ABED-MERAIM 2017a - Revision history

Scipedia at 16:27, 27 April 2017

2017-04-27T16:27:38Z

Scipedia at 16:26, 27 April 2017

2017-04-27T16:26:10Z

Scipedia at 16:24, 27 April 2017

2017-04-27T16:24:39Z

Scipedia at 16:24, 27 April 2017

2017-04-27T16:24:07Z

Scipediacontent at 13:10, 27 April 2017

2017-04-27T13:10:46Z

Scipedia at 08:50, 27 April 2017

2017-04-27T08:50:52Z

Scipedia at 08:49, 27 April 2017

2017-04-27T08:49:57Z

Scipedia at 08:35, 27 April 2017

2017-04-27T08:35:18Z

Scipedia at 08:30, 27 April 2017

2017-04-27T08:30:58Z

Scipedia at 08:27, 27 April 2017

2017-04-27T08:27:44Z

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 |}
 </div>
-<span style="text-align: center; font-size: 75%;">(a)</span>            <span style="text-align: center; font-size: 75%;">(b)</span>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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 |}
 '''</div>
-(a)          <span style="text-align: center; font-size: 75%;">(b)</span>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

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 |}
 '''</div>
-<span style="text-align: center; font-size: 75%;">(a) </span> <span style="text-align: center; font-size: 75%;">(b)</span>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

@@ Line 571: / Line 571: @@
 <span style="text-align: center; font-size: 75%;">'''Figure 6''': Effect of the hardening exponent <math display="inline">n</math> on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the UT strain path, in the case of stress-controlled nucleation</span></div>
-==12 CONCLUSIONS==
+==5 CONCLUSIONS==
 In this work, The GTN ductile damage model has been coupled with bifurcation theory to predict the occurrence of strain localization for metallic materials. The resulting approach has been implemented into the finite element software ABAQUS/standard in the framework of large plastic strains and a fully three-dimensional formulation. Ductility limits of metallic materials are then predicted using the proposed approach. A parametric study with respect to damage and hardening parameters has been conducted in order to determine the most influential parameters on strain localization. The analysis showed that the damage parameters have a significant impact on the predicted ductility limits. With regard to hardening, it is shown that the choice of void nucleation mechanism has an important influence on the sensitivity of the predicted ductility limits to strain hardening. Indeed, in the case of strain-controlled nucleation, the predicted limit strains were found almost insensitive to strain hardening for most strain paths, while a significant influence was observed in the case of stress-controlled nucleation. The latter leads to non-normality in the plastic flow rule, which plays a destabilizing role that promotes early strain localization. This work also discussed the use of a micromechanics-based calibration for the GTN <math display="inline">q</math>-parameters in the case of strain-controlled nucleation, which is shown to allow accounting for hardening effects on strain localization predictions.

← Older revision		Revision as of 16:24, 27 April 2017
Line 1:		Line 1:
−
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>'''HARDENING EFFECTS ON FORMABILITY LIMIT PREDICTION BASED ON GURSON-TYPE DAMAGE MODELS AND BIFURCATION ANALYSIS'''</big></div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~'''H. CHALAL* AND F. ABED-MERAIM*,†'''</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<span style="text-align: center; font-size: 75%;">*</span> Laboratoire d’Étude des Microstructures et de Mécanique des Matériaux</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~LEM3, UMR CNRS 7239, Arts et Métiers ParisTech</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~4 rue Augustin Fresnel, 57078 Metz Cedex 3, France</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~e-mail: [mailto:farid.abed-meraim@ensam.eu farid.abed-meraim@ensam.eu], [mailto:hocine.chalal@ensam.eu hocine.chalal@ensam.eu], [http://www. www.]ensam.eu</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<sup>†</sup> Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~DAMAS, Université de Lorraine</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~Ile du Saulcy 57045 Metz, Cedex 1, France</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~e-mail: [mailto:farid.abed-meraim@ensam.eu farid.abed-meraim@ensam.eu], [http://www. www.]ensam.eu</div>~~
−
−	~~'''Key words:''' Bifurcation analysis, GTN-type damage model, Formability limits, Hardening effects, Void nucleation mechanism, Micromechanics-based calibration.~~
−
	'''Abstract.''' In this work, ductility limits of metallic materials, associated with the occurrence of strain localization, are predicted using the GTN damage model coupled with bifurcation theory. The resulting approach is implemented into the finite element code ABAQUS within the framework of large plastic strains and a fully three-dimensional formulation. A parametric study with respect to damage and hardening parameters is conducted in order to identify the most influential material parameters on strain localization. The analysis shows that the damage parameters have a significant impact on the predicted ductility limits, while the effect of hardening parameters on strain localization depends on the choice of void nucleation mechanism.		'''Abstract.''' In this work, ductility limits of metallic materials, associated with the occurrence of strain localization, are predicted using the GTN damage model coupled with bifurcation theory. The resulting approach is implemented into the finite element code ABAQUS within the framework of large plastic strains and a fully three-dimensional formulation. A parametric study with respect to damage and hardening parameters is conducted in order to identify the most influential material parameters on strain localization. The analysis shows that the damage parameters have a significant impact on the predicted ductility limits, while the effect of hardening parameters on strain localization depends on the choice of void nucleation mechanism.

@@ Line 445: / Line 445: @@
 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;"
 |-
-|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">q</math>-parameter
-<math display="inline">q</math>
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">n=0.025</math>
 -parameter
 |  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
 <math display="inline">n=0.025</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">n=0.05</math>
 <math display="inline">n=0.05</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">n=0.10</math>
 <math display="inline">n=0.10</math>
 |-
-|  style="border-top: 1pt solid black;border-left: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-left: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">q_1</math>
 <math display="inline">q_1</math>
-|  style="border-top: 1pt solid black;text-align: center;"|1.74
+|  style="border-top: 1pt solid black;text-align: center; width: 150px;"|1.74
-|  style="border-top: 1pt solid black;text-align: center;"|1.48
+|  style="border-top: 1pt solid black;text-align: center; width: 150px;"|1.48
-|  style="border-top: 1pt solid black;border-right: 1pt solid black;text-align: center;"|1.29
+|  style="border-top: 1pt solid black;border-right: 1pt solid black;text-align: center; width: 150px;"|1.29
 |-
-|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">q_2</math>
 <math display="inline">q_2</math>
-|  style="border-bottom: 1pt solid black;text-align: center;"|1.013
+|  style="border-bottom: 1pt solid black;text-align: center; width: 150px;"|1.013
-|  style="border-bottom: 1pt solid black;text-align: center;"|1.013
+|  style="border-bottom: 1pt solid black;text-align: center; width: 150px;"|1.013
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;"|0.982
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center; width: 150px;"|0.982
 |}
@@ Line 478: / Line 471: @@
 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;"
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|Material
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|Material
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">f_0</math>
 <math display="inline">f_0</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">s_N</math>
 <math display="inline">s_N</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">{\epsilon }_N</math>
 <math display="inline">{\epsilon }_N</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">f_N</math>
 <math display="inline">f_N</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">f_{cr}</math>
 <math display="inline">f_{cr}</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|<math display="inline">{\delta }_{GTN}</math>
 <math display="inline">{\delta }_{GTN}</math>
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|Steel
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|Steel
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.001
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|0.001
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.1
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|0.1
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.3
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|0.3
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.05
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|0.05
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.04
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|0.04
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|5
+|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center; width: 150px;"|5
 |}

@@ Line 428: / Line 428: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 3''': Effect of the hardening exponent <math display="inline">n</math>on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the uniaxial tensile strain path</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 3''': Effect of the hardening exponent <math display="inline">n</math> on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the uniaxial tensile strain path</span></div>
 ==4.2 Calibration of the GTN <math display="inline">q</math>-parameters==

@@ Line 388: / Line 388: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 1''': Effect of the initial void volume fraction
+<span style="text-align: center; font-size: 75%;">'''Figure 1''': Effect of the initial void volume fraction <math display="inline">f_0</math> (a), and the nucleation parameter <math display="inline">f_N</math> (b) on the ductility limit predictions for Al5754 aluminum</span></div>
 <math display="inline">f_0</math>
 (a), and the nucleation parameter
 <math display="inline">f_N</math>
 (b) on the ductility limit predictions for Al5754 aluminum</span></div>
-Figure 1 shows the impact of varying one damage parameter at a time on the prediction of the ductility limits for Al5754 aluminum. Concerning the effect of the initial void volume fraction
+Figure 1 shows the impact of varying one damage parameter at a time on the prediction of the ductility limits for Al5754 aluminum. Concerning the effect of the initial void volume fraction <math display="inline">f_0</math> (Figure 1(a)), large values for this parameter (e.g., <math display="inline">f_0=0.01</math>) imply that the material has already entered the coalescence stage, which dramatically lowers the predicted ductility limits. However, for very small values for parameter <math display="inline">f_0</math>, the ductility limit predictions are only slightly affected, which suggests that at such low void volume fraction levels, void growth is not the predominant mechanism for damage evolution. For the nucleation parameter <math display="inline">f_N</math>, the predicted ductility limits are lowered as this parameter increases. This trend is consistent with the physical meaning of this parameter (volume fraction of void-nucleating particles), as larger values for the latter tend to precipitate damage, thus promoting early plastic flow localization (see Eq. (6)).
 <math display="inline">f_0</math>
 (Figure 1(a)), large values for this parameter (e.g.,
 <math display="inline">f_0=0.01</math>
 ) imply that the material has already entered the coalescence stage, which dramatically lowers the predicted ductility limits. However, for very small values for parameter
 <math display="inline">f_0</math>
 , the ductility limit predictions are only slightly affected, which suggests that at such low void volume fraction levels, void growth is not the predominant mechanism for damage evolution. For the nucleation parameter
 <math display="inline">f_N</math>
 , the predicted ductility limits are lowered as this parameter increases. This trend is consistent with the physical meaning of this parameter (volume fraction of void-nucleating particles), as larger values for the latter tend to precipitate damage, thus promoting early plastic flow localization (see Eq. (6)).
 ==4.1.2 Effect of the hardening exponent <math display="inline">n</math>==
@@ Line 424: / Line 412: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 2''': Effect of the hardening exponent
+<span style="text-align: center; font-size: 75%;">'''Figure 2''': Effect of the hardening exponent <math display="inline">n</math>, associated with the Swift law, on the ductility limit predictions for Al5754 aluminum with strain-controlled nucleation</span></div>
 <math display="inline">n</math>
 , associated with the Swift law, on the ductility limit predictions for Al5754 aluminum with strain-controlled nucleation</span></div>
 Similar trends have been observed in [9], where the GTN model was used with strain-controlled nucleation and coupled with the bifurcation theory. Indeed, in such a modeling approach, strain localization is mainly controlled by damage-induced softening, as shown in Figure 3(a) for the uniaxial tensile (UT) strain path, where it can be seen that flow localization occurs at strongly negative hardening moduli. Moreover, the evolution of void volume fraction based on strain-controlled nucleation for this particular loading path (UT) is shown to be insensitive to the strain hardening of the dense matrix material (see Figure 3(b)).
@@ Line 442: / Line 428: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 3''': Effect of the hardening exponent
+<span style="text-align: center; font-size: 75%;">'''Figure 3''': Effect of the hardening exponent <math display="inline">n</math>on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the uniaxial tensile strain path</span></div>
 <math display="inline">n</math>
 on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the uniaxial tensile strain path</span></div>
 ==4.2 Calibration of the GTN <math display="inline">q</math>-parameters==
@@ Line 610: / Line 594: @@
 |}
-Figure 5 illustrates the effect of the hardening exponent
+Figure 5 illustrates the effect of the hardening exponent <math display="inline">n</math> on the prediction of limit strains for the studied steel material. It is clearly shown that the consideration of non-normality in the GTN model, due to stress-controlled nucleation, allows for a significant effect of strain hardening on the limit strains. Indeed, the predicted limit strains increase as the hardening exponent <math display="inline">n</math> increases, which is consistent with the literature findings (see, e.g., [11]). The effect of the hardening exponent <math display="inline">n</math> on the evolution of the Cauchy stress and the void volume fraction until localization for the UT strain path is shown in Figure 6. It can be seen that, in contrast to the case of strain-controlled nucleation (see Figure 3(b)), the evolution of void volume fraction is significantly affected by the hardening exponent <math display="inline">n</math>, which allows accounting for strain hardening effects on strain localization. Moreover, the Cauchy stress evolution reveals that the hardening modulus at localization is not strongly negative, as compared to that obtained in the case of strain-controlled nucleation. This is caused by the non-normality of the plastic flow, which plays a destabilizing role in the localization bifurcation analysis.
 <math display="inline">n</math>
 on the prediction of limit strains for the studied steel material. It is clearly shown that the consideration of non-normality in the GTN model, due to stress-controlled nucleation, allows for a significant effect of strain hardening on the limit strains. Indeed, the predicted limit strains increase as the hardening exponent
 <math display="inline">n</math>
 increases, which is consistent with the literature findings (see, e.g., [11]). The effect of the hardening exponent
 <math display="inline">n</math>
 on the evolution of the Cauchy stress and the void volume fraction until localization for the UT strain path is shown in Figure 6. It can be seen that, in contrast to the case of strain-controlled nucleation (see Figure 3(b)), the evolution of void volume fraction is significantly affected by the hardening exponent
 <math display="inline">n</math>
 , which allows accounting for strain hardening effects on strain localization. Moreover, the Cauchy stress evolution reveals that the hardening modulus at localization is not strongly negative, as compared to that obtained in the case of strain-controlled nucleation. This is caused by the non-normality of the plastic flow, which plays a destabilizing role in the localization bifurcation analysis.
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 626: / Line 602: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 5''': Effect of the hardening exponent
+<span style="text-align: center; font-size: 75%;">'''Figure 5''': Effect of the hardening exponent <math display="inline">n</math> on the prediction of ductility limits for the studied steel material with stress-controlled nucleation</span></div>
 <math display="inline">n</math>
 on the prediction of ductility limits for the studied steel material with stress-controlled nucleation</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 642: / Line 616: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 6''': Effect of the hardening exponent
+<span style="text-align: center; font-size: 75%;">'''Figure 6''': Effect of the hardening exponent <math display="inline">n</math> on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the UT strain path, in the case of stress-controlled nucleation</span></div>
 <math display="inline">n</math>
 on: (a) the Cauchy stressstrain curve, and (b) void volume fraction, until localization along the UT strain path, in the case of stress-controlled nucleation</span></div>
 ==12 CONCLUSIONS==
-In this work, The GTN ductile damage model has been coupled with bifurcation theory to predict the occurrence of strain localization for metallic materials. The resulting approach has been implemented into the finite element software ABAQUS/standard in the framework of large plastic strains and a fully three-dimensional formulation. Ductility limits of metallic materials are then predicted using the proposed approach. A parametric study with respect to damage and hardening parameters has been conducted in order to determine the most influential parameters on strain localization. The analysis showed that the damage parameters have a significant impact on the predicted ductility limits. With regard to hardening, it is shown that the choice of void nucleation mechanism has an important influence on the sensitivity of the predicted ductility limits to strain hardening. Indeed, in the case of strain-controlled nucleation, the predicted limit strains were found almost insensitive to strain hardening for most strain paths, while a significant influence was observed in the case of stress-controlled nucleation. The latter leads to non-normality in the plastic flow rule, which plays a destabilizing role that promotes early strain localization. This work also discussed the use of a micromechanics-based calibration for the GTN
+In this work, The GTN ductile damage model has been coupled with bifurcation theory to predict the occurrence of strain localization for metallic materials. The resulting approach has been implemented into the finite element software ABAQUS/standard in the framework of large plastic strains and a fully three-dimensional formulation. Ductility limits of metallic materials are then predicted using the proposed approach. A parametric study with respect to damage and hardening parameters has been conducted in order to determine the most influential parameters on strain localization. The analysis showed that the damage parameters have a significant impact on the predicted ductility limits. With regard to hardening, it is shown that the choice of void nucleation mechanism has an important influence on the sensitivity of the predicted ductility limits to strain hardening. Indeed, in the case of strain-controlled nucleation, the predicted limit strains were found almost insensitive to strain hardening for most strain paths, while a significant influence was observed in the case of stress-controlled nucleation. The latter leads to non-normality in the plastic flow rule, which plays a destabilizing role that promotes early strain localization. This work also discussed the use of a micromechanics-based calibration for the GTN <math display="inline">q</math>-parameters in the case of strain-controlled nucleation, which is shown to allow accounting for hardening effects on strain localization predictions.
 <math display="inline">q</math>
 -parameters in the case of strain-controlled nucleation, which is shown to allow accounting for hardening effects on strain localization predictions.
 ==REFERENCES==

@@ Line 457: / Line 457: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Table 2''': Calibrated
+<span style="text-align: center; font-size: 75%;">'''Table 2''': Calibrated <math display="inline">q</math>-parameters</span></div>
 <math display="inline">q</math>
 -parameters</span></div>
 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;"
@@ Line 531: / Line 529: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Figure 4''': Effect of the hardening exponent
+<span style="text-align: center; font-size: 75%;">'''Figure 4''': Effect of the hardening exponent <math display="inline">n</math> on the prediction of ductility limits using the calibrated <math display="inline">q</math>-parameters</span></div>
 <math display="inline">n</math>
 on the prediction of ductility limits using the calibrated
 <math display="inline">q</math>
 -parameters</span></div>
-<span id='_GoBack'></span>Figure 4 shows the effect of the hardening exponent
+<span id='_GoBack'></span>Figure 4 shows the effect of the hardening exponent <math display="inline">n</math> of the power law on the prediction of ductility limits for the studied steel material based on the calibration of the <math display="inline">q</math>-parameters and strain-controlled nucleation model. The predicted ductility limits clearly show sensitivity to strain hardening for all strain paths, thanks to the use of micromechanics-based calibrated <math display="inline">q</math>-parameters.
 <math display="inline">n</math>
 of the power law on the prediction of ductility limits for the studied steel material based on the calibration of the
 <math display="inline">q</math>
 -parameters and strain-controlled nucleation model. The predicted ductility limits clearly show sensitivity to strain hardening for all strain paths, thanks to the use of micromechanics-based calibrated
 <math display="inline">q</math>
 -parameters.
 ==4.3 Stress-controlled nucleation model==

@@ Line 550: / Line 550: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 <span style="text-align: center; font-size: 75%;">'''Table 4''': Hardening and damage parameters for the studied steel material</span></div>
 {| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"