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Damage evolution in TWIP and standard austenitic steel by means of 3D X ray tomography
Materials Science & Engineering A 579 (2013) 92–98
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Damage evolution in TWIP and standard austenitic steel by means of 3D X ray tomography D. Fabrègue a,b,n, C. Landron a,b, O. Bouaziz c, E. Maire a,b a
Université de Lyon, CNRS, F-69621 Villeurbanne, France INSA-Lyon, MATEIS UMR5510, F-69621 Villeurbanne, France c ArcelorMittal Research, Voie Romaine-BP30320, F-57283 Maizières les Metz, France b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 October 2012 Received in revised form 5 May 2013 Accepted 8 May 2013 Available online 15 May 2013
The evolution of ductile damage of Fe–22Mn–0.6C austenitic TWIP steel by means of 3D X ray tomography in-situ tensile tests is reported for the first time. The comparison with another fully austenitic steel (316 stainless steel) is also carried out. The damage process of TWIP steel involves intense nucleation of small voids combined with the significant growth of the biggest cavities whereas macroscopical triaxiality remains constant. Due to this high nucleation rate, the average cavity diameter remains constant unlike the 316 stainless steel. & 2013 Elsevier B.V. All rights reserved.
Keywords: Austenitic steel 3D characterization Damage initiation Modeling Tomography
1. Introduction The high manganese austenitic steels discovered by Sir Hadfield in 1880 constitute one of the most attractive materials for structural application since they exhibit a unique combination of strength and ductility [1]. However, to widen their application areas, some issues must be tackled. Their fracture behavior is one of them. Indeed, despite the fact that they have been studying for some time, the mechanisms leading to their fracture are still unclear. Reports often discuss the macroscopical features of the fracture. For example, it has been shown that high manganese steels break in the uniform elongation range before necking at room temperature as shown in TWIP 940 by Chung [2]. Moreover, the fracture surface exhibits a slant surface whatever the stress state was [3]. However the microscopic features that lead to fracture are still under discussion. This subject has been addressed in very few papers and mainly on Hadfield steel (i.e. with a Mn content between 12% and 14%). Bayraktar [4] suggested that the fracture occurs by microvoid coalescence without necking. On the other hand, Abbasi [5] proposed a fracture mechanism based more on the occurrence of intense nucleation. According to that study,
n Correspondence to: INSA-Lyon, MATEIS, 25 Avenue Jean Capelle, UMR5510, F-69621 Villeurbanne, France. Tel.: +33 472438179. E-mail address: [email protected] (D. Fabrègue).
0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.05.013
nucleation events can take place on inclusions or carbides (but the steel considered in our study does not contain that type of carbides) for large dimples whereas small dimples stem from Mn–C couples and the dynamic strain aging process characteristics of this type of steel [1]. X ray tomography has recently been used [6] on the same steel after fracture of a butterfly specimen. These authors showed the presence of strings of elongated voids aligned in the rolling direction. The authors linked this elongated shape to the plastic anisotropy measured (Lankford coefficients) saying that the deformation of the voids is dictated by the surrounding material when the cavity growth is limited (low triaxiality level). They also showed that the volume fraction of these large elongated voids is not sufficient to explain the fracture of high manganese austenitic steel. They then suggested that a large amount of nucleation events of small voids accompanied by rapid coalescence was responsible for the final fracture. This study gives a lot of information on the damage mechanisms of high manganese TWIP steel but it is restricted to post fracture analysis. Thus an “in situ” experiment for different deformation states is required in order to have access to the evolution of the voids in terms of numbers and size as a function of strain. This will permit to get a better insight on the phenomena responsible for the fracture of TWIP steels. Thus in situ tensile experiments are reported in this paper. For the sake of comparison, the study is realized on one hand on the TWIP steel and on the other hand on a classic (i.e. containing no Mn) austenitic stainless steel (namely a 316L). The evolution of the
D. Fabrègue et al. / Materials Science & Engineering A 579 (2013) 92–98
number of voids, of their size and of their different shape factors is given. Then the experimental results are compared to analytical models describing the evolution of the diameter of the cavities.
2. Experiment This study compares the damage evolution in two austenitic steels namely 316L and TWIP steel. The first is a stainless steel with a composition of 0.02% C, 16% Cr, 11% Ni, and 2% Mo (balance iron). TWIP steel is high Mn steel where the austenite is stable at room temperature. No other phase such as carbides is observed at room temperature. Its composition is 22% Mn–0.6% C (composition in weight percent, balance iron) with an average grain size diameter of about 2–3 mm, supplied by ArcelorMittal. A sample from both steels was cut from a 1 mm thick sheet obtained by hot rolling and annealing thermal treatment. Micro tensile specimens were machined by electrical discharge machining according to the shape shown in Fig. 1. X ray microtomography was used to quantify damage during in situ tensile tests. The tomography set-up is located at the ID15 beam line at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Tomography acquisition was performed with a voxel size of 1.6 mm3. Thus the void diameters are calculated with an accuracy of 71.6 mm. Initial reconstructed volumes were median filtered and simply thresholded to differentiate the material from the voids by absorption difference. Refer to [7] for more information on the precise experimental procedure. In order to correlate the distribution of voids with the strain, local values of the strain are obtained by considering the minimum section area Smin and using the relationship: S0 ϵloc ¼ ln Smin
ð1Þ
where S0 is the initial section of the sample. The local strain is then calculated at each step. Using this equation implies that the volume fraction of the voids considered is small enough to keep the total volume unchanged. The beginning of the tensile test is carried out with a stress triaxiality of 0.33. After some deformation, triaxiality may evolve if the sample shape changes. This is considered by using the Bridgman formula [8] modified by Wierzbicki [9] considering the curvature radius of the surface of the sample RS: 1 pffiffiffi a T ¼ þ 2ln 1 þ 3 2RS
93
3. Results and discussion 3.1. Tensile behavior Mechanical behavior (stress/strain) was monitored during the 3D X ray tomography experiment. The tensile curves obtained for the two steels are given in Fig. 2. It is worth noticing that these curves were calculated considering the minimum section of the sample, thus the stress and strain values are very high since they constitute local values and not macroscopic ones. These values may be considered as ultimate ones. It is clearly seen that both steels behave completely differently. 316L presents a yield strength of about 400 MPa and then exhibits a large strain hardening up to maximal strength of about 1600 MPa at a strain of 1.6. On the other hand, the mechanical properties of TWIP steel are outstanding with an ultimate tensile strength of about 2600 MPa and a fracture strain of about 0.55. The yield strength seems to be about 800 MPa but this is a rough estimation since no points were recorded at very low strain. Reports state a value of 400 MPa for yield strength at room temperature [10]. The strain hardening is very high compared to other steel grades and thus larger than the strain hardening of the 316L. The values observed here are in accordance with other studies [10] and explain the current interest in these steels and their promising use in safety parts in the automotive industry. It is important to note that the deformation modes of these two steels have been observed to be the same i.e. twinning and dislocation glide [10,11]. Fig. 3 shows the 3D tomography of the two investigated steels just before final fracture. For the sake of comparison, the same state for DP steel from [12] is also shown. In these three images the voids are underlined by an opaque red color surface while the outside surface of the sample is transparent. The differences in fracture behavior between the DP steel with a ferritic matrix, the austenitic steels 316L and TWIP are obvious in this figure. When the first ones exhibit an important necking and a very high void density before fracture, TWIP steel shows no localization of the deformation and only a small void density. It is important to note that in the case of 316L the cavities seem to be aligned in the direction of the tensile stress. This could be due to heterogeneities in the material such as chemical segregation [13]. Figs. 2 and 3 underline that even if the underlying deformation mechanisms are the same in 316L and TWIP (i.e. mechanical twinning and dislocation glide), they lead to different mechanical macroscopic behaviors and that damage behavior must be also 3000
ð2Þ
TWIP
2500
316L
a being the width of the minimum section. Stress (MPa)
2000
1500
1000
500
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Strain Fig. 1. Specimen geometry for in situ X ray tomography.
Fig. 2. Stress/strain curve obtained during the in situ tensile test.
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Fig. 3. 3D views of the damage at the center of the specimen strained at (a) TWIP steel at εloc ¼ 0.5 (just before fracture), (b) DP steel just before fracture [12] and (c) 316L just before fracture.
105
105 density of voids in TWIP(n/mm3)
density of voids in TWIP(n/mm3)
density of voids in 316L(n/mm3) Density of voids(n/mm3)
Density of voids(n/mm3)
density of voids in 316L(n/mm3) 104
1000
100
104
1000
100 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
local strain
0.4
0.6
0.8
1
1.2
local strain/fracture strain
Fig. 4. (a) Void density as a function of the local strain for TWIP and 316 steel and (b) void density as a function of the local strain divided by the fracture strain for the same two steels.
different. The next part will now focus on the quantification of the different damage features in these austenitic steels. 3.2. Void nucleation Fig. 4a shows the evolution of the number of voids as a function of the local strain and Fig. 4b reports the same parameter as a function of the normalized strain (equal to the local strain divided by the fracture strain) for the two austenitic steels under study. In the case of 316L austenitic steel, the void density evolves as follows: after a plateau from normalized strain of 0 to normalized strain of about 0.2, the number of voids increases exponentially with the normalized strain. In the case of TWIP steel, no plateau can be observed, the number of voids increases like that in 316L. This result indicates that no sudden and large nucleation takes place to explain the fracture of TWIP steel at least at the scale observable with X ray tomography: nucleation occurs during the entire deformation process in a continuous manner. The high nucleation rate observed in the two cases could be due to a high number of nucleation sites. Nucleation could possibly take place at the grain boundaries or at the twin interface or at least, the twins could promote void nucleation due to the high density of dislocation in their vicinity. This fact is hard to check since the twins are usually very narrow (especially in TWIP steel) and thus only post mortem techniques can give access to that. It is also worth noticing
that the number of voids at fracture is very low in the case of these two homogeneous samples (especially for the TWIP steel) compared to all other types of ductile material at fracture already published with a similar imaging resolution [14–16]. Considering the average diameter of the cavities (see next section) these numbers correspond to volume fraction of voids at fracture of about 3 10−3 and 0.2 for the TWIP and the 316L respectively. The fraction for the TWIP is very low compared to the usual values found in the literature. 3.3. Void growth X ray tomography allows the separation of void nucleation from void growth. This leads to a better understanding of the damage phenomenon. The volume (number of voxels) of each cavity was measured and its equivalent diameter was then calculated assuming a spherical shape. The absolute error in this measurement is constant (of the order of one voxel). As a consequence, the relative error in measurement of the dimensions depends on the dimensions themselves: it is rather large (100%) when the cavities nucleate (due to very small size) but then quickly becomes negligible as the growth occurs. Fig. 5 presents the evolution of the average void diameter and the evolution of the average diameter of the 20 biggest cavities in order to eliminate nucleation on the average diameter for the two steels considered.
D. Fabrègue et al. / Materials Science & Engineering A 579 (2013) 92–98
18 total population of voids 20 largest voids
16
Average equivalent diameter (µm)
Average equivalent diameter (µm)
18
95
14 12 10 8 6 4 2
Total population of voids 20 largest voids
16 14 12 10 8 6 4 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.2
0.4
local strain
0.6
0.8
1
1.2
1.4
1.6
local strain
Fig. 5. Evolution of the average void diameter and of the 20 biggest voids for (a) TWIP and (b) 316L.
1
0.8
Triaxiality
The results reported in Fig. 5 highlight the effect of nucleation and growth using the methodology detailed in [7]. Considering the average diameter of the total population, in TWIP steel the diameter is almost constant during all the deformation process with a value of about 4.5 mm. However growth is experienced by the largest cavities. This growth is very large since the diameter of the largest cavities almost doubled at a fracture strain of 0.5. In the case of 316L, the value of the average diameter is almost the same as that of the biggest ones up to 0.5 strain. The average diameter of all cavities increases when the local strain attains 1 to reach about 6 mm at fracture. This is due to an increase in the growth rate observed from this value of local strain when considering the 20 largest cavities. That leads to the conclusion that the growth of cavities is counter balanced by the nucleation of new ones in the TWIP whereas growth outbalances nucleation in 316L when local strain increases. In order to get a better insight into the growth process, the evolution of stress triaxiality has to be taken into account. Fig. 6 shows the evolution of this triaxiality with the local strain. Stress triaxiality during the tensile test of TWIP steel remains constant and equal to 1/3. This is consistent with the fact that no necking appears (Fig. 3). On the other hand, in the case of 316L, triaxiality stays constant up to a strain of 0.3 and then a large increase in triaxiality is observed. Then the cavity growth rate in the two steels can be clarified. As mentioned in previous reports, growth is mainly governed by the triaxiality level [17,18]. Increasing triaxiality thus accelerates cavity growth. From Figs. 5 and 6, it is clear that the growth rate is much larger in TWIP compared to 316L since the diameter of the 20 largest cavities increases more in TWIP than in 316L. In fact, the diameter of these voids is almost equal to 8.5 mm in TWIP at a strain¼0.5 whereas it is equal to 5 mm in 316L for the same strain (knowing that the initial diameter is almost the same in both cases). This large increase is more noticeable since stress triaxiality remains constant for TWIP whereas it increases to 0.42 in 316L at a strain of 0.5. This large cavity growth rate could be due to a change in cavity shape and/or to the outstanding kinematic hardening of TWIP. It has been shown [19] that higher kinematic hardening leads to an enhancement of void growth. These data lead to the conclusion that nucleation is intense in the TWIP sample because it counter balances the increase in size of the biggest voids due to growth/ coalescence. It seems that the fracture of TWIP sample is not controlled only by a higher nucleation rate than in other austenitic steels as suggested by Abbasi [5] talking about Hadfield steels but also by some coalescence because the biggest cavities grow faster
0.6
0.4
Triaxiality 316L Triaxiality TWIP
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Strain Fig. 6. Evolution of stress triaxiality as a function of the strain for TWIP and 316L steels.
in TWIP than in other steels as proposed by Bayraktar [4] (also about Hadfield steels). 3.4. Modeling 3.4.1. Growth Rice and Tracey proposed a model for cavity growth [17], based on the hypothesis of a spherical cavity in a perfectly plastic infinite matrix. This model shows that if a homogeneous strain field is applied, growth rate is described by: dR 3 ¼ αRT exp T ð3Þ Rdϵ 2 where T is the stress triaxiality and αRT is a constant. This relationship was modified by Huang [18] in order to extend the approach to low values of triaxiality: dR 3 ¼ αH T 0:25 exp T ð4Þ Rdϵ 2 This relationship has been demonstrated to give good results on different types of steel with a ferritic type matrix (ferrite or martensite) [20] provided that αH is identified from experimental results. This model is used here for both austenitic steels. The results of the experiments and of the model are shown in Fig. 7.
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18
18 20 largest voids Model
20 largest voids Model
16 Average equivalent diameter (µm)
Average equivalent diameter (µm)
16 14 12 10 8 6 4
14 12 10 8 6 4
2
2 0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.2
0.4
local strain
0.6
0.8
1
1.2
1.4
1.6
local strain
Fig. 7. Evolution of the diameter of the 20 largest cavities experimentally and prediction using Huang's model on (a) TWIP and (b) 316L.
Table 1 Factor of Huang's law for cavity growth. The values for DP, IF and martensitic steels are from [20,21].
Value of αH
TWIP
316L
DP steel
IF steel
Martensitic steel
0.9
0.5
0.55
0.22
1.6
9 total population of voids Model
8
Average equivalent diameter (µm)
Average equivalent diameter (µm)
9
7 6 5 4 3
Total population of voids Model
8 7 6 5 4 3
0
0.1
0.2
0.3
0.4
0.5
0.6
Local strain
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Local strain
Fig. 8. Evolution of the average equivalent diameter as a function of the local strain for (a) TWIP and (b) 316L. Experiments (squares) and prediction using Eq. (5) (circles).
The values of the pre exponential term giving the best fit for these two austenitic steels are given in Table 1. Some values obtained in [20,21] for different ferritic matrix steels are also given in this table for the purpose of comparison. The value of αH for TWIP is very high compared to that of DP steel but lower to that of fully martensitic steel. Until now, this trend has not been explained and must be investigated in detail in the future. However, one possible explanation could be the influence of the viscosity on cavity growth rate as suggested by Budiansky [22].
3.4.2. Description of the total average diameter Huang's relationship has been demonstrated to accurately describe the evolution of the diameter of the 20 largest voids i.e. it describes the growth regime. In order to describe the evolution
of the average diameter of the whole cavity population, nucleation should also be taken into account. The approach proposed by Bouaziz [23] is used in the following. The equation permitting to describe the average radius is given by: dRg 1 dN dR ¼ ðR−R0 Þ − dε dε N dε
ð5Þ
dRg =dε is the evolution of the void radius due to growth (Huang's law), N is the number of cavities by mm3 and R0 the radius of cavity at nucleation. More details on this law can be found in [23]. It has been shown [23] that this law is able to reproduce the evolution of the average radius for different steels. Fig. 8 presents the evolution of the average diameter measured thanks to 3D X ray tomography experiments compared to the above model.
D. Fabrègue et al. / Materials Science & Engineering A 579 (2013) 92–98
5
5 Rz/Rx
Rz/Rx 4.5
Ry/Rx
4
4
3.5
3.5
Shape factor
Shape factor
4.5
3 2.5
Ry/Rx
3 2.5
2
2
1.5
1.5
1
97
1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.1
local strain
0.2
0.3
0.4
0.5
0.6
local strain
Fig. 9. Evolution of the shape factors of the 20 biggest cavities for (a) 316L and (b) TWIP.
The radius at nucleation was fixed at 1.8 mm for TWIP and 316L. As can be seen from Fig. 8, the model predicts a constant average equivalent diameter for TWIP steel which is consistent with the experiments. In the case of 316L, the average diameter seems to increase with the strain. This trend is captured by the model. However, the model underestimates the growth. This discrepancy can be explained by an increase in radius at nucleation due to nucleation events appearing successively at different sites at twin boundaries and at grain boundaries. This point has to be checked by more detailed experiments. It is important to note that the large increase at high strain level can also be due to the phenomenon of coalescence which is not taken into account in this approach. 3.5. Evolution of the cavity shape factors Since shape of the cavities could play a major role in the phenomenon of coalescence and thus could govern the ultimate strain in a damage process, the shape factors are going to be addressed. Tomography enables us to measure them as a function of strain as shown in Fig. 9. Fig. 9 gives more information about the damage behavior in both steels. In the case of 316L, for low strains, the cavities are quite spherical as the two shape factors are less than 2. However as the strain increases, the ratio Rz/Rx raises significantly at strains above 0.6 up to a value of 4 just before fracture whereas Ry/Rx remains constant. This means that the cavities become elongated in the strain direction. This result can be also qualitatively deduced from Fig. 3c. The evolution of the cavities in the TWIP steel is different. Indeed, initially the cavities are somewhat elongated in the direction perpendicular to the strain: Ry/Rx is equal to 2 71.2, which is reached for Rz/Rx around the strain of 0.4 increasing from 1.370.5. This could be due to nucleation of the cavities at the twin interfaces. Then as usual, when the strain increases, the Rz/Rx ratio increases. At fracture, the cavities are ellipsoidally elongated as the two shape factors become around 2. This evolution of the shape factors suggests that the cavities grow by blunting like precrack. This explains the possibility of large cavity growth observed in Fig. 5 whereas the stress triaxiality stays equal to 0.33. 3.6. Observation of shear bands When looking at the surface of the TWIP sample just before fracture, some bands may be observed as shown in Fig. 10. These bands were not present at the beginning of loading. They are
Fig. 10. 3D picture of TWIP steel just before fracture showing shear bands.
oriented with an angle equal to about 451 with the loading direction. They could be Portevin-Le Chatelier (PLC) bands emanating at the surface of the sample due to deformation. This phenomenon has already been observed in this material [3]. These authors show that these PLC bands are responsible for the initiation of final fracture. This is in accordance with our study where the bands are can be detected only at the last deformation step. However, no obvious influence of these bands on growth or cavity nucleation in the vicinity is shown.
4. Conclusions 3D X ray tomography has been used to get a better insight into the damage process of austenitic TWIP steel. It shows that although triaxiality remains constant equal to 0.33, cavity growth is experienced and seems to be important. However as the average cavity diameter remains constant, it means that the nucleation rate is very high. Thus the fracture of this steel is due to both
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intense nucleation and considerable cavity growth. It seems that the large increase of cavity diameter comes from a blunting type growth. This increase in cavity diameter could be also due to the presence of shear bands involving an increase in local triaxiality and permitting cavity growth. However due to the small size of the cavities, tomography with higher resolution is needed to have a better idea of the real number of voids and to suggest a clear scenario for the final fracture. In addition, the comparison in damage evolution with 316L austenitic stainless steel with a lower strain-hardening and a lower flow stress showed clear differences: in this case growth is not compensated for by nucleation events leading to an increase in the average cavity diameter. The respective possible roles of twins, of kinematic hardening and of early coalescence have been suggested but they will need to be investigated in detail in future experiments. References [1] O. Bouaziz, S. Allain, C.P. Scott, P. Cugy, D. Barbier, Curr. Opin. Solid State Mater. Sci. 15 (2011) 141–168. [2] K. Chung, K. Ahn, D.H. Yoo, K.H. Chung, M.H. Seo, S.H. Park, Int. J. Plast. 27 (2011) 52–81. [3] G. Scavino, F. D’Aiuto, P. Matteis, P.R. Spena, D. Firrao, Metall. Mater. Trans. A 41 (2010) 1493–1501. [4] E. Bayraktar, F.A. Khalid, C. Levaillant, J. Mater. Process. Technol. 147 (2004) 145–154.
[5] M. Abbasi, S. Kheirandish, Y. Kharrazi, J. Hejazi, Mater. Sci. Eng. A 72 (2009) 513–514. [6] J. Lorthios, F. Nguyen, A.F. Gourgues, T.F. Morgeneyer, P. Cugy, Scr. Mater. 63 (2010) 1220–1223. [7] E. Maire, O. Bouaziz, M. Di Michiel, C. Verdu, Acta Mater. 56 (2008) 4954–4964. [8] P.W. Bridgman, Rev. Mod. Phys. 17 (1945) 3–14. [9] T. Wierzbicki, Y. Bao, Bridgman Revisited: On the History Effect on Ductile Fracture, Massachusetts Institute of Technology, Cambridge, MA, 2004. [10] S. Allain, J.P. Château, D. Dahmoun, O. Bouaziz, Mater. Sci. Eng. A 387 (2004) 143–147. [11] X. Wu, X. Pan, J.C. Mabon, M. Li, JF. Stubbins, J. Nucl. Mater. 356 (2006) 70–77. [12] C. Landron, O. Bouaziz, E. Maire, Scr. Mater. 63 (2010) 973–976. [13] D. Gentet, X. Feaugas, M. Risbet, Y. Lejeail, P. Pilvin, Mater. Sci. Eng. A 528 (2011) 7696–7707. [14] C.F. Martin, C. Josserond, L. Salvo, J.J. Blandin, P. Cloetens, E. Boller, Scr. Mater. 42 (2000) 375–381. [15] E. Maire, S. Zhou, J. Adrien, M. Di Michiel, Eng. Frac. Mech. 15 (2011) 2679–2690. [16] F. Bron, J. Besson, A. Pineau, Mater. Sci. Eng. A 380 (2004) 356–364. [17] J.R. Rice, D.M. Tracey, J. Mech. Phys. Solids 17 (1969) 201–217. [18] Y. Huang, J. Appl. Mech. 58 (1991) 1084–1086. [19] J. Besson, C. Guillemer-Neel, Mech. Mater. 35 (2003) 1–18. [20] C. Landron, E. Maire, O. Bouaziz, J. Adrien, L. Lecarme, A. Barregi, Acta Mater. 59 (2011) 7564–7573. [21] A. Bareggi, E. Maire, O. Bouaziz, M. Di Michiel, Int. J. Frac. 174 (2012) 217–227. [22] B. Budiansky, J.W. Hutchinson, S. Slutsky, Mechanism of Sol, Pergamon Press, Oxford, 1982. [23] O. Bouaziz, E. Maire, M. Giton, J. Lamarre, Y. Salingue, M. Di Michiele, Revue de métallurgie-cahier d'informations techniques 105 (2008) 102–107.
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Damage evolution in TWIP and standard austenitic steel by means of 3D X ray tomography D. Fabrègue a,b,n, C. Landron a,b, O. Bouaziz c, E. Maire a,b a
Université de Lyon, CNRS, F-69621 Villeurbanne, France INSA-Lyon, MATEIS UMR5510, F-69621 Villeurbanne, France c ArcelorMittal Research, Voie Romaine-BP30320, F-57283 Maizières les Metz, France b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 October 2012 Received in revised form 5 May 2013 Accepted 8 May 2013 Available online 15 May 2013
The evolution of ductile damage of Fe–22Mn–0.6C austenitic TWIP steel by means of 3D X ray tomography in-situ tensile tests is reported for the first time. The comparison with another fully austenitic steel (316 stainless steel) is also carried out. The damage process of TWIP steel involves intense nucleation of small voids combined with the significant growth of the biggest cavities whereas macroscopical triaxiality remains constant. Due to this high nucleation rate, the average cavity diameter remains constant unlike the 316 stainless steel. & 2013 Elsevier B.V. All rights reserved.
Keywords: Austenitic steel 3D characterization Damage initiation Modeling Tomography
1. Introduction The high manganese austenitic steels discovered by Sir Hadfield in 1880 constitute one of the most attractive materials for structural application since they exhibit a unique combination of strength and ductility [1]. However, to widen their application areas, some issues must be tackled. Their fracture behavior is one of them. Indeed, despite the fact that they have been studying for some time, the mechanisms leading to their fracture are still unclear. Reports often discuss the macroscopical features of the fracture. For example, it has been shown that high manganese steels break in the uniform elongation range before necking at room temperature as shown in TWIP 940 by Chung [2]. Moreover, the fracture surface exhibits a slant surface whatever the stress state was [3]. However the microscopic features that lead to fracture are still under discussion. This subject has been addressed in very few papers and mainly on Hadfield steel (i.e. with a Mn content between 12% and 14%). Bayraktar [4] suggested that the fracture occurs by microvoid coalescence without necking. On the other hand, Abbasi [5] proposed a fracture mechanism based more on the occurrence of intense nucleation. According to that study,
n Correspondence to: INSA-Lyon, MATEIS, 25 Avenue Jean Capelle, UMR5510, F-69621 Villeurbanne, France. Tel.: +33 472438179. E-mail address: [email protected] (D. Fabrègue).
0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.05.013
nucleation events can take place on inclusions or carbides (but the steel considered in our study does not contain that type of carbides) for large dimples whereas small dimples stem from Mn–C couples and the dynamic strain aging process characteristics of this type of steel [1]. X ray tomography has recently been used [6] on the same steel after fracture of a butterfly specimen. These authors showed the presence of strings of elongated voids aligned in the rolling direction. The authors linked this elongated shape to the plastic anisotropy measured (Lankford coefficients) saying that the deformation of the voids is dictated by the surrounding material when the cavity growth is limited (low triaxiality level). They also showed that the volume fraction of these large elongated voids is not sufficient to explain the fracture of high manganese austenitic steel. They then suggested that a large amount of nucleation events of small voids accompanied by rapid coalescence was responsible for the final fracture. This study gives a lot of information on the damage mechanisms of high manganese TWIP steel but it is restricted to post fracture analysis. Thus an “in situ” experiment for different deformation states is required in order to have access to the evolution of the voids in terms of numbers and size as a function of strain. This will permit to get a better insight on the phenomena responsible for the fracture of TWIP steels. Thus in situ tensile experiments are reported in this paper. For the sake of comparison, the study is realized on one hand on the TWIP steel and on the other hand on a classic (i.e. containing no Mn) austenitic stainless steel (namely a 316L). The evolution of the
D. Fabrègue et al. / Materials Science & Engineering A 579 (2013) 92–98
number of voids, of their size and of their different shape factors is given. Then the experimental results are compared to analytical models describing the evolution of the diameter of the cavities.
2. Experiment This study compares the damage evolution in two austenitic steels namely 316L and TWIP steel. The first is a stainless steel with a composition of 0.02% C, 16% Cr, 11% Ni, and 2% Mo (balance iron). TWIP steel is high Mn steel where the austenite is stable at room temperature. No other phase such as carbides is observed at room temperature. Its composition is 22% Mn–0.6% C (composition in weight percent, balance iron) with an average grain size diameter of about 2–3 mm, supplied by ArcelorMittal. A sample from both steels was cut from a 1 mm thick sheet obtained by hot rolling and annealing thermal treatment. Micro tensile specimens were machined by electrical discharge machining according to the shape shown in Fig. 1. X ray microtomography was used to quantify damage during in situ tensile tests. The tomography set-up is located at the ID15 beam line at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Tomography acquisition was performed with a voxel size of 1.6 mm3. Thus the void diameters are calculated with an accuracy of 71.6 mm. Initial reconstructed volumes were median filtered and simply thresholded to differentiate the material from the voids by absorption difference. Refer to [7] for more information on the precise experimental procedure. In order to correlate the distribution of voids with the strain, local values of the strain are obtained by considering the minimum section area Smin and using the relationship: S0 ϵloc ¼ ln Smin
ð1Þ
where S0 is the initial section of the sample. The local strain is then calculated at each step. Using this equation implies that the volume fraction of the voids considered is small enough to keep the total volume unchanged. The beginning of the tensile test is carried out with a stress triaxiality of 0.33. After some deformation, triaxiality may evolve if the sample shape changes. This is considered by using the Bridgman formula [8] modified by Wierzbicki [9] considering the curvature radius of the surface of the sample RS: 1 pffiffiffi a T ¼ þ 2ln 1 þ 3 2RS
93
3. Results and discussion 3.1. Tensile behavior Mechanical behavior (stress/strain) was monitored during the 3D X ray tomography experiment. The tensile curves obtained for the two steels are given in Fig. 2. It is worth noticing that these curves were calculated considering the minimum section of the sample, thus the stress and strain values are very high since they constitute local values and not macroscopic ones. These values may be considered as ultimate ones. It is clearly seen that both steels behave completely differently. 316L presents a yield strength of about 400 MPa and then exhibits a large strain hardening up to maximal strength of about 1600 MPa at a strain of 1.6. On the other hand, the mechanical properties of TWIP steel are outstanding with an ultimate tensile strength of about 2600 MPa and a fracture strain of about 0.55. The yield strength seems to be about 800 MPa but this is a rough estimation since no points were recorded at very low strain. Reports state a value of 400 MPa for yield strength at room temperature [10]. The strain hardening is very high compared to other steel grades and thus larger than the strain hardening of the 316L. The values observed here are in accordance with other studies [10] and explain the current interest in these steels and their promising use in safety parts in the automotive industry. It is important to note that the deformation modes of these two steels have been observed to be the same i.e. twinning and dislocation glide [10,11]. Fig. 3 shows the 3D tomography of the two investigated steels just before final fracture. For the sake of comparison, the same state for DP steel from [12] is also shown. In these three images the voids are underlined by an opaque red color surface while the outside surface of the sample is transparent. The differences in fracture behavior between the DP steel with a ferritic matrix, the austenitic steels 316L and TWIP are obvious in this figure. When the first ones exhibit an important necking and a very high void density before fracture, TWIP steel shows no localization of the deformation and only a small void density. It is important to note that in the case of 316L the cavities seem to be aligned in the direction of the tensile stress. This could be due to heterogeneities in the material such as chemical segregation [13]. Figs. 2 and 3 underline that even if the underlying deformation mechanisms are the same in 316L and TWIP (i.e. mechanical twinning and dislocation glide), they lead to different mechanical macroscopic behaviors and that damage behavior must be also 3000
ð2Þ
TWIP
2500
316L
a being the width of the minimum section. Stress (MPa)
2000
1500
1000
500
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Strain Fig. 1. Specimen geometry for in situ X ray tomography.
Fig. 2. Stress/strain curve obtained during the in situ tensile test.
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Fig. 3. 3D views of the damage at the center of the specimen strained at (a) TWIP steel at εloc ¼ 0.5 (just before fracture), (b) DP steel just before fracture [12] and (c) 316L just before fracture.
105
105 density of voids in TWIP(n/mm3)
density of voids in TWIP(n/mm3)
density of voids in 316L(n/mm3) Density of voids(n/mm3)
Density of voids(n/mm3)
density of voids in 316L(n/mm3) 104
1000
100
104
1000
100 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
local strain
0.4
0.6
0.8
1
1.2
local strain/fracture strain
Fig. 4. (a) Void density as a function of the local strain for TWIP and 316 steel and (b) void density as a function of the local strain divided by the fracture strain for the same two steels.
different. The next part will now focus on the quantification of the different damage features in these austenitic steels. 3.2. Void nucleation Fig. 4a shows the evolution of the number of voids as a function of the local strain and Fig. 4b reports the same parameter as a function of the normalized strain (equal to the local strain divided by the fracture strain) for the two austenitic steels under study. In the case of 316L austenitic steel, the void density evolves as follows: after a plateau from normalized strain of 0 to normalized strain of about 0.2, the number of voids increases exponentially with the normalized strain. In the case of TWIP steel, no plateau can be observed, the number of voids increases like that in 316L. This result indicates that no sudden and large nucleation takes place to explain the fracture of TWIP steel at least at the scale observable with X ray tomography: nucleation occurs during the entire deformation process in a continuous manner. The high nucleation rate observed in the two cases could be due to a high number of nucleation sites. Nucleation could possibly take place at the grain boundaries or at the twin interface or at least, the twins could promote void nucleation due to the high density of dislocation in their vicinity. This fact is hard to check since the twins are usually very narrow (especially in TWIP steel) and thus only post mortem techniques can give access to that. It is also worth noticing
that the number of voids at fracture is very low in the case of these two homogeneous samples (especially for the TWIP steel) compared to all other types of ductile material at fracture already published with a similar imaging resolution [14–16]. Considering the average diameter of the cavities (see next section) these numbers correspond to volume fraction of voids at fracture of about 3 10−3 and 0.2 for the TWIP and the 316L respectively. The fraction for the TWIP is very low compared to the usual values found in the literature. 3.3. Void growth X ray tomography allows the separation of void nucleation from void growth. This leads to a better understanding of the damage phenomenon. The volume (number of voxels) of each cavity was measured and its equivalent diameter was then calculated assuming a spherical shape. The absolute error in this measurement is constant (of the order of one voxel). As a consequence, the relative error in measurement of the dimensions depends on the dimensions themselves: it is rather large (100%) when the cavities nucleate (due to very small size) but then quickly becomes negligible as the growth occurs. Fig. 5 presents the evolution of the average void diameter and the evolution of the average diameter of the 20 biggest cavities in order to eliminate nucleation on the average diameter for the two steels considered.
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18 total population of voids 20 largest voids
16
Average equivalent diameter (µm)
Average equivalent diameter (µm)
18
95
14 12 10 8 6 4 2
Total population of voids 20 largest voids
16 14 12 10 8 6 4 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.2
0.4
local strain
0.6
0.8
1
1.2
1.4
1.6
local strain
Fig. 5. Evolution of the average void diameter and of the 20 biggest voids for (a) TWIP and (b) 316L.
1
0.8
Triaxiality
The results reported in Fig. 5 highlight the effect of nucleation and growth using the methodology detailed in [7]. Considering the average diameter of the total population, in TWIP steel the diameter is almost constant during all the deformation process with a value of about 4.5 mm. However growth is experienced by the largest cavities. This growth is very large since the diameter of the largest cavities almost doubled at a fracture strain of 0.5. In the case of 316L, the value of the average diameter is almost the same as that of the biggest ones up to 0.5 strain. The average diameter of all cavities increases when the local strain attains 1 to reach about 6 mm at fracture. This is due to an increase in the growth rate observed from this value of local strain when considering the 20 largest cavities. That leads to the conclusion that the growth of cavities is counter balanced by the nucleation of new ones in the TWIP whereas growth outbalances nucleation in 316L when local strain increases. In order to get a better insight into the growth process, the evolution of stress triaxiality has to be taken into account. Fig. 6 shows the evolution of this triaxiality with the local strain. Stress triaxiality during the tensile test of TWIP steel remains constant and equal to 1/3. This is consistent with the fact that no necking appears (Fig. 3). On the other hand, in the case of 316L, triaxiality stays constant up to a strain of 0.3 and then a large increase in triaxiality is observed. Then the cavity growth rate in the two steels can be clarified. As mentioned in previous reports, growth is mainly governed by the triaxiality level [17,18]. Increasing triaxiality thus accelerates cavity growth. From Figs. 5 and 6, it is clear that the growth rate is much larger in TWIP compared to 316L since the diameter of the 20 largest cavities increases more in TWIP than in 316L. In fact, the diameter of these voids is almost equal to 8.5 mm in TWIP at a strain¼0.5 whereas it is equal to 5 mm in 316L for the same strain (knowing that the initial diameter is almost the same in both cases). This large increase is more noticeable since stress triaxiality remains constant for TWIP whereas it increases to 0.42 in 316L at a strain of 0.5. This large cavity growth rate could be due to a change in cavity shape and/or to the outstanding kinematic hardening of TWIP. It has been shown [19] that higher kinematic hardening leads to an enhancement of void growth. These data lead to the conclusion that nucleation is intense in the TWIP sample because it counter balances the increase in size of the biggest voids due to growth/ coalescence. It seems that the fracture of TWIP sample is not controlled only by a higher nucleation rate than in other austenitic steels as suggested by Abbasi [5] talking about Hadfield steels but also by some coalescence because the biggest cavities grow faster
0.6
0.4
Triaxiality 316L Triaxiality TWIP
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Strain Fig. 6. Evolution of stress triaxiality as a function of the strain for TWIP and 316L steels.
in TWIP than in other steels as proposed by Bayraktar [4] (also about Hadfield steels). 3.4. Modeling 3.4.1. Growth Rice and Tracey proposed a model for cavity growth [17], based on the hypothesis of a spherical cavity in a perfectly plastic infinite matrix. This model shows that if a homogeneous strain field is applied, growth rate is described by: dR 3 ¼ αRT exp T ð3Þ Rdϵ 2 where T is the stress triaxiality and αRT is a constant. This relationship was modified by Huang [18] in order to extend the approach to low values of triaxiality: dR 3 ¼ αH T 0:25 exp T ð4Þ Rdϵ 2 This relationship has been demonstrated to give good results on different types of steel with a ferritic type matrix (ferrite or martensite) [20] provided that αH is identified from experimental results. This model is used here for both austenitic steels. The results of the experiments and of the model are shown in Fig. 7.
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18
18 20 largest voids Model
20 largest voids Model
16 Average equivalent diameter (µm)
Average equivalent diameter (µm)
16 14 12 10 8 6 4
14 12 10 8 6 4
2
2 0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.2
0.4
local strain
0.6
0.8
1
1.2
1.4
1.6
local strain
Fig. 7. Evolution of the diameter of the 20 largest cavities experimentally and prediction using Huang's model on (a) TWIP and (b) 316L.
Table 1 Factor of Huang's law for cavity growth. The values for DP, IF and martensitic steels are from [20,21].
Value of αH
TWIP
316L
DP steel
IF steel
Martensitic steel
0.9
0.5
0.55
0.22
1.6
9 total population of voids Model
8
Average equivalent diameter (µm)
Average equivalent diameter (µm)
9
7 6 5 4 3
Total population of voids Model
8 7 6 5 4 3
0
0.1
0.2
0.3
0.4
0.5
0.6
Local strain
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Local strain
Fig. 8. Evolution of the average equivalent diameter as a function of the local strain for (a) TWIP and (b) 316L. Experiments (squares) and prediction using Eq. (5) (circles).
The values of the pre exponential term giving the best fit for these two austenitic steels are given in Table 1. Some values obtained in [20,21] for different ferritic matrix steels are also given in this table for the purpose of comparison. The value of αH for TWIP is very high compared to that of DP steel but lower to that of fully martensitic steel. Until now, this trend has not been explained and must be investigated in detail in the future. However, one possible explanation could be the influence of the viscosity on cavity growth rate as suggested by Budiansky [22].
3.4.2. Description of the total average diameter Huang's relationship has been demonstrated to accurately describe the evolution of the diameter of the 20 largest voids i.e. it describes the growth regime. In order to describe the evolution
of the average diameter of the whole cavity population, nucleation should also be taken into account. The approach proposed by Bouaziz [23] is used in the following. The equation permitting to describe the average radius is given by: dRg 1 dN dR ¼ ðR−R0 Þ − dε dε N dε
ð5Þ
dRg =dε is the evolution of the void radius due to growth (Huang's law), N is the number of cavities by mm3 and R0 the radius of cavity at nucleation. More details on this law can be found in [23]. It has been shown [23] that this law is able to reproduce the evolution of the average radius for different steels. Fig. 8 presents the evolution of the average diameter measured thanks to 3D X ray tomography experiments compared to the above model.
D. Fabrègue et al. / Materials Science & Engineering A 579 (2013) 92–98
5
5 Rz/Rx
Rz/Rx 4.5
Ry/Rx
4
4
3.5
3.5
Shape factor
Shape factor
4.5
3 2.5
Ry/Rx
3 2.5
2
2
1.5
1.5
1
97
1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.1
local strain
0.2
0.3
0.4
0.5
0.6
local strain
Fig. 9. Evolution of the shape factors of the 20 biggest cavities for (a) 316L and (b) TWIP.
The radius at nucleation was fixed at 1.8 mm for TWIP and 316L. As can be seen from Fig. 8, the model predicts a constant average equivalent diameter for TWIP steel which is consistent with the experiments. In the case of 316L, the average diameter seems to increase with the strain. This trend is captured by the model. However, the model underestimates the growth. This discrepancy can be explained by an increase in radius at nucleation due to nucleation events appearing successively at different sites at twin boundaries and at grain boundaries. This point has to be checked by more detailed experiments. It is important to note that the large increase at high strain level can also be due to the phenomenon of coalescence which is not taken into account in this approach. 3.5. Evolution of the cavity shape factors Since shape of the cavities could play a major role in the phenomenon of coalescence and thus could govern the ultimate strain in a damage process, the shape factors are going to be addressed. Tomography enables us to measure them as a function of strain as shown in Fig. 9. Fig. 9 gives more information about the damage behavior in both steels. In the case of 316L, for low strains, the cavities are quite spherical as the two shape factors are less than 2. However as the strain increases, the ratio Rz/Rx raises significantly at strains above 0.6 up to a value of 4 just before fracture whereas Ry/Rx remains constant. This means that the cavities become elongated in the strain direction. This result can be also qualitatively deduced from Fig. 3c. The evolution of the cavities in the TWIP steel is different. Indeed, initially the cavities are somewhat elongated in the direction perpendicular to the strain: Ry/Rx is equal to 2 71.2, which is reached for Rz/Rx around the strain of 0.4 increasing from 1.370.5. This could be due to nucleation of the cavities at the twin interfaces. Then as usual, when the strain increases, the Rz/Rx ratio increases. At fracture, the cavities are ellipsoidally elongated as the two shape factors become around 2. This evolution of the shape factors suggests that the cavities grow by blunting like precrack. This explains the possibility of large cavity growth observed in Fig. 5 whereas the stress triaxiality stays equal to 0.33. 3.6. Observation of shear bands When looking at the surface of the TWIP sample just before fracture, some bands may be observed as shown in Fig. 10. These bands were not present at the beginning of loading. They are
Fig. 10. 3D picture of TWIP steel just before fracture showing shear bands.
oriented with an angle equal to about 451 with the loading direction. They could be Portevin-Le Chatelier (PLC) bands emanating at the surface of the sample due to deformation. This phenomenon has already been observed in this material [3]. These authors show that these PLC bands are responsible for the initiation of final fracture. This is in accordance with our study where the bands are can be detected only at the last deformation step. However, no obvious influence of these bands on growth or cavity nucleation in the vicinity is shown.
4. Conclusions 3D X ray tomography has been used to get a better insight into the damage process of austenitic TWIP steel. It shows that although triaxiality remains constant equal to 0.33, cavity growth is experienced and seems to be important. However as the average cavity diameter remains constant, it means that the nucleation rate is very high. Thus the fracture of this steel is due to both
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intense nucleation and considerable cavity growth. It seems that the large increase of cavity diameter comes from a blunting type growth. This increase in cavity diameter could be also due to the presence of shear bands involving an increase in local triaxiality and permitting cavity growth. However due to the small size of the cavities, tomography with higher resolution is needed to have a better idea of the real number of voids and to suggest a clear scenario for the final fracture. In addition, the comparison in damage evolution with 316L austenitic stainless steel with a lower strain-hardening and a lower flow stress showed clear differences: in this case growth is not compensated for by nucleation events leading to an increase in the average cavity diameter. The respective possible roles of twins, of kinematic hardening and of early coalescence have been suggested but they will need to be investigated in detail in future experiments. References [1] O. Bouaziz, S. Allain, C.P. Scott, P. Cugy, D. Barbier, Curr. Opin. Solid State Mater. Sci. 15 (2011) 141–168. [2] K. Chung, K. Ahn, D.H. Yoo, K.H. Chung, M.H. Seo, S.H. Park, Int. J. Plast. 27 (2011) 52–81. [3] G. Scavino, F. D’Aiuto, P. Matteis, P.R. Spena, D. Firrao, Metall. Mater. Trans. A 41 (2010) 1493–1501. [4] E. Bayraktar, F.A. Khalid, C. Levaillant, J. Mater. Process. Technol. 147 (2004) 145–154.
[5] M. Abbasi, S. Kheirandish, Y. Kharrazi, J. Hejazi, Mater. Sci. Eng. A 72 (2009) 513–514. [6] J. Lorthios, F. Nguyen, A.F. Gourgues, T.F. Morgeneyer, P. Cugy, Scr. Mater. 63 (2010) 1220–1223. [7] E. Maire, O. Bouaziz, M. Di Michiel, C. Verdu, Acta Mater. 56 (2008) 4954–4964. [8] P.W. Bridgman, Rev. Mod. Phys. 17 (1945) 3–14. [9] T. Wierzbicki, Y. Bao, Bridgman Revisited: On the History Effect on Ductile Fracture, Massachusetts Institute of Technology, Cambridge, MA, 2004. [10] S. Allain, J.P. Château, D. Dahmoun, O. Bouaziz, Mater. Sci. Eng. A 387 (2004) 143–147. [11] X. Wu, X. Pan, J.C. Mabon, M. Li, JF. Stubbins, J. Nucl. Mater. 356 (2006) 70–77. [12] C. Landron, O. Bouaziz, E. Maire, Scr. Mater. 63 (2010) 973–976. [13] D. Gentet, X. Feaugas, M. Risbet, Y. Lejeail, P. Pilvin, Mater. Sci. Eng. A 528 (2011) 7696–7707. [14] C.F. Martin, C. Josserond, L. Salvo, J.J. Blandin, P. Cloetens, E. Boller, Scr. Mater. 42 (2000) 375–381. [15] E. Maire, S. Zhou, J. Adrien, M. Di Michiel, Eng. Frac. Mech. 15 (2011) 2679–2690. [16] F. Bron, J. Besson, A. Pineau, Mater. Sci. Eng. A 380 (2004) 356–364. [17] J.R. Rice, D.M. Tracey, J. Mech. Phys. Solids 17 (1969) 201–217. [18] Y. Huang, J. Appl. Mech. 58 (1991) 1084–1086. [19] J. Besson, C. Guillemer-Neel, Mech. Mater. 35 (2003) 1–18. [20] C. Landron, E. Maire, O. Bouaziz, J. Adrien, L. Lecarme, A. Barregi, Acta Mater. 59 (2011) 7564–7573. [21] A. Bareggi, E. Maire, O. Bouaziz, M. Di Michiel, Int. J. Frac. 174 (2012) 217–227. [22] B. Budiansky, J.W. Hutchinson, S. Slutsky, Mechanism of Sol, Pergamon Press, Oxford, 1982. [23] O. Bouaziz, E. Maire, M. Giton, J. Lamarre, Y. Salingue, M. Di Michiele, Revue de métallurgie-cahier d'informations techniques 105 (2008) 102–107.