γi {αi }∆ˆ ǫpl + ∆θ{α} 3 Ci dθ (2b) where ǫˆpl is the accumulated plastic strain.CHABOCHE. elastic modulus.5.Sheldon’s ansys. The input consists of defining the elastic properties (e. one may note that the first term is the hardening modulus. while the material parameters for that temperature are entered via the TBDATA command.g. Any temperature-dependent group of constants are preceded with the TBTEMP command defining the temperature.NUXY. The second and third material constants are C1 and γ1 — these may be followed by additional pairs of Ci and γi . Poisson’s ratio) via MP.n. depending on the number n of kinematic models requested. or the yield stress of the material — this value may be overridden if an isotropic hardening model is added. The first constant is R.ntemp. 2 . θ is temperature.. the second term of the evolution of the back stress is a “recall term” that produces a nonlinear effect.EX and MP. and Ci and γi are the Chaboche material parameters for n number of pairs. then issuing TB. On the other hand. In Equation (2b). the parameter C1 will describe the slope of stress versus equivalent plastic strain. while the nonlinear constitutive model fully defines the plastic behavior. as the elastic modulus and Poisson’s ratio completely describe the elastic behavior. these terms describe the relationship between stress and equivalent total strain. However.1 One can also reproduce the same behavior with the bilinear kinematic hardening model (TB.BKIN). where some.CHABOCHE as well as TB.BKIN is based on total strain.Sheldon’s ansys. use equivalent total strain. 1 The term “linear” refers to the relationship of stress and equivalent plastic strain.net Tips 2. The Chaboche model in ANSYS uses equivalent plastic strain. This would represent a linear kinematic hardening model. showing identical results. and others.2 Chaboche Nonlinear Kinematic Hardening Initial Hardening Modulus The material parameter Ci is the initial hardening modulus. if γ1 is set to zero. For a single kinematic hardening model (n = 1). use equivalent plastic strain.PLASTIC. 3 . Figure 1: Linear Kinematic Hardening as shown in Figure 1 — plots of a linear kinematic hardening model with TB. Using a stress value σ ′ > R. whereas Ci in TB. such as TB. the relationship between Ci and Etan is expressed in Equation (3): 1 800 720 640 560 480 400 320 240 160 80 (x10**-3) 0 0 2 1 Etan = σ ′ −R C1 σ′ − R σ′ + Eelastic − R Eelastic 4 3 6 5 (3) It is worth pointing out that there are various plasticity models in ANSYS. such as TB.CHABOCHE is based on equivalent plastic strain.BKIN are superimposed.BKIN. and the author prefers this approach. For ANSYS plasticity models such as “bilinear” or “multilinear” kinematic hardening. one should note that the tangent modulus Etan specified in TB. Despite the Chaboche hardening modulus initially being the same as the linear kinematic model. 1 1 1250 2000 1125 1800 1000 1600 875 1400 750 1200 625 1000 500 800 375 600 250 400 125 200 (x10**-2) 0 0 . controls the rate at which the hardening modulus decreases with increasing plastic strain. C1 = 140. For this case.6 1. Since the limiting value of the back stress {α} is Cγ11 = 370.6 1.8 (b) Larger Strain Figure 2: Comparison of Linear and Nonlinear Kinematic Hardening 4 . which manifests itself as a hardening modulus of zero at large plastic strains. One can see the hardening modulus for the Chaboche model decreasing to zero.2 1 1.3 Chaboche Nonlinear Kinematic Hardening Nonlinear Recall Parameter The second material parameter.Sheldon’s ansys. A comparison of TB. 600 while γ1 = 380.7 (x10**-2) 0 1 0 . For the Chaboche model. γi .2 (a) Small Strain . which matches with the results shown in Figure 2(b). By examining Equation ˙ becomes lower as plastic (2b). Figure 2(b) is the same plot but at larger strains.1 . including a nonzero γ1 parameter. Specifically.6 .5 .net Tips 2. the rate of which is defined by γi .4 2 1.4 .3 .4 . the yield stress was assumed to be 520 MPa.8 . indicating that the yield surface cannot translate anymore. the hardening modulus decreases.9 . a limiting value of Ci /γi exists.2 . one can see that the back stress increment {α} strain increases.BKIN with the Chaboche model.8 . is shown in Figure 2(a). one would expect that the asymptotic value would be R + α = 890 MPa. 8 . a single kinematic model may not be sufficient to describe the complex response of a given material.2 1 1.Sheldon’s ansys. This superposition of simple modFigure 3: Two Kinematic Models els is another salient feature of the Chaboche model.8 Combined Hardening The main usage of the Chaboche model is for cyclic loading applications. as will be discussed shortly.MISO: Multilinear isotropic hardening.4 . For example. and the elastic domain remains unchanged. discussed in Subsections 2.5 . respectively. Similar to other kinematic hardening models in ANSYS. The Chaboche kinematic hardening model. can be used in conjunction with an isotropic hardening model. Figure 3 shows a comparison of three results of stress versus plastic strain — single kinematic models “CHAB 1A” and “CHAB 1B” are plotted with a two-model version “CHAB 2”. However.2 and CHAB_1B 2.6 1. C1 and CHAB_1A γ1 . however. and the user can select from one of the following: • TB. where the user specifies up 5 . so with TB.3. where σmax is the maximum stress prior to unloading — this is the well-known Bauschinger effect. when the loading is reversed for a simple tensile specimen. The “CHAB 2” model contains the same material parameters as “CHAB 1A” and “CHAB 1B. The isotropic hardening model describes the change in the elastic domain for the material.CHABOCHE.4 Chaboche Nonlinear Kinematic Hardening Multiple Kinematic Hardening Models A single nonlinear kinematic hardening model is described by the CHAB_2 two material parameters.BISO: Bilinear isotropic hardening. up to n = 5 kinematic models may be superimposed. yielding is assumed to occur at σmax − 2R.net Tips 2.” 1 2000 1800 1600 1400 1200 1000 800 600 400 200 (x10**-2) 0 0 .4 2 1. the yield surface translates in principal stress space.2 2.6 1. where the user specifies the yield stress and tangent modulus (with respect to total strain) • TB. and this idea will be expanded upon in the next subsection.” and one may be able to visualize how these two kinematic models are superimposed (assuming constant yield stress R) to create the more complex stress-strain response of “CHAB 2. 1 (a) Voce Hardening . and “VOCE 2” is the complete model..5 . proportional loading.3 ..5 2 1. where R = k + Ro ǫˆpl + pl R∞ 1 − e−bˆǫ . “VOCE 1B” is the asymptotic term. plastic strain data points • TB.NLISO. Ro . Material parameters k. Compare Figure 4(a) with Figure 3.. R∞ . total strain data points • TB. γ2 = 0.MISO: Nonlinear plasticity.5 3 2.Sheldon’s ansys. The initial yield stress k should be defined the same for both.4 .” sample models are compared in Figure 4(b).9 (b) Voce and Chaboche Figure 4: Combined Hardening The Voce hardening law can be thought ofas containing two terms — a pl −bˆ ǫ pl linear term Ro ǫˆ and an asymptotic term R∞ 1 − e .6 . and for the Chaboche model. R∞ = Cγ11 describes the limiting. in order to better understand the material parameters. where “VOCE 1A” is the linear term.7 1 . It may be useful to compare the Voce and two-term nonlinear kinematic hardening models further for monotonic loading situations. where the user specifies up to 100 stress vs. the present discussion comparing the two constitutive models is meant to aid the reader in obtaining a better 6 . A sample model is shown in Figure 4(a).5 (x10**-2) 0 5 0 4. asymptotic value that is added to the initial yield stress k.PLASTIC.. 1 1 400 2000 VOCE_2 360 1800 320 1600 280 VOCE_1A 1400 240 VOCE_1B 1200 200 1000 160 800 120 600 80 400 40 200 (x10**-2) 0 0 1 . and one can see that the stressstrain response is the same. The hardening modulus at very large strains is defined by Ro = C2 .VOCE: Voce hardening law. Two “equivalent. Of course.2 ..8 . and note the similarities for the case of monotonic. The rate at which this decays is defined by b = γ1 .net Tips Chaboche Nonlinear Kinematic Hardening to 100 stress vs.5 . and b are supplied by the user.5 4 3.. For cyclic hardening. The combination of kinematic and isotropic hardening in the Chaboche references2 typically use a form similar to the Voce hardening law but without the constant hardening modulus term: pl R = k + Q 1 − e−bˆǫ (4) To understand how this isotropic hardening term is used. and b. as the elastic domain does not keep increasing indefinitely. R∞ = Q... [3].CHABOCHE) assumes a von Mises yield surface by default. To use calibrated material constants. please keep in mind that the value of R∞ = Q is typically a function of the strain range. Note that with progressive number of cycles. The Hill anisotropic yield function (TB. Also. Also. the change in the elastic domain stabilizes. so the material parameters should be reflective of the expected operational strain ranges. consider the case of a strain-controlled loading of a test specimen. [4]. although details of this will not be covered in the present memo.CHABOCHE.VOCE) with the Chaboche nonlinear kinematic hardening model — use of other isotropic hardening models is permitted.. the elastic domain expands — this difference in the change in the yield stress is described by the addition of Figure 5: Cyclic Hardening the isotropic hardening term. The user is not limited to using the Voce hardening law (TB. 1 2000 1600 1200 800 400 0 -400 -800 -1200 -1600 -2000 0 8 4 2. the stress response of a sample model is shown in Figure 5.Sheldon’s ansys. note that with more cycles.net Tips Chaboche Nonlinear Kinematic Hardening understanding of the material parameters rather than to incorrectly imply that the two models are interchangeable for cyclic loading applications. 2 See Refs. and [5] 7 . simply specify k.NLISO.6 16 12 24 20 32 28 40 36 Anisotropic Yield Function It is worth pointing out that the Chaboche model in ANSYS (TB. This is why the Ro term is neglected for many situations.HILL) may be used in conjunction with TB. with ±ǫmax loading. 42%. the slope after the yield stress can be taken as an estimate of C1 .Sheldon’s ansys. For a given test. straincontrolled tests can be used to obtain the material parameters k. estimate the asymptotic value corresponding to Cγ11 . 520 Figure 6: Strain-Controlled Test MPa. 3.001 0. C1 .0005 0.0015 -0. Determine the yield stress k from the elastic domain (k is usually half the elastic domain size). tension-compression tests of symmetric.0015 0. Using the expression ∆σ 2 − k = γ1 tanh(γ1 2 ).0005 0 0. Since C1 is the initial hardening modulus.1 Example Using Several Stabilized Cycles An example of stress versus plastic strain is shown in Figure 6. the Cyclic Test 1000 800 600 400 Stress 200 0 -0. cyclic tests are performed. 2. 3. Looking at the stabilized hysteresis loop. ∆ǫpl 2 2 = 0.0025 -0. and γ1 . The following steps (taken from the above reference) can be used to determine the material parameters: 1. so the yield stress k can be estimated as 530 MPa. an initial value of γ1 can be obtained. determine the plastic strain range ∆ǫpl . pl ∆ǫ 4. This can be done. fit the results to solve for C1 and γ1 . 060 MPa. When two additional strain-controlled. For a given test. The stabilized hysteresis loops corresponding to different strain amplitudes should be used. The plastic strain range ∆ǫpl is estimated to be 2×.net Tips 3 Chaboche Nonlinear Kinematic Hardening Calibration of Single Model Per page 226 of Reference [5]. The stress range ∆σ is around 2 × 760 = 1. determine the stress range ∆σ. one can see that the elastic domain is roughly (300 + 760) = 1.0025 -200 -400 -600 -800 -1000 Plastic Strain following sets of data were obtained: (a) k2 = 500.002 -0.001 -0. pl C1 ∆ǫ 5. in Microsoft Excel using the Solver Add-In. for example. 8 ∆σ2 2 = 840.37% . Note that curve-fitting procedures often benefit from reasonable initial values.21% = 0. where the test specimen is loaded ±ǫa . Plotting ∆σ 2 −k against 2 for the multiple tests. and through the relation of Cγ11 determined in Step 4.002 0. material indentification can still be performed. and k3 values. 3. and the parameters could be refined further. with the abscissa being ∆ǫ2 and the ordinate being ∆σ 2 −k. the asymptotic value was estimated to be ≤ 400 MPa. so γ1 = 410 — these were used as starting values. All three data points are pl plotted as red dots in Figure 7. Using Step 4 in Section 3.009 0.008 0. The constants C1 = 122.000 MPa. From Figure 6. 000 MPa and γ1 = 314 were then obtained — the curve-fit data matches reasonably well with the presented data.001 0. k2 . although the derived parameters are best suited for that particular strain range.net Tips Chaboche Nonlinear Kinematic Hardening Curve-Fit Data 450 400 350 ∆σ/2 .Sheldon’s ansys.k 300 250 Data from 3 Test Curve-Fit Parameters 200 150 100 50 0 0 0.2 Example Using Single Stabilized Cycle If only data from a single strain-controlled test is available.5625%. 9 .006 0. 2 = 0. The yield stress k was estimated to be 530 MPa based on k1 . if needed. the initial hardening modulus C1 was estimated as 164.01 ∆εpl/2 Figure 7: Curve-Fit Data ∆ǫpl 3 3 and (b) k3 = 540.002 0. ∆σ 2 = 900.004 0.003 0.005 0.007 0. Stress Points There are two items that need adjustment: the plastic strain values ǫpl should start from zero. then subtract the stress by the yield stress. The considerations noted earlier about using the slope after yielding as the initial value of C1 .0042 αi (MPa) -230 -50 60 150 230 Table 2: Corrected Plastic Strain vs.0 0.0021 -0.1 that this was determined to be 1060 MPa). and the stress needs to be converted to the back stress {α}.Sheldon’s ansys. αi = 10 .001 0.0011 0.001 0. assume that stress and plastic strain points for the stabilized loop are tabulated as follows: ǫpl i -0. as these will aid any curve-fitting procedure.0 0. pl C1 −γ ǫ stress relation αi = γ1 1 − e 1 i . still apply. along with estimating an initial value of γ1 from the asymptotic value. the first data point is used in the following equation to solve for the other pairs: pl pl C1 1 − e−γ1 ǫi + α1 e−γ1 ǫi for i > 1 (5) γ1 One can use Microsoft Excel’s Solver Add-In or any other means to perform a fit to determine C1 and γ1 . The resulting values are shown below: ǫpl i 0.0021 σi (MPa) 300 480 590 680 760 Table 1: Sample Plastic Strain vs.0021 0. although this expression assumes zero initial back stress.net Tips Chaboche Nonlinear Kinematic Hardening Using the example shown in Figure 6.0031 0. or half the value of the elastic domain. Back Stress To determine the C one may consider using the back 1 and γ1 constants. For the back stress. Shift the plastic strain values such that the first point starts from zero. determine the elastic range (recall from Subsection 3. Since the model contains non-zero back stress. Stress for First Curve As with the situation presented in Subsection 3. Plastic strain and stress data will be estimated from the first curve of Figure 6.1. 500 MPa and γ1 = 280. {αi } values for each data point can be matched against the following equation: αi = pl C1 1 − e−γ1 ǫi γ1 (6) The user can solve for values of Ci and γi — the same procedure to obtain initial values and to perform the curve-fit as explained above still apply for the uniaxial tension case.0022 σi (MPa) 520 580 630 680 720 Table 3: Plastic Strain vs. and (b) the curve-fit data obtained from several stabilized cycles should be more representative of behavior over a wider strain range.0015 0. As an alternative or supplement to the option presented in Subsection 3. 000 MPa and γ1 = 314 obtained in Subsection 3. For this particular set of data that 11 . which would represent a single tensile loading case: ǫpl i 0. and this scalar value should be subtracted from the stress values in Table 3.2% offset).Sheldon’s ansys. the author obtained values of C1 = 120.0 0. 3.2.1.3 Example Using Single Tension Curve A user may only have data from a single tension test but may wish to use the nonlinear kinematic hardening model. The yield surface should be determined (for example. such an approach may be suitable for situations dealing with a few cycles. visual estimates. While these values differ slightly from C1 = 122. please note that (a) the author did not use a digitizer to determine the data points but used rough. by using 0.net Tips Chaboche Nonlinear Kinematic Hardening For this example. the stress needs to be converted to the back stress. Then.0005 0.001 0. the user may also use the approach outlined in this subsection to manually obtain C1 and γ1 coefficients for various strain ranges to better understand the variation of the material parameters that may be present. While this is not recommended since there is no cyclic test data with which to correlate the material parameters. This is accessed via adding TB. despite some variation in the C1 and γ1 values. the parameters derived from multiple stabilized cycles would be expected to give the best correlation for different strain ranges.001 0. a rate-dependent form of the Chaboche model was introduced.1 and 3.2. 4 Cyclic Test 1000 800 600 400 Stress 200 0 -0. As noted above..RATE. With a simple model. 3.001 -0.0025 TEST MULT_STAB SING_STAB UNIAX -200 -400 -600 -800 -1000 Plastic Strain Figure 8: Comparison of Data Rate-Dependent Version In ANSYS 12.CHABOCHE definition without isotropic hardening. the results for this strain range match reasonably well for all three cases.0.0015 -0. the lack of cyclic test data prevents validation of the derived material parameters for general cyclic applications. the author obtained C1 = 135.0015 0.002 -0. As one can see. the user can see what the numerical cyclic behavior will be for the set of material parameters.0005 0.002 0. the author strongly recommends creating a simple one-element model to simulate the cyclic behavior of the calculated material paramters — in this way..0025 -0. However.4 Comparison of Results Figure 8 shows the original data compared with the three sets of calculated material parameters.0005 0 0. The use of the single uniaxial test data for this specific example resulted in higher estimates of C1 and γ1 compared with those obtained in Subsections 3. The rate-dependent additions to the Chaboche model (Equations (1) and (2)) are shown below: ǫ˙pl = σ−R K 1 m (7) R = K0 + R0 ǫˆpl + R∞ 1 − e−bˆǫpl 12 (8) . although the asymptotic value Cγ11 = 356 MPa is actually the smallest of the three.net Tips Chaboche Nonlinear Kinematic Hardening was approximated from the first curve of Figure 6. the user may also vary the loading for larger strain ranges in order to understand the response in strain ranges for which no test data is present. 300 MPa and γ1 = 380.Sheldon’s ansys.CHABOCHE in conjunction with the regular TB.. Consequently.1. CHABOCHE are K0 . including [1]. Theory Reference for ANSYS and ANSYS Workbench 11. First edition. From looking at the above equations. These 6 material constants are input via the TBDATA command. Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity. 2009): Updated for 12. one can see that the rate-dependent Chaboche model incorporates a strain-rate dependent term (similar to Peirce or Perzyna options) with Voce hardening. and b (isotropic hardening Equation (8)) while the last 2 constants are m and K (rate-dependent Equation (7)). mean stress relaxation. the information in this memo may serve as a starting point for users wishing to combine multiple nonlinear kinematic hardening models or to define combined (isotropic and kinematic) hardening laws. International Journal of Plasticity. 1989. 2005.0. Inc.1. Advanced Structural Nonlinearities. 5 Conclusion Background information on the Chaboche material model was presented in this memo. added Section 4.0. Revisions to this Document • STI0805A (November 14.0. [2] ANSYS.. shakedown. While phenomena such as rachetting.RATE. References [1] ANSYS. The reader is encouraged to review the cited references. for additional details on cyclic plasticity and its definition and usage in ANSYS. [3] Jean-Louis Chaboche. Inc. R∞ . Added additional information concerning input in Section 1.. R0 . 13 . Inventory Number: 002205. ANSYS 9. 5:247–302.net Tips Chaboche Nonlinear Kinematic Hardening where the first 4 constants for TB. 2007.. and cyclic softening were not introduced. Updated Equation (2).Sheldon’s ansys. along with a basic discussion of the calibration of the material constants for a single nonlinear kinematic hardening model. 1990. 14 . 7:661–678. Mechanics of Solid Materials. International Journal of Plasticity.Sheldon’s ansys. On Some Modifications of Kinematic Hardening to Improve the Description of Ratchetting Effects.net Tips Chaboche Nonlinear Kinematic Hardening [4] Jean-Louis Chaboche. English edition. Cambridge University Press. 1991. [5] Jean Lemaitre and Jean-Louis Chaboche. 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