improved low cycle fatigue analysis for nickle based
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Improved low cycle fatigue analysis for Nickle basedturbine nozzles
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Citation for published version (APA):Maggiani, G., Roy, M., & Withers, P. (2018). Improved low cycle fatigue analysis for Nickle based turbine nozzles.In ASME TurboExpo 2018: Turbomachinery Technical Conference amd Exposition (Vol. 7A: Structures andDynamics). American Society of Mechanical Engineers.
Published in:ASME TurboExpo 2018
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Download date:14. Jan. 2022
DRAFT - IMPROVED LOW CYCLE FATIGUE ANALYSIS FOR NI-BASED TURBINENOZZLES
Gianluca Maggiani∗
BHGEvia F. Matteucci 2, Florence
50127 - ItalyEmail: [email protected]
Matthew J. RoySchool of Mechanical, Aerospace and Civil Engineering
University of ManchesterManchester, UK, M13 9PL
Email: [email protected]
Simone ColantoniBHGE
via F. Matteucci 2, Florence50127 - Italy
Email: [email protected]
Philip J. WithersSchool of Materials
University of ManchesterManchester, UK, M13 9PL
Email: [email protected]
ABSTRACTThe requirements for cleaner energy have driven in-
dustrial gas turbines manufacturers to increase firing tem-peratures and improve cooling of nozzles. The applica-tion of high temperature alloys having adequate thermo-mechanical requirements is critical, as assessed by low cy-cle fatigue performance. The effect of higher firing tem-peratures combined with higher cooling efficiencies haslead to operating cycles where the level of plastic strainimparted define component life. The capability of ma-terial models to account for non-linear effects such asratchetting or shakedown, cyclic hardening or softening aswell as Bauschinger or relaxation effects have been high-lighted in this context. Neglecting these effects can leadto over and under-conservative life assessment analysis,while accounting for them using standard multilinear ma-terial models lead to convergence issues in finite elementanalysis. In this paper, Chaboche viscoplastic model hasbeen applied to a transient structural of a first stage gasturbine nozzle. Fitting of the model based on experimen-
∗Address all correspondence to this author.
tal mechanical test data on MAR-M-247 alloy will be de-scribed, followed by an overview of how the model maybe implemented to a benchmark nozzle thermo-mechanicaltransient analysis. Finally the details how the Chaboche-type model has provided up to 50% decrease in computa-tion time when compared to using a standard multi-linearmaterial modelling approach.
INTRODUCTIONIndustrial gas turbine design has seen numerous devel-
opments in the last decade, with advances comprising bothmaterials employed as well as design optimization leadingto more efficient and higher performance. These changeshave been driven by environmental requirements for loweremissions. Recent improvements allowing low NOx con-figurations include the materials available, better coolingof high temperature hot gas path components, increasedfiring temperatures driven by higher pressure ratios. In-dustrial gas turbines are now seeing different applicationscompared to the past, mostly driven by mechanical driverunits (gas turbine installed mainly in oil and gas pipelines
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as centrifugal compressors drivers). Nowadays, industrialgas turbines are integrated into power generation grids aswell as providing back-up solutions for renewable energyplants to allow for supply oscillations within the grid it-self. These new applications have modified life require-ments and assessments of gas turbine components, espe-cially those lying on the hot gas path. This section of thegas turbine now needs to withstand higher temperatures andmore severe cyclic loading leading to thermo-mechanicalfatigue. Nickel/cobalt-based superalloys are predominantlyemployed, such as MAR-M-247 and Rene 41. Within thisframework the capability to assess the life of gas turbinecomponents in a reliable way has become more and morecritical, due to the complexities arising from the higherloads, higher temperatures and cyclic loads with dwell ef-fects leading to aggressive low cycle fatigue (LCF).
Start-up and shut-down cycling during operation canresult in crack initiation due to LCF and subsequentcrack growth under non-linear, strain-controlled condi-tions, which for nozzles, may decrease engine perfor-mance, increase cost of repair and could possibly pro-duce risk of forced outage. Standard LCF characterisationmethodology makes use of isothermal strain range versuscyclic life data in the calculation of the number of cycles tofailure. However, LCF in nozzles is typically produced as aresult of thermally-driven stress cycles. In these cycles, thestress range is generated by continuous changes in metaltemperature throughout the start-up and shutdown portionsof the operating cycle. Thermal stresses are created bymetal temperature variation within a connected structure,where the thermal expansion of hot material is constrainedby regions of cooler material. Predominantly in simplysupported nozzles, applied mechanical stresses are a sec-ondary contributor to LCF. Cantilevered nozzles may havehigher contribution from mechanical stresses in LCF anal-ysis by virtue of complex 3D stress fields that vary through-out the operating range, see Fig. 1.
It is therefore necessary to develop analysis techniquesto characterize the material under specific loading condi-tions based on a transient analysis according to operabilitycurves and embed them in a finite element analysis (FEA)framework. Furthermore, there is also scope to address thecalculation methodology by implementing different mate-rial models within an LCF life assessment, that can accountfor secondary non-linear effects, such as ratchetting, shake-down, hardening or softening or relaxation effects as well
FIGURE 1. Example of the two different nozzles design. Onthe left figure, a simply supported nozzle is shown. On the rightpicture a cantilevered nozzle configuration is depicted.
as Bauschinger effects.Nickel/cobalt superalloys will be first discussed and
relevant data on MAR-M-247 will be presented. Anoverview on different material models for most commonFEA tools will be discussed in context of this data and willdiscuss the capabilities of each. The Chaboche unified vis-coplasticity theory will be introduced, providing the capa-bility for handle nonlinear kinematic hardening and creepeffects within a unique low cycle fatigue material model.The Chaboche parameters will be evaluated for MAR-M-247 and applied to a transient analysis. A methodologyfor fitting the model with available monotonic stress-straincurves at different temperatures will be demonstrated.
NICKEL/COBALT-BASED SUPERALLOYSNickel/cobalt-based superalloys are widely used in gas
turbines where high temperature resistance, toughness andhigh stress and strain resistance are expected. MAR-M-247 have been used for applications where average metaltemperatures are 70% of the alloys melting temperature.
Blades and nozzles in gas turbines are mostly manufac-tured by investment casting, allowing for control of grainsize and alloy microstructure. Both of them are funda-mentals for providing good mechanical behaviour. Grainboundaries are critical locations for crack nucleation andinitiation, and single crystal alloys are mostly used in firststages of high pressure turbines, while equiaxed alloys areused in the later stages. Face-centered cubic (FCC) nickelis the primary structure of these alloys, while other 5 to 10chemical elements constitute up to 40% in weight of the
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alloy, as is the case for one of the more commonly speci-fied variants: MAR-M-247. Mechanical performance dataon MAR-M-247 are available in the open literature, mostrecently from Brindley, Halford, Kaufman and Martin-Meizoso [1], [2], [3] and [4]. Tensile properties are moststrongly related to γ’ precipitates and the strength of the γ
matrix, see Fig. 2.Yield, as well as tensile strength are inversely propor-
tional to the size of γ’ precipitates. Unlike many other al-loys, the yield stress does not decrease as it would be ex-pected with increasing temperatures: an increase in yieldstrength is instead appreciable to a temperature of∼ 800◦C,after which a large drop in yield can be observed, as perFig. 3. This behaviour is related to the strength of the γ’phase which improves with temperature up to that point.Liao and Gasko [5] and [6] have studied the influence onthe microstructure on the MAR-M-247 mechanical prop-erties and the influence of the eutectic γ-γ’ phase. Fig. 3depicts these aspects.
For what has been depicted in this section, MAR-M-247 in its applications at high temperatures, as for first stagenozzles, will show incipient plastic effects which will tendto be more and more important within a small incrementin temperature or stress. This is a very important aspect toreproduce in any assessment and that can have large impacton component life predictions.
CONSTITUTIVE MODELSThe elasto-plastic materials behaviour as effect of ex-
ternal loads has been widely studied in materials science,leading to different constitutive models. Most models areaccurate enough to predict stresses and strains in case ofmonotonic loading and in case of isotropic or kinematichardening rules. However, to account for cyclic plasticity,two approaches have been used. The first one, has the hy-pothesis that the actual state of material only depends on thepresent value of observable variables and adopts a furtherset of internal-state variables. The other approach insteadaccounts for the time history of the component.
Whenever local stress is below the yield limit, then therelation between local stress and local strains can be con-sidered to be rate independent, meaning that the materialbehaviour is still elastic and Hookes law is valid. Oncethis limit is crossed, a certain amount of strain will be re-tained despite stress relief. The total strain of the material
can therefore be decomposed into an elastic and a plasticpart. In case the local strain evolution is function of thetime and the yielding limit is past, the material behaviour iscalled viscoplastic. The typical yield stress definition can-not be used in the viscoplasticity model because the yieldstress value is higher than an applied stress in a creep testwhich produces creep strain. The model represents plasticand creep strain in one parameter namely the inelastic strain[7]. This is in fact the starting assumption of the majorityof constitutive models: the separation of the total strain intoa reversible, elastic part denoted as εe and an irreversible,inelastic part denoted as ε p, e.g.ε = εe + ε p.
In a standard analysis, plastic effects may be taken intoaccount by applying the Multilinear Kinematic Hardening(MKIN) material model which will be shortly depicted inthe next section.
Multi-linear approachIn most commercial FEA tools, two approaches are
readily available for studying plasticity effects includingkinematic and isotropic hardening. In the former, the cen-tre of the yield surface of material simply translates towardsthe principal axes, without any change in size of the yieldlocus. In the latter, the centre of the yield surface remainsconstant but the radius will change. Two simple modelsallow for studying such cases. The bilinear kinematic hard-ening model can be applied whenever there is a paucity ofdata, or whenever a simple model that gives only a first ap-proximation of the behavior of metals subjected to cyclicloading is required, while a multilinear kinematic harden-ing model is applied in case the monotonic stress-straincurves of the material are well characterized. As the elas-tic and inelastic domains are distinctly separated, the in-troduction of a stress level that defines these two fields isdemonstrated by the following:
Eσ = {σ ∈ R| f (σ) = |σ |−σy ≤ 0} (1)
where the function f : R −→ R is the yield functionand is used to compare against the yield condition, wheref (σ) < 0 is the elastic domain and f (σ) = 0 is inelastic.Adding a rigid motion that acts as as translation operatorto the elasticity domain, Eσ , mean introduces a kinematichardening model:
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FIGURE 2. Temperature distribution from FEA versus the microstructure at specific points. It is noteworthy the influence of temper-ature on the γ− γ ′ matrix structure.
Eσ = {σ ∈ R| f (σ) = |σ −X |−σy ≤ 0} (2)
where X is the kinematic hardening variable. In orderto account for changes to the yield surface, a further vari-able P is introduced into Eqn 1 and 2. This is referred to asthe isotropic hardening variable.
Eσ = {σ ∈ R| f (σ) = |σ −X |− (σy +P)≤ 0} (3)
Chaboche approachThe Chaboche model as presented by Hart et al. [8]
and Chaboche [9], uses a flow rule to describe the materialinelastic flow, a backstress equation describing the evolu-tion of the yield stress surface center, and governing equa-tions for yield surface size evolution. This model containsan elastic domain and needs to have material parametersfit directly as they change with temperature. Chaboche ex-plicitly accounts for isotropic hardening through means of
an isotropic hardening resistance parameter that describesthe change in size of the yield surface. A temperature rateterm is included within the model back stress formulation.The basic difference with plasticity models previously pre-sented is that within Chaboche model (as well for other vis-coplastic models), there is no explicit stress state inequality.
According to Hart et al. [8], the most simple viscoplas-tic model is represented by a friction element with constantσy connected in parallel with a non linear dashpot given bythe equation:
σ = Jε1m (4)
where the constants J and m are free parameters. Thisviscoplastic model is valid in the event that only overstressis responsible for the evolution of plastic strain. The flowrule which models the plastic strain is given by:
g(σ) = εp =
(f (σ ,σy)
J )m sgn(σ), |σ |> σy
0(5)
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FIGURE 3. MAR-M-247 Tensile Properties from available lit-erature review, [1], [2], [3] and [4]. Both Yield and UTS lookconstant up to a a temperature of ∼ 800◦C after which they dropdrastically to about 1/4 of their maximum value.
with f (σ ,σy) = |σ |−σy. This means that, within vis-coplasticity framework, the plastic flow is determined bythe overstress. Whenever f > 0, then plastic strain occurs.
From theory it is known that the yield stress is not aconstant value but it is dependent on the loading history[8]. This means that hardening or softening effects takeplace whenever a tension-compression cycle is performed.If tension is held in first step, then the material hardens intension and softens in compression and vice-versa. Thisresults in lowering the yield stress, that is, a different elas-tic threshold has been set. This limit will be higher thanthe yield stress which would have been expected in a com-pression test only. This phenomena is recognized as theBauschinger effect.
The overall result of the above depicted effect is thatthe elastic domain does not only change in size (isotropichardening), but it can also be translated (kinematic harden-ing) [8]. The kinematic hardening behaviour is dependenton the plastic strain, whereas the isotropic hardening is afunction of the accumulated plastic strain. For a materialwhich has not undergone any load history the center of theelastic domain is initially zero with the limits σy in tensionand −σy in compression. Whenever hardening effects arein place, the incipient plastic flow defined by f > 0 is rep-
resented by the equation
f (σ ,X ,P) = |σ −X |−P−σy (6)
where X and P are kinematic and isotropic hardeningparameters, respectively. High temperatures materials mayshow another phenomenon to be taken into account by ma-terial model. The softening of the material as function oftime may be observed in relaxation tests. This is a recov-ery process, which is time dependant. This effect has to betaken into account only for high temperatures. Within theevolution equations for both kinematic and isotropic hard-ening variables, this is translated into a decrease in the sizeof the elastic region as function of the time. Therefore, theflow rule of a Chaboche model [8] is defined by equation5 with hardening variables as in equation 6. Thus, the flowrule of this model is given by:
εp = 〈
|σ −X |−P−σy
J〉m sgn(σ −X) (7)
with 〈 |σ−X |−P−σyJ 〉 = max(0,x) and the isotropic hard-
ening variable P is defined by the differential equation:
P = b(q−P)p (8)
The variable p take into account the accumulated plas-tic strain, i.e. p = |ε p|. The parameter b is the termthat evaluates the speed of stabilization. The asymptoticvalue, as per the evolution of the isotropic hardening, isthe value of the parameter q. To obtain a more accu-rate model of kinematic hardening the approach of Harth[8] is to consider more than only one variable. We applya sum of three non-linear kinematic hardening variablesX = X1 +X2 +X3.Their evolution is given by Armstrongand Frederick [9] type equations:
Xi = ciεp−αiXi p,(i = 1,2,3) (9)
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TABLE 1. Chaboche constitutive behaviour
Strain: ε(t) = εe(t)+ ε p(t)
Elastic Law: σ(t) = Eεe(t)
Flow Rule:ε(t) =
(1J(|σ(t)−X(t)|− . . .
P(t)−σy)
)m
sgn(σ(t)−X(t)
)
Hardening:X(t) = X1(t)+X2(t)+X3(t)
Xi(t) = ciεp(t)−αiXi(t)|ε p(t)|
P(t) = b(q−P(t)
)|ε p(t)|
Initial Conditions: εp(0) = 0,X1(0) = 0,X2(0)
X3(0) = 0,P(0) = 0
Parameters: σy (Yield Stress)
J,m (Flow Rule)
a1,a2,a3,c1,c2,c3,b,q (Hardening)
The values of the parameters α1, α2 and α3 denote thespeed of saturation and according to these values the pa-rameters c1, c2 and c3 are asymptotic values of the kine-matic hardening variables [8]. A complete depiction of themodel is given in Table 1.
The superposition of three kinematic hardening vari-ables leads to a more accurate description, as each vari-able covers its own strain range. This means that, whileone variable describes hardening for rather large strains, theother one permits the transition from elastic to plastic do-mains. The third one instead is added to optimize the accu-racy at lower strain ranges. The complete Chaboche modelis presented in Table 1. It has 11 material parameters. Inthis framework, it has been considered the hypothesis thatthe Young’s modulus is a fixed value. The inelastic part ofthe models consists of a system of four ordinary differentialequations.
MATERIAL MODEL DEVELOPMENTIn describing the material model development for
MAR-M-247, only kinematic hardening effects will be aco-cunted for, both because they are easily obtained frommonotonic stress-strain curves and because they are readilyimplemented into the majority of available FEA tools. Todetermine parameters at different temperatures of interest,the first step is to calibrate the coefficients from appropriateflow stress curves at different temperatures. It is thereforeimportant to define an elastic limit. In this study two ap-proaches have been implemented. A 0.2% yield point wasfirst used and then a 1E-6 difference in strain deciding thislimit. This second approach has been implemented in orderto take into account for yield stress effects. 0.2% yield isa conventional way to define this parameter. According tothe data for monotonic stress-strain curves of MAR-M-247available within proprietary material database, a differentapproach has been evaluated. It has been calculated the dif-ference between total and elastic strains and 1E-6 has beentaken as threshold for yield stress definition, due to datapoint accuracy. The elastic limit is fundamental to evaluatethe backstress, that is the parameter that takes into accountfor the translation of the yield surface in stress space. Itis defined as the difference between the stress in a uniaxialstress state of a tension specimen and the initial yield stressat the elastic limit, previously defined. So that once thebackstress is defined, the inelastic strain curves are defined(Fig 4).
According to Kalnins et al. [10], a minimum of threeChaboche parameters can interpolate the backstress curvesat the different temperatures. Once the backstress curvesare available, the parameters can be fit with a a Levenberg-Marquardt (LM) algorithm. First, the αNLK backstress isdescribed by the following equations:
αk = (ak
ck)[1− exp(−ckε
p)] (10)
αNLK =N
∑k=1
ck (11)
where K refers to each of the N components applied,and ε p is the uniaxial plastic strain. ak and ck are the
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FIGURE 4. Example of the the different curves at 980◦C. Themonotonic stress-strain curve is the curve as found in Baker andHughes material database. The monotonic stress-plastic straincurve is the one applied commonly in MKIN anlysis. The back-stress curve is the curve applied in the paper for obtaining theChaboche parameters as per Kalnins et al. [10].
Chaboche parameters that are determined by the calibra-tion process (Fig. 5).
APPLICATIONTo demonstrate and benchmark material models, the
first stage nozzle of a high pressure gas turbine was anal-ysed with a sequentially-coupled thermomechnical FEA.Representative start-up of the nozzle up to the steady-stateconditions was modelled followed by shut-down. The full-size component is made up of 22 identical sectors that areseparated by a small gap where seals are mounted. This al-lows for simplified modelling efforts as only one single sec-tor needs to be considered. A tetrahedral mesh was then ap-plied to a simplified geometry of this single sector (Fig. 6).The mesh resulted contained 800000 nodes and more than1000000 elements.This is made by simply providing theFEA tool with monotonic stress-plastic strains curves. Theresults obtained through this model have been then used asthe baseline to compare the Chaboche model with.
Boundary conditions were applied to the chordal hingeof the nozzle, which, in a complete model would represent
FIGURE 5. Example of the the different curves at 640◦C. Pic-ture here is showing the good fitting found through the applica-tion of the three Chaboche components.
the contact surface with the inner ring and the gas turbinecasing, see Fig.6.
Temperature fields were directly applied stemmingfrom a parallel gas flow analysis. These were employedas input conditions for pressure and temperature. In Fig.7 and 8, pressures and temperatures are given as contourplots.
RESULTS AND DISCUSSIONThe analysis described above provided 20 points in
time along the duty cycle to compare material model per-formance (Fig. 9).
The nozzle mesh has been kept constant in eachof the model applied (bilinear kinematic hardening andChaboche), as well as the constrains (simply supportednozzle). Cyclic symmetry has been applied to the gas tur-bine casing and to the inner ring, and nonlinear contactshave been used to simulate the contacts between these el-ements and the nozzle itself. No appreciable differencesbetween Chaboche and MKIN were noted as per Fig. 10which plots the difference in stresses resolved by each ma-terial model.
The main advantages demonstrated by the Chabochemodel were as follows:
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FIGURE 6. Mesh applied to the nozzle geometry and cordalhinge contacts. These will simulate the displacements imposedto the nozzle from inner ring and outer casing.
FIGURE 7. Nozzle airfoils pressure map at steady state con-dition.
1. Expedited convergence of the FEA. It took in fact veryfew steps for the entire nozzle model to converge andthis showed to be much faster, saving up to 50% ofcomputational time as compared to MKIN.
2. The results with both models applied (the model con-sidering the 0.2% yield stress and the model with recal-culation of the yield value), provided comparable re-sults. Fig 11 highlights stress and strain results for onesingle node at the trailing edge of the nozzle in a crit-
FIGURE 8. Nozzle airfoils temperature map at steady statecondition.
ical zone and there is only one time step where largerdifferences (up to 5% in stress) are shown.
3. While with bilinear or multilinear kinematic hardeningmodel it has shown to be hard to evaluate more than onecycle, an analysis with eight cycles easily converged,see Fig 12. This may be related to the specific applica-tion considered in the present paper, specifically to theboundary conditions, loads and specific material itself.The simple bilinear models fails in considering cyclichardening and ratchetting effects while the multilin-ear kinematic hardening showed stability issues parti-coularly in the second cycle analysis.
4. The Chaboche model showed the capability of takinginto account for such effects as ratchetting or shake-down and hardening or softening of the material, seeFig. 13. It is evident here that the hardening of the ma-terial results in a drop in the stress of the component.It is also evident that the hysteresis cycle has not yetcome to stabilization.
It is also noteworthy that after 8 cycles a drop of about5% in stress is evident. This may lead to strongly influ-enced crack propagation assessments due to a reduction ofalternate stress effects as differing predicted LCF lives dueto a reduction in mean stress, which is of predominant im-portance in crack nucleation.
FUTURE DEVELOPMENTSAfter this first application of the Chaboche model
through monotonic stress-strain curves and the interestingresults obtained by considering only the kinematic harden-ing part of the model itself on both calculation time reduc-tion and accuracy capability, in the next future it will beworth to investigate the following:
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FIGURE 9. Points selected for the transient structural assessment on the entire engine mission. These points have been selected bytaking into account for peaks and drops as per the rainflow methodology.
1. Application of cyclic stress-strain curves in order to re-evaluate the kinematic hardening part by the applica-tion of more appropriate data and to include isotropiceffects;
2. Application of relaxation data in order to include theviscous part of the model and to evaluate the capabilityof taking into account for cyclic and time based effects;
3. Add further cycle steps to the analysis in order to eval-uate when the hysteresis cycle of the material reachesstabilization and evaluate the whole hardening effects;and
4. Evaluate the model capability through LCF tests withlong (10 minutes or 30 minutes) holds to test the accu-racy towards defined solutions
ACKNOWLEDGMENT
The authors would like to thank Baker Hughes, a GECompany, for supporting this research.
REFERENCES[1] Brindley, K. A., 2014. “Thermomechanical fatigue of
MAR-M-247: extension of a unified constitutive andlife model to higher temperatures”. Master of science,Georgia Institute of Technology.
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[3] Kaufman, M., 1982. Properties of Cast MAR-M-247for Turbine Blisk Applications. Tech. rep., GeneralElectric, USA.
[4] Martın-meizoso, A., 2011. Thermomechanical fa-tigue behaviour and life prediction of MAR-M-247nickel based superalloy. Tech. rep., CEIT.
[5] Liao, J.-h., Bor, H.-y., Wei, C.-n., Chao, C.-g., andLiu, T.-f., 2012. “Influence of microstructure andits evolution on the mechanical behavior of modifiedMAR-M247 fine-grain superalloys at 871C”. Materi-
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FIGURE 10. Comparison in stress results between MKIN and Chaboche model. The MKIN model taken into account in the analysisis based half cycle monotonic stress-strain curves. It is worth to notice as the peak is found towards 0 value meaning that there is goodaccordance between MKIN and Chaboche models.
FIGURE 11. Comparison of results among the 3 methodolo-gies applied. Results are in good accordance. MKIN providesthe lowest stress results, with higher delta strain within first cy-cle. This may result in a more conservative LCF life assessment.
als Science & Engineering A, 539, pp. 93–100.[6] Gasko, K. L., Janowski, G. M., and Pletka, B. J.,
1988. “The Influence of Eutectic on the Mechani-
FIGURE 12. Results on 8 cycles. Chaboche model showedthe capability to easily run many different cycles.
cal Properties of Conventionally Cast MAR-M-247”.Materials Science and Engineering, 104, pp. 1–8.
[7] Saad, A. A., 2012. “Cyclic plasticity and creep ofpower plant materials.”. Phd, University of Notting-ham.
[8] Harth, T., 2007. “Identification of Material Parame-ters for Inelastic Constitutive Models Using Stochas-tic Methods”. Wiley-VCH Verlag GmbH & Co.,
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FIGURE 13. Detail of steady state results on stress strain plot.Hardening tendency is evident through the 8 cycles. 1st cyclesteady state value is the one on the top left of the picture and thesteady state value is then following the path towards the bottomright value. It is important to define when this hardening beahv-ior will find saturation.
429(2), pp. 409–429.[9] Chaboche, J. L., 2008. “A review of some plasticity
and viscoplasticity constitutive theories”. Elsevier In-ternational Journal of Plasticity, 24, pp. 1642–1693.
[10] Kalnins, Arturs; Rudolph, Jurgen; Willuweit, A.,2015. “Using the Nonlinear Kinematic Harden-ing Material Model of Chaboche for Elastic Plas-tic Ratcheting Analysis”. Journal of Pressure VesselTechnology, 137(June 2015), p. 10.
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