Visco-plastic Chaboche model for nickel-based alloys under anisothermal cyclic loading



==VISCO-PLASTIC CHABOCHE MODEL FOR NICKEL-BASED ALLOYS UNDER ANISOTHERMAL CYCLIC LOADING==

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'''HÉLÈNE MORCH*, LAURENT DUCHÊNE* AND ANNE-MARIE HABRAKEN*†'''</div>

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<span style="text-align: center; font-size: 75%;">*</span> Dpt ArGEnCo, MS²F-MSM</div>

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University of Liège</div>

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Allée de la Découverte 9, 4000 Liège, Belgium</div>

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e-mail: [mailto:helene.morch@ helene.morch@]ulg.ac.be; [mailto:l.duchene@ulg.ac.be l.duchene@ulg.ac.be]; [mailto:anne.habraken@ulg.ac.be anne.habraken@ulg.ac.be] </div>

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[http://www.uee.ulg.ac.be/cms/c_2672632/fr/mecanique-des-solides-et-des-materiaux-msm http://www.uee.ulg.ac.be/cms/c_2672632/fr/mecanique-des-solides-et-des-materiaux-msm]</div>

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<sup>†</sup> Fund for Scientific Research (FRS-FNRS)</div>

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[http://www.fnrs.be www.fnrs.be]</div>

'''Key words:''' Constitutive modeling, Nickel-based superalloys, Cyclic plasticity, Thermo-mechanical modeling

'''Summary:''' ''' '''The mechanical behavior of visco-plastic materials such as nickel-based alloys is highly dependent on temperature. Some characteristics such as viscosity, hardening, static recovery, dynamic recovery have more or less influence on the overall behavior depending on the considered temperature. The unified constitutive model developed by Chaboche [1] is very efficient in representing this complexity as it is very adaptable and can contain many features. A basic Chaboche model contains a viscosity law and one or several hardening equations. Within these hardening equations, it is possible to add several features that will represent the complex behavior of the material.

The aim of this study is to understand the role of the different parameters and the influence of the different features in an advanced Chaboche model adapted to cyclic anisothermal loading. This specific model was also developed in [2],[3]. However, part of this study is based on particular cases where different features of the model are analyzed [4]–[6].

==1 INTRODUCTION==

The use of nickel-based superalloys at high temperature requires advanced visco-plastic models to accurately represent the material behavior. The Chaboche model is very efficient for representing complex behavior as it can include various features such as isotropic hardening, kinematic hardening, static recovery, or thermo-mechanical behavior. The drawback of this type of model is the number of parameters. Indeed, a high level of accuracy of the model requires a high number of parameters. As a consequence, the determination of these parameters can be an arduous task. To facilitate this important step in the completion of the model, a sensitivity study can provide valuable information. The sensitivity analysis reveals which tests are more relevant for the determination of each parameter, but also how the parameters can impact the model and what physical meaning they have.

==2 VISCO-PLASTIC MODELING==

The visco-plastic model used in this study is a Chaboche-type constitutive model as developed in Ahmed, 2013 [2] for superalloy Haynes 230 at high temperture. This model, described hereafter, was implemented in the 3D finite element code Lagamine [7] developed at the University of Liège. For simplification, the model is described here as a 1D model, since only uniaxial tests are used in this study.

===2.1 Visco-plasticity===

The mechanical strain can be decomposed in an elastic contribution and a visco-plastic contribution:

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>\epsilon ={\epsilon }^{e}+{\epsilon }^{vp}</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(1) </span>
|}

The elastic strain and the stress are related through Hooke's law, where E  is the Young's modulus:

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>\sigma =E{\epsilon }^{e}</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(2) </span>
|}

The yield locus is defined by the von-Mises criterion, with <math display="inline">X</math> the back-stress, <math display="inline">{\sigma }_{0}</math>the initial yield strength, and <math display="inline">R</math> the isotropic hardening variable:

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>f=\left| \sigma -X\right| -{\sigma }_{0}-R</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(3) </span>
|}

The viscosity is modeled through Norton's equation (equation (4)), with viscous parameters <math display="inline">n</math> and <math display="inline">K</math>.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>\overset{\cdot}{p}=\left| {\overset{\cdot}{\epsilon }}^{vp}\right| ={\left\langle \frac{f}{K}\right\rangle }^{n}</math>

<math>where\, \left\langle x\right\rangle =\left\{ \begin{matrix}x\, if\, x\geq 0\\0\, if\, x<0\end{matrix}\right.</math> 
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(4) </span>
|}

===2.2 Hardening equations===

The evolution of the isotropic variable <math display="inline">R</math>, described by equation (5) depends on the plastic strain rate <math display="inline">\overset{\cdot}{p}</math> and on two parameters <math display="inline">b</math> and <math display="inline">Q</math>. <math display="inline">b</math> can be understood as the rate at which variable <math display="inline">R</math> will reach its saturation value <math display="inline">Q</math>.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>\overset{\cdot}{R}=b\left( Q-R\right) \overset{\cdot}{p}</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(5) </span>
|}

The back-stress <math display="inline">X</math> is composed of one or several back stresses <math display="inline">{X}_{i}</math>. Each of these <math display="inline">{X}_{i}</math> obeys a non linear kinematic hardening rule defined through an Armstrong-Frederick equation [8]. Following the work of Yaguchi et al., 2002 [4], [5] a state variable <math display="inline">{Y}_{i}</math> can be added in the equation in order to model the evolution of the mean stress and a temperature-dependency term is used for anisothermal modeling. The variable <math display="inline">{Y}_{i}</math> is controlled by parameters <math display="inline">{\alpha }_{b,i}</math>and <math display="inline">{Y}_{st,i}</math>, both positive. The equation of a back stress <math display="inline">{X}_{i}</math> therefore consists of 4 terms:

:* strain hardening, controlled by parameter <math display="inline">{C}_{i}</math><span style="text-align: center; font-size: 75%;">;</span>

:* dynamic recovery, controlled by parameter <math display="inline">{\gamma }_{i}</math><span style="text-align: center; font-size: 75%;">;</span>

:* static recovery, controlled by parameters <math display="inline">{b}_{i}</math> and <math display="inline">{r}_{i}</math><span style="text-align: center; font-size: 75%;">;</span>

:* temperature rate, with the influence of parameter <math display="inline">{C}_{i}</math><span style="text-align: center; font-size: 75%;">.</span>

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"| <math>X=\sum _{i=1}^{n}{X}_{i}</math><span style="text-align: center; font-size: 75%;"><br/></span> <math>{\overset{\cdot}{X}}_{i}={C}_{i}{\overset{\cdot}{\epsilon }}^{vp}-{\gamma }_{i}\left( {X}_{i}-\right. </math><math>\left. {Y}_{i}\right) \overset{\cdot}{p}-{b}_{i}\, {\left| {X}_{i}\right| }^{{r}_{i}}{X}_{i}+</math><math>\frac{1}{{C}_{i}}\frac{\partial {C}_{i}}{\partial T}\overset{\cdot}{T}{X}_{i}</math><span style="text-align: center; font-size: 75%;"><br/></span> <math>with\, {\overset{\cdot}{Y}}_{i}=-{\alpha }_{b,i}\left( {Y}_{st,i}\frac{{X}_{i}}{\left| {X}_{i}\right| }+\right. </math><math>\left. {Y}_{i}\right) {\left| {X}_{i}\right| }^{{r}_{i}}</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(6) </span>
|}

Cyclic hardening is represented by the evolution of the dynamic recovery parameter <math display="inline">{\gamma }_{i}</math>. The rate of evolution of this parameter is controlled by the plastic strain rate <math display="inline">\overset{\cdot}{p}</math> and a parameter <math display="inline">{D}_{{\gamma }_{i}}</math>. The parameter <math display="inline">{\gamma }_{i}</math>evolves towards a saturation value <math display="inline">{\gamma }_{i}^{0}</math> which depends on the radius of the strain memory surface <math display="inline">q</math>. The strain memory surface <math display="inline">{g}_{M}</math> is defined by equation (8), where <math display="inline">H(x)</math> is the Heaviside step function.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>{\overset{\cdot}{\gamma }}_{i}={D}_{{\gamma }_{i}}\left( {\gamma }_{i}^{0}-{\gamma }_{i}\right) \overset{\cdot}{p}</math>

<math>with\, \, {\gamma }_{i}^{0}={a}_{{\gamma }_{i}}+{b}_{{\gamma }_{i}}{e}^{-{c}_{{\gamma }_{i}}q}</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(7) </span>
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>{g}_{M}\left( {\epsilon }^{vp}-\zeta \right) =\left| {\epsilon }^{vp}-\zeta \right| -</math><math>q</math>

<math>\overset{\cdot}{\zeta }=\left( 1-\eta \right) H\left( {g}_{M}\right) \overset{\cdot}{p}</math>

<math>\overset{\cdot}{q}=\eta ~H\left( {g}_{M}\right) \overset{\cdot}{p}</math>
|  style="text-align: right;vertical-align: top;"|<span id='_Ref480451178'></span><span style="text-align: center; font-size: 75%;">(8) </span>
|}

===2.3 Influence of the maximum temperature===

In the case of anisothermal cyclic loading on Haynes 230, experiments show that the maximum temperature of the cycle has an influence on the overall behavior of the material [2]. This is modeled by a variation of the Young's modulus, expressed as a weighted average of the initial Young's modulus at temperature <math display="inline">T</math> and the Young's modulus at maximum temperature <math display="inline">{T}_{max}</math>:

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"|
|  style="vertical-align: top;"| <math>E={f}_{E}E+\left( 1-{f}_{E}\right) {E}_{{T}_{max}}</math>

<math>{\overset{\cdot}{f}}_{E}={b}_{E}\left( {f}_{E}^{S}-{f}_{E}\right) \overset{\cdot}{p}</math>
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">(9) </span>
|}

The weighted average factor <math display="inline">{f}_{E}</math> represents the weight of the initial Young's modulus. A small value of <math display="inline">{f}_{E}</math> corresponds to a significant influence of the maximum temperature. <math display="inline">{f}_{E}</math> evolves at a rate <math display="inline">{b}_{E}</math> towards a saturated value <math display="inline">{f}_{E}^{S}</math>.

==3 METHOD==

The sensitivity study was conducted using reference sets of parameters available in the literature. These sets of parameters came from models that did not contain as many features as the model hereinbefore presented. However, using different sub-models allowed to study the sensitivity of the model to each of the parameters.

===3.1 Numerical tests===

The sensitivity study was conducted numerically on each parameter by performing a cyclic test and modifying one parameter at a time. Different sets of reference parameters were used to perform the simulations. <span id='cite-_Ref480463625'></span>[[#_Ref480463625|Table 1]] summarizes the reference articles used depending on the tested parameters.

<div id="_Ref480463625" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Table 1 : '''Reference articles used to obtain parameters</span></div>

{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;"|'''Parameter studied'''
|  style="border: 1pt solid black;text-align: center;"|'''Set of reference parameters used'''
|-
|  style="border: 1pt solid black;vertical-align: top;"| <math>K,n,\, b,\, Q,\, {C}_{1},\, {C}_{2},\, {\gamma }_{1},{\gamma }_{2}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Zhan and Tong, 2007 [6]
|-
|  style="border: 1pt solid black;vertical-align: top;"| <math>{b}_{1},\, {r}_{1},{\alpha }_{b,1},\, {Y}_{st,1},{D}_{{\gamma }_{1}},{a}_{{\gamma }_{1}},{b}_{{\gamma }_{1}},{c}_{{\gamma }_{1}},\eta</math> 
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Yaguchi et al., 2002a [4]
|-
|  style="border: 1pt solid black;vertical-align: top;"| <math>{b}_{E},\, {f}_{E}^{s}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Yaguchi et al., 2002b [5]
|}

The first model used [6] is an isothermal model describing the behavior of Alloy X at 650°C with two back-stresses for kinematic hardening, where no static recovery effect is taken into account. To study the influence of static recovery, variable Y, and cyclic hardening, a different model was used [4]. The latter describes the behavior of IN738LC at 850°C. Kinematic hardening is modeled using only one back-stress, with an internal variable Y. The influence of parameters <math display="inline">{b}_{E}</math> and <math display="inline">{f}_{E}^{s}</math> was tested on an anisothermal model [5] identical to [4].

It is to be noted that no reference was available for parameters <math display="inline">{D}_{{\gamma }_{1}},{a}_{{\gamma }_{1}},{b}_{{\gamma }_{1}},{c}_{{\gamma }_{1}},\eta ,\, {b}_{E}</math> and <math display="inline">{f}_{E}^{s}</math>, therefore, a reference value was determined for each of these parameters through trial and error, to obtain results that seemed coherent with the experimental data available (although the parameters were not determined to fit experimental curves). These reference values are summarized in <span id='cite-_Ref480465345'></span>[[#_Ref480465345|Table 2]].

<div id="_Ref480465345" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Table 2 : '''Reference parameters</span></div>

{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"| <math>{D}_{{\gamma }_{1}}</math>
|  style="border: 1pt solid black;vertical-align: top;"| <math>{a}_{{\gamma }_{1}}</math>
|  style="border: 1pt solid black;vertical-align: top;"| <math>{b}_{{\gamma }_{1}}</math>
|  style="border: 1pt solid black;vertical-align: top;"| <math>{c}_{{\gamma }_{1}}</math>
|  style="border: 1pt solid black;vertical-align: top;"| <math>\eta</math> 
|  style="border: 1pt solid black;vertical-align: top;"| <math>{b}_{E}</math>
|  style="border: 1pt solid black;vertical-align: top;"| <math>{f}_{E}^{s}</math>
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|300
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|100
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.2
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1000
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.2
|}

The sensitivity of the model on the different parameters was tested on two strain-controlled cyclic tests. A schematic representation of one period of the cyclic tests is given in <span id='cite-_Ref480467113'></span>[[#_Ref480467113|Figure 1]] (a). Anisothermal tests (<span id='cite-_Ref480467113'></span>[[#_Ref480467113|Figure 1]] (b)) were used to evaluate the sensitivity of the model to parameters <math display="inline">{b}_{E}</math> and <math display="inline">{f}_{E}^{s}</math>. For both Test 1 and Test 2, the strain amplitude is <math display="inline">\Delta \epsilon =</math><math>2\%</math>. The strain rates <math display="inline">\overset{\cdot}{\epsilon }</math> and hold times <math display="inline">{t}_{H}</math> of Test 1 and 2 are respectively {0.1%/s, 20s} and {0.001%/s, 1000s}. Each of these tests were performed for 50 cycles, which is enough for the stress-strain hysteresis loop to reach its saturation value using the reference parameters.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="text-align: center;vertical-align: top;"|'''(a)'''

''' [[Image:draft_Morch_621684164-image1-c.png|258px]] '''
|  style="text-align: center;vertical-align: top;"|'''(b)'''

''' [[Image:draft_Morch_621684164-image2-c.png|222px]] '''
|}

<div id="_Ref480467113" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 1: '''(a)''' '''Strain-controlled isothermal test; (b) Strain-controlled anisothermal test</span></div>

===3.2 Sensitivity criteria===

Different criteria <math display="inline">{F}_{i}</math> were used to determine the sensitivity of the model to the different parameters:

:* <math display="inline">{F}_{1}:</math> The tensile stress before the first hold time (equivalent to the stress at the end of a tensile test);

:* <math display="inline">{F}_{2}:</math> The amount of stress relaxation during the first hold time (equivalent to a relaxation test);

:* <math display="inline">{F}_{3}:</math> The stress amplitude <math display="inline">{\sigma }_{amp}=</math><math>\frac{{\sigma }_{max}-{\sigma }_{min}}{2}</math> at the 50<sup>th</sup> cycle;

:* <math display="inline">{F}_{4}:</math> The mean stress <math display="inline">{\sigma }_{mean}=</math><math>\frac{{\sigma }_{max}+{\sigma }_{min}}{2}</math> at the 50<sup>th</sup> cycle.

The mean stress and the stress amplitude were also used as criteria to study the sensitivity over the cycles.

For each criterion <math display="inline">{F}_{i}</math>, the sensitivity to a parameter <math display="inline">P</math> is expressed as <math display="inline">\frac{\partial {F}_{i}}{\partial P}</math>. The criterion <math display="inline">{F}_{i}</math> is computed for three values of the parameter <math display="inline">P</math> and the sensitivity <math display="inline">\frac{\partial {F}_{i}}{\partial P}</math> is calculated as the slope of the <math display="inline">{F}_{i}(P)</math> line obtained from a linear regression.

==4 RESULTS==

===4.1 Viscous parameters <math display="inline">\mathit{\boldsymbol{K,n}}</math>===

The viscous parameters <math display="inline">K</math> and <math display="inline">n</math> represent the relation between the stress and the strain rate. <span id='cite-_Ref480790052'></span>[[#_Ref480790052|Figure 2]] shows the sensitivity of criteria <math display="inline">{F}_{1},\, {F}_{2},\, {F}_{3}</math> and <math display="inline">{F}_{4}</math>. Both parameters have an influence on the tensile stress ( <math display="inline">{F}_{1}</math>) and on the stabilized stress amplitude ( <math display="inline">{F}_{3}</math>). The stress relaxation is also influenced by these viscous parameters but to a smaller extent.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image3.png|300px]] 
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image4.png|276px]] 
|}

<div id="_Ref480790052" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 2: '''(a) Sensitivity to the drag stress </span> <math display="inline">\mathit{\boldsymbol{K}}</math><span style="text-align: center; font-size: 75%;">; (b) Sensitivity to the viscous exponent</span> <math display="inline">\mathit{\boldsymbol{\, n}}</math></div>

===4.2 Isotropic hardening parameters <math display="inline">\mathit{\boldsymbol{b,Q}}</math>===

The isotropic hardening parameters <math display="inline">b</math> and <math display="inline">Q</math> control the growth of the yield surface. Knowing this, it is predictable that these parameters will influence mostly the tensile stress and the stress amplitude. <span id='cite-_Ref480790858'></span>[[#_Ref480790858|Figure 3]] shows that these parameters influence mainly the stress amplitude over the cycles, with <math display="inline">b</math> reaching a peak around the 10th cycle, where the stress amplitude starts to stabilize. The small sensitivity of the tensile stress to these isotropic hardening parameters is due to the test itself, which only reaches a 1% strain, therefore not allowing isotropic hardening to fully develop. In practise, a tensile test to rupture would show as much sensitivity to isotropic hardening as the cyclic tests that were performed here.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image5.png|276px]] 
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image6.png|288px]] 
|}

<div id="_Ref480790858" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3 - '''(a) Sensitivity to parameter </span> <math display="inline">\mathit{\boldsymbol{b}}</math><span style="text-align: center; font-size: 75%;">; (b) Stress amplitude and mean stress sensitivity to </span> <math display="inline">\mathit{\boldsymbol{b}}</math><span style="text-align: center; font-size: 75%;"> over cycles; (c) Sensitivity to parameter </span> <math display="inline">\mathit{\boldsymbol{Q}}</math></div>

===4.3 Kinematic hardening parameters <math display="inline">{\mathit{\boldsymbol{C}}}_{\mathit{\boldsymbol{i}}}</math>===

Kinematic hardening is characterized by a change in the position of the yield surface. As for isotropic hardening, the parameters controlling kinematic hardening influence the tensile stress and the stress amplitude, as seen in <span id='cite-_Ref480791358'></span>[[#_Ref480791358|Figure 4]]. As shown in Tong et al. [9], the use of two parameters <math display="inline">{C}_{1}</math> and <math display="inline">{C}_{2}</math> allows the description of a transient region with a fast growing back-stress <math display="inline">{X}_{1}</math> and a steady-state behaviour once <math display="inline">{X}_{1}</math> has reached its stabilized value, with back-stress <math display="inline">{X}_{2}</math> growing at a quasi-steady pace. The parameters <math display="inline">{C}_{1}</math> and <math display="inline">{C}_{2}</math> also have an influence on the shape of the hysteresis loop, and more particularly on its curvature in the visco-platic domain.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image7.png|288px]] 
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image8.png|276px]] 
|}

<div id="_Ref480791358" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 4 - '''(a) Sensitivity to parameter </span> <math display="inline">{\mathit{\boldsymbol{C}}}_{\mathit{\boldsymbol{1}}}</math><span style="text-align: center; font-size: 75%;">; (b) Sensitivity to parameter </span> <math display="inline">{\mathit{\boldsymbol{C}}}_{\mathit{\boldsymbol{2}}}</math></div>

===4.4 Dynamic recovery parameters <math display="inline">{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{i}}}</math>===

The dynamic recovery parameters <math display="inline">{\gamma }_{1}</math> and <math display="inline">{\gamma }_{2}</math> have the opposite effect of <math display="inline">{C}_{1}</math> and <math display="inline">{C}_{2}</math>. <span id='cite-_Ref480792783'></span>[[#_Ref480792783|Figure 5]] shows that <math display="inline">{F}_{1}</math> and <math display="inline">{F}_{3}</math> show a negative sensitivity to both <math display="inline">{\gamma }_{1}</math> and <math display="inline">{\gamma }_{2}</math>, which means an increase in <math display="inline">{\gamma }_{i}</math> will lead to a decrease in <math display="inline">{F}_{j}</math>. Similarly to the kinematic hardening parameters,  <math display="inline">{\gamma }_{1}</math> and <math display="inline">{\gamma }_{2}</math> also have an effect on the curvature of the hysteresis loop.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image9.png|276px]] 
|  style="vertical-align: top;"| [[Image:draft_Morch_621684164-image10.png|252px]] 
|}

<div id="_Ref480792783" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 5 - '''(a) Sensitivity to parameter </span> <math display="inline">{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{1}}}</math><span style="text-align: center; font-size: 75%;">; (b) Sensitivity to parameter </span> <math display="inline">{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{2}}}</math></div>

===4.5 Static recovery parameters <math display="inline">{\mathit{\boldsymbol{b}}}_{\mathit{\boldsymbol{i}}}\mathit{\boldsymbol{,\, \, }}{\mathit{\boldsymbol{r}}}_{\mathit{\boldsymbol{i}}}</math>===

To study the influence of the static recovery parameters, the internal variable <math display="inline">Y</math> was not taken into account. Indeed, parameter <math display="inline">{r}_{i}</math> is used both in the static recovery term of the back-stress evolution and in the evolution of internal variable <math display="inline">Y</math> - equation (6). <span id='cite-_Ref480794470'></span>[[#_Ref480794470|Figure 6]] shows the sensitivity to static recovery parameters <math display="inline">{b}_{1},\, \, {r}_{1}</math> without considering the effect of <math display="inline">{r}_{1}</math> on <math display="inline">Y</math>. It is apparent that Test 2, which is slower and has longer hold times than Test 1, is much more impacted by the variation of these parameters. This is an expected result considering static recovery is a phenomenon that takes place at a constant strain. The influence is maximal on the stress relaxation (criterion <math display="inline">{F}_{2}</math>), but also appears on the other criteria. This can be explained by equation (6): a higher value of <math display="inline">{b}_{1}</math> or <math display="inline">{r}_{1}</math> will lead to a greater decrease in the back-stress <math display="inline">{X}_{1}</math>, therefore lowering the tensile stress and stress amplitude.

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<div id="_Ref480794470" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 6 - '''(a) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{b}}}_{\mathit{\boldsymbol{1}}}</math><span style="text-align: center; font-size: 75%;">; (b) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{r}}}_{\mathit{\boldsymbol{1}}}</math><span style="text-align: center; font-size: 75%;"> (without considering the internal variable Y)</span></div>

===4.6 Mean stress evolution parameters <math display="inline">{\mathit{\boldsymbol{\alpha }}}_{\mathit{\boldsymbol{b,i\, }}}\mathit{\boldsymbol{,\, }}{\mathit{\boldsymbol{Y}}}_{\mathit{\boldsymbol{st,i}}}\mathit{\boldsymbol{,\, }}{\mathit{\boldsymbol{r}}}_{\mathit{\boldsymbol{i}}}</math>===

The internal variable <math display="inline">{Y}_{1}</math> that controls the evolution of the mean stress over cycles is calculated using three parameters <math display="inline">{\alpha }_{b,1\, },\, {Y}_{st,1},\, {r}_{1}</math>. <span id='cite-_Ref480812930'></span>[[#_Ref480812930|Figure 7]] (a) represents the sensitivity to parameter <math display="inline">{\alpha }_{b,1}</math>'''. '''

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<div id="_Ref480812930" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 7 - '''(a)''' '''Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{\alpha }}}_{\mathit{\boldsymbol{b}}\mathit{\boldsymbol{,1}}}</math><span style="text-align: center; font-size: 75%;">; (b) Sensitivity to modified </span> <math display="inline">{\mathit{\boldsymbol{\alpha }}}_{\mathit{\boldsymbol{b}}\mathit{\boldsymbol{,1}}}</math></div>

The sensitivity computed with Test 1 seems coherent: the parameter influences only the mean stress. However, sensitivities computed with Test 2 give rather unexpected results, and show a bigger influence on the stress amplitude than on the mean stress. The reason for this can be found by considering the evolution of variable <math display="inline">{Y}_{1}</math> over time, described by equation (6). The variable <math display="inline">{Y}_{1}</math> varies within <math display="inline">-</math><math>{Y}_{st,1}</math> and <math display="inline">+{Y}_{st,1}</math>. Therefore, according to (6), <math display="inline">{\overset{\cdot}{Y}}_{1}</math> is positive when <math display="inline">{X}_{1}>0</math> and negative when <math display="inline">{X}_{1}<0</math> unless <math display="inline">{Y}_{1}</math> has reached its upper bound or lower bound. During a cycle, if <math display="inline">{\alpha }_{b,1}</math> is small enough, <math display="inline">{Y}_{1}</math> decreases slowly while <math display="inline">{X}_{1}>0</math> (tensile deformation), and then increases when <math display="inline">{X}_{1}<0</math> (compressive deformation). The time spent in compressive deformation is much smaller than in tension, therefore, the variable <math display="inline">{Y}_{1}</math> globally decreases over cycles. However, if <math display="inline">{\alpha }_{b,1}</math> is too big compared to the frequency of the test, the variable <math display="inline">{Y}_{1}</math> reaches its upper and lower bound during the first cycle. This is an unwanted effect because <math display="inline">{Y}_{1}</math> will assume positive values, meaning the mean stress can increase over cycles, which is physically incoherent.

As a result, in order to represent the mean stress evolution properly, <math display="inline">{\alpha }_{b,1}</math> should be chosen accordingly to the frequency of the cyclic test. Particularly, the value of <math display="inline">{\alpha }_{b,1}</math> should be bigger for bigger frequencies. <span id='cite-_Ref480812930'></span>[[#_Ref480812930|Figure 7]] (b) shows the sensitivity to <math display="inline">{\alpha }_{b,1}</math>, using <math display="inline">{\alpha }_{b,1}=</math><math>0.01\ast {\alpha }_{b,1}</math> for Test 2. It appears clearly that <math display="inline">{\alpha }_{b,1}</math> then only influences the mean stress.

The influence of parameters <math display="inline">{Y}_{st,1}</math> and <math display="inline">{r}_{1}</math> is tested using the modified value of <math display="inline">{\alpha }_{b,1}</math> for Test 2. The results are shown in <span id='cite-_Ref480834530'></span>[[#_Ref480834530|Figure 8]]. <math display="inline">{Y}_{st,1}</math> has an influence on the mean stress, as expected. <math display="inline">{r}_{1}</math> has an influence on both stress relaxation and mean stress since this parameter is used both for the variable <math display="inline">{Y}_{1}</math> and for the static recovery term in the equation of the back-stress.

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<div id="_Ref480834530" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 8 - '''(a) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{Y}}}_{\mathit{\boldsymbol{st,1}}}</math><span style="text-align: center; font-size: 75%;">; (b) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{r}}}_{\mathit{\boldsymbol{1}}}</math></div>

===4.7 Cyclic hardening parameters <math display="inline">{\mathit{\boldsymbol{D}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{i}}}}\boldsymbol{,\, }{\mathit{\boldsymbol{a}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{i}}}}\boldsymbol{,\, }{\mathit{\boldsymbol{b}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{i}}}}\boldsymbol{,}{\mathit{\boldsymbol{c}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{i}}}}\boldsymbol{,\, }\mathit{\boldsymbol{\eta }}</math>===

Cyclic hardening is described using five parameters that impact the value of parameter <math display="inline">{\gamma }_{i}</math> over the cycles. Parameters <math display="inline">{D}_{{\gamma }_{i}}</math> and <math display="inline">\eta</math>  control respectively the rate of cyclic hardening and the rate of growth of the plastic strain memorization surface. <span id='cite-_Ref480799055'></span>[[#_Ref480799055|Figure 9]] (a) and (b) show the sensitivity of the stress amplitude and mean stress to <math display="inline">{D}_{{\gamma }_{1}}</math> and <math display="inline">\eta</math> . In both cases, the sensitivity grows to a peak value, and then decreases towards 0 as the values of <math display="inline">{\gamma }_{1}</math> (for <math display="inline">{D}_{{\gamma }_{1}}</math>) and <math display="inline">q</math> (for <math display="inline">\eta</math> ) stabilize. <math display="inline">{a}_{{\gamma }_{i}},\, {b}_{{\gamma }_{i}}</math> and <math display="inline">{c}_{{\gamma }_{i}}</math> control the stabilized value of <math display="inline">{\gamma }_{i}</math>. Therefore, it is expected that these parameters only influence the stabilized value of the stress amplitude, as seen in <span id='cite-_Ref480799055'></span>[[#_Ref480799055|Figure 9]]. Parameters <math display="inline">{a}_{{\gamma }_{i}}</math> and <math display="inline">{b}_{{\gamma }_{i}}</math> have a negative influence while <math display="inline">{c}_{{\gamma }_{i}}</math> has a positive influence on the stress amplitude. These parameters also have a small influence on the mean stress, which can be explained by the fact that the evolution law of <math display="inline">{Y}_{1}</math> depends on the back-stress <math display="inline">{X}_{1}</math>, which is itself dependent on <math display="inline">{\gamma }_{1}</math>. The sensitivity observed for Test 2 on the mean stress should however not be taken into account, as it is the result of the inadequacy of <math display="inline">{\alpha }_{b,1\, }</math> mentioned in the previous paragraph.

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<div id="_Ref480799055" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 9 - '''(a) Cyclic sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{D}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{1}}}}</math><span style="text-align: center; font-size: 75%;">; (b) Cyclic sensitivity to </span> <math display="inline">\mathit{\boldsymbol{\eta }}</math><span style="text-align: center; font-size: 75%;">; (c) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{a}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{1}}}}</math><span style="text-align: center; font-size: 75%;">; (d) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{b}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{1}}}}</math><span style="text-align: center; font-size: 75%;">; (e) Sensitivity to </span> <math display="inline">{\mathit{\boldsymbol{c}}}_{{\mathit{\boldsymbol{\gamma }}}_{\mathit{\boldsymbol{1}}}}</math></div>

===4.8 Maximum temperature parameters <math display="inline">{\mathit{\boldsymbol{b}}}_{\mathit{\boldsymbol{E}}}\mathit{\boldsymbol{,\, }}{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{E}}}^{\mathit{\boldsymbol{S}}}</math>===

Parameters <math display="inline">{b}_{E}</math> and <math display="inline">{f}_{E}^{S}</math>, that represent the influence of the maximum temperature, do not have a visible and clear influence on the criteria used previously. <span id='cite-_Ref480875058'></span>[[#_Ref480875058|Figure 10]] shows the stabilized hysteresis loop of Test 1 for three values of <math display="inline">{f}_{E}^{S}</math>. Small values of <math display="inline">{f}_{E}^{S}</math> - i.e. substantial influence of the maximum temperature - lead to a decrease in the slope of the stress-strain curve in the small temperature domain (corresponding to compression strain). This is an expected result considering that the Young's modulus, which partly controls the slope of the stress-strain curve, decreases with temperature.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:draft_Morch_621684164-image22.png|396px]] </div>

<div id="_Ref480875058" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 10 - '''Influence of parameter </span> <math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{E}}}^{\mathit{\boldsymbol{S}}}</math><span style="text-align: center; font-size: 75%;"> on the stabilized hysteresis loop</span></div>

The influence of <math display="inline">{b}_{E}</math> is difficult to show, as <math display="inline">{b}_{E}</math> represents the rate at which the slope decreases over time. Ahmed, 2013 [2] suggests that <math display="inline">{b}_{E}</math> should be chosen to make <math display="inline">{f}_{E}</math> evolve at a quick rate, so that the saturation value <math display="inline">{f}_{E}^{S}</math> is reached for a cumulative plastic strain <math display="inline">p</math> of approximately 0.2% for Haynes 230.

==5 CONCLUSION==

Based on numerical tests, the sensitivity of the model to each of its parameters was determined based on various criteria. The results of this sensitivity study were analyzed to narrow down the number and type of tests necessary for the determination of each parameter. The study shows that tensile tests, relaxation tests, and cyclic tests are necessary for the determination of the parameters of this model. A special attention should be given to parameter <math display="inline">{\alpha }_{b,1}</math> for the representation of the mean stress evolution. Indeed, the sensitivity study revealed that this parameter must be chosen with consideration to the frequency of the test simulated. Particularly, low-frequency cyclic tests should be modeled with a small value of <math display="inline">{\alpha }_{b,1}</math>.

Further work needs to be made to determine a dependency law between <math display="inline">{\alpha }_{b,1}</math> and the test frequency, based on experimental tests. The sensitivity study can also serve as a basis to establish a method for the determination of the parameters.

==ACKNOWLEDGEMENTS==

The authors acknowledge Cockerill Maintenance & Ingénierie (CMI) and the Interuniversity Attraction Poles Program - Belgian State - Belgian Science Policy (P6-24). A.M. Habraken and L. Duchêne acknowledge the Belgian Fund for Scientific Research FRS-FNRS for its support.

==REFERENCES==

[1] J. L. Chaboche, “A review of some plasticity and viscoplasticity constitutive theories,” ''Int. J. Plast.'', vol. 24, no. 10, pp. 1642–1693, 2008.

[2] R. Ahmed, “Constitutive Modeling for Very High Temperature Thermo-Mechanical Fatigue Responses,” North Carolina State University, 2013.

[3] R. Ahmed, M. Menon, and T. Hassan, “Constitutive Model Development for Thermo-Mechanical Fatigue Response Simulation of Haynes 230,” in ''Proceedings of the ASME 2012 Pressure Vessels & Piping Conference'', 2012, pp. 171–179.

[4] M. Yaguchi, M. Yamamoto, and T. Ogata, “A viscoplastic constitutive model for nickel-base superalloy, part 1: Kinematic hardening rule of anisotropic dynamic recovery,” ''Int. J. Plast.'', vol. 18, no. 8, pp. 1083–1109, 2002.

[5] M. Yaguchi, M. Yamamoto, and T. Ogata, “A viscoplastic constitutive model for nickel-base superalloy, part 2: Modeling under anisothermal conditions,” ''Int. J. Plast.'', vol. 18, no. 8, pp. 1111–1131, 2002.

[6] Z. L. Zhan and J. Tong, “A study of cyclic plasticity and viscoplasticity in a new nickel-based superalloy using unified constitutive equations. Part II: Simulation of cyclic stress relaxation,” ''Mech. Mater.'', vol. 39, no. 1, pp. 73–80, 2007.

[7] “Lagamine code, [http://www.uee.ulg.ac.be/cms/c_2383455/en/lagamine.” http://www.uee.ulg.ac.be/cms/c_2383455/en/lagamine.”] .

[8] P. J. Armstrong and C. O. Frederick, “A Mathematical Representation of the Multi Axial Bauschinger Effect,” CEGB Report RD/B/N 731,Central Electricity Generating Board, 1966.

[9] J. Tong, Z. L. Zhan, and B. Vermeulen, “Modelling of cyclic plasticity and viscoplasticity of a nickel-based alloy using Chaboche constitutive equations,” ''Int. J. Fatigue'', vol. 26, no. 8, pp. 829–837, 2004.