a. Microstructures of the interphase-precipitated carbide and the fibrous carbide
The microstructural features of IP are revealed by TEM. The quantitative measurements of microstructure are summarized in Table 6-1, showing that the sheet spacing and carbide size decrease with decreasing the transformation temperature but the particle spacing at every transformation temperature is approximately the same.
Table 6-1 The features of interphase precipitation measured by TEM
Transformation Temperature
Features unit 670 oC 650 oC 630 oC
sheet spacing nm 34.1 4.7± 19.3±2.1 12.3 1.1± particle spacing nm 42.6 3.8± 41.4 1.0± 40.2±5.6 particle size nm 6.8±0.2 4.2±0.1 3.2±0.1
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Figure 6-5 gives representative structure of IP at different transformation temperatures and representative corresponding high-resolution TEM images for carbide size determinations.
Figure 6-5 TEM micrographs of interphase precipitation and corresponding HRTEM lattice images taken on the samples transformed isothermally at: (a, d) 630 oC, (b, e) 650 oC, and (c, f) 670 oC.
The fibrous precipitates are also observed in some ferrite grains of the steel studied, as shown in Figure 6-6. Direct TEM evidences have indicated that the nucleation sites for fibrous carbides and for interphase-precipitated carbides are different [101], and the former precipitate mode is favored by both a slowing down of the ferrite/austenite interface [5] and an incoherent interface [2]
143
Figure 6-6 The TEM micrographs of typical carbide fiber microstructure isothermally transformed at (a) 630 oC, (b) 650 oC, and (c) 670 oC
As far as the crystallographic aspect is concerned, a semi-coherent interface following Kurdjurmov-Sachs orientation relationships was accounted to be the interface for ledge mechanism operation of IP. However, Yen [8] and Okamoto [13] and Miyamoto [27] have shown that the ledged interface of IP does not have to
but actually be close to be , , and . Their powerful TEM and EBSD evidences indicate that the operation of ledge mechanism of
transformation do not have to be associated with a semi-coherent interface. Therefore, according to the present study, it would be suggested that the kinetics of ferrite/austenite interface is the key factor to determine the development of fibrous carbide.
b. Modeling results
Figure 6-7 shows the calculated evolutions of particle spacing and sheet spacing with time, i.e. with progressing of transformation. The sheet spacing decreases as transformation proceeds; however, the particle spacing demonstrates an opposite trend. These features are easy to be interpreted by normal metallurgical recognitions. By the use of mass balance consideration, the available solute content (vanadium) for
α γ
{
1 1 0}
α{
2 1 1}
α{
1 1 1}
α{
2 1 0}
αγ →α
γ →α
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further precipitation becomes less after the formation of each row of carbide, leading to a wider particle spacing. On the other hand, the sheet spacing is related to the interface velocity. During the transformation, the interface velocity decreases with because of the lowering driving force for transformation (see Figure 6-2(b)), the sheet spacing becomes finer at the later stage of transformation.
Figure 6-7 The calculated evolutions of (a) particle spacing and (b) sheet spacing with time at different transformation temperatures, showing that the sheet spacing becomes finer but particle spacing exhibit an opposite tendency as transformation proceeds.
6.6 Discussions
a. The effect of carbide precipitation on the overall interface velocity
The overall interface velocity is also calculated and additional samples were isothermally transformed at 650 oC for 30s, 90s, 180s, and 600s respectively to show the availability of the calculated results, as shown in Figure 6-8. The thickness transformed at different time step was measured from individual micrograph, which is presented in Figure 6-8(b). Each data point was measured at least 10 ferrite grains. It shows that the thickness calculated by our approach is consistent with that measured from optical micrographs.
The interface velocity of the case of interface precipitation is a function of α
α
α 145
nucleation rate, transformation driving force, and particle spacing (see equation (6-3)).
The latter increases with time (See Figure 6-7(a)) whereas the former two ones are known to decrease with time during ferrite transformation. However, we have to keep in mind that all these parameters are tightly related (equations (6-1) to (6-3)). The overall interface velocity thus results from the combination of these various aspects. That is why, we have decided to plot the evolution of υ υαγ / α, where υα is the velocity for the ferrite/austenite interface migrating without the carbides on the interface in order to show the contribution of interphase precipitation on interface velocity. As shown in Figure 6-9, the ratio of υ υαγ / α is found to decrease with the progressive of transformation and smaller than one and in the order of 10-2, indicating that IP has a strong slowing impact on interface velocity. It is not surprising because carbides precipitated can have a non-negligible effect on pinning the ferrite/austenite interface.
Figure 6-8 (a) the calculated evolutions of overall interface velocity with time, and (b) the calculated evolution of ferrite thickening isothermally at 650 oC showing a good agreement with experimental data.
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Figure 6-9 The ratio of υ υαγ / α at different transformation temperatures with the progressive of austenite-to-ferrite transformation of the steel studied
It is interesting to see that the present results show the sheet spacing becomes finer as the transformation proceeds. This feature is consistent with the observation of Murakami [16] but contradicted with the results of Lagneborg and Zajac [11]. It can be shown that the interface velocity and sheet spacing can be simply related to the diffusion coefficient by combining equation (6-2) and (6-3):
2 crit
p
c D
αγ c
ν λ
= α (6-12)
This relation clearly shows that for a given temperature, a slower interface velocity is associated with a wider sheet spacing when the approximation ccrit =cpα is fulfilled. In that specific case, we have:
D ναγ
= λ (6-13)
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The results of Lagneborg and Zajac [11] reveal a similar tendency where the sheet spacing increases during the thickening of ferrite growth. However, as the carbon content increases, the effect of carbon enrichment in γ on the interface velocity is no longer neglected. The ccrit would be reduced with increasing γ →α transformation because of carbide periodically precipitated on the interface. Finally ccrit deviates from cα in great extents at the later stage of transformation. Therefore, it is more reasonable to use equation (6-12) instead of equation (6-13). The magnitude of
(
ccrit cαp)
2 thenbecomes smaller while decreasing ccrit during the progressive of γ →α transformation. In that case, the interface velocity decreases with time and the calculated evolution of α thickness is in very good agreement with that one measured (see Figure 6-8) and consistent with the TEM observations of Murakami [16].
As far as the role of interface velocity is concerned, a simple intuitive interface explanation for the formation of fibrous carbide can be proposed, as shown in Figure 6-10. The carbide nucleated at the ledge can grow along the side of ledge by a fast diffusion path. If the carbide growth exceeds the ledge growth, the unpinning even will never occur and fibrous carbide would be produced. This will of course be promoted by lower driving force for transformation. A rational consequence is that the fibrous carbide will appear preferably in the later stage of the transformation [48].
According to Figure 6-10, the development of carbide fiber occurs when the pinning carbide grows along the pinned superledge. The particle spacing of interphase-precipitated carbides and the distance between carbide fibers are measured by TEM and shown in Figure 6-11. Compared to the particle spacing of IP, the carbide distance of carbide fiber is generally larger than that of interphase-precipitated carbides,
γ →α
148
implying that the fibrous carbide occurs at the later stage of austenite-to-ferrite transformation. A region showing the transition is given in Figure 6-12.
Figure 6-10 The proposed model for the formation of carbide fiber. (a) a unit ledge is pinned by carbides; (b) the growth of carbide exceeds that of ferrite, growing along the side of ledge, and (c) the ledge never unpins
from pinning carbides, resulting in carbide fibers.
149
Figure 6-11 A comparison of measured particle spacing of different carbide aggregates measured at the samples isothermally transformed at 650 oC for 1h. The particle spacing of interphase precipitation and carbide fiber show distinctive two groups, indicating that the carbide fiber develops at the later stage of
austenite-to-ferrite transformation.
Figure 6-12 TEM micrograph showing the transition of interphase precipitation to fibrous carbide; the previous ledged interface can be well-illustrated.
150
b. The minimum sheet spacing of interphase precipitation at a given transformation temperature
For the austenite-to-ferrite transformation accomplished by ledge mechanism, Bhadeshia [51] proposed that there is a minimum ledge height for ledge mechanism of
transformation, , which is determined by interfacial energy, , and the free-energy change of formation nucleus unit volume,
(6-14)
In his work, the sheet spacing of interphase precipitation was used to support this proposition because ledge mechanism has been well-recognized for IP and the results show that the measured sheet spacing of IP was found to be greater than in most cases. In superledge model, the sheet spacing is proposed to result from the combined interaction between and carbide nucleation rates, transformation driving force, and particle spacing. It has been shown in Figure 6-7(b) that the sheet spacing becomes finer with time. However, the carbon enrichment in normally makes the nucleation more difficult, which means the decreases with time (see Figure 6-2(a)). It is suggested to use equation (6-14) to define the minimum sheet spacing of interphase precipitation that can be observed as the transformation temperature is given.
The predicted sheet spacing by either equations (6-2) and (6-14) are shown in Figure 6-13. The experimental data are found to be within the range of feature predicted by superledge model, exhibiting good agreements with calculated results. The results presented above are very promising because it accounts for the evolution of features of
γ →α hmin α γ σα
IP with time.
Figure 6-13 The boundary of the predicted sheet spacing coupled with experimental data (the solid points) measured at different transformation temperatures
6.7 Conclusions
The evolution of features of IP with progressive transformation has been analyzed using a superledge model coupled with a mass balance approach to composition evolution. The work of Murakami is incorporated and discussed with the present model. The main perspectives are highlighted as follows:
The characteristic features of IP, the sheet spacing and particle spacing, evaluated by this approach are given by a range, instead of a discrete value and the experimental data exhibits good agreements with the calculated results. It is suggested that the
γ →α
γ
152
variation in the characteristic feature of interphase precipitation becomes significant as the carbon and solute contents increase.
The present superledge model enables us to predict the kinetics of austenite-to-ferrite transformation. The calculated overall interface velocity decreases as the transformation temperature is lowered, which is consistent with the metallurgical understandings. The thickening of ferrite from the present calculations is in agreement with the results obtained from dilatometer measurements.
The present model takes the carbide pinning into consideration. It is noticed that the magnitude of interface velocity is considerably reduced because of carbide precipitating on the interface.
The conditions for the development of carbide fiber are reviewed and clarified by the calculated results. It is suggested that a wider particle spacing combined with carbide growth along the superledge would favor the formation of fibrous carbide.
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Chapter 7
The effect of the interphase-precipitated carbide on the strengthening contribution to ferrite
7.1 Introduction and context of the study
The control of microstructure in physical metallurgy is very important because it directly affects the mechanical properties. In order to characterize the mechanical properties, tensile test has been commonly used because it provides an efficient way to access to macroscopic quantities needed by engineer, such as the yield strength, tensile strength, Young’s modulus…etc., of material. The elasto-plastic behavior of material can be also obtained by reading the generated stress-strain curve after each test.
However, because the specimen of tensile test is usually in a millimeter scale , the generated stress-strain curve is actually a combination of all the strengthening contributions, for example, grain boundary strengthening, precipitation hardening, solid solution strengthening, and the effect of the second allotropic phase…etc. As a consequence, it is difficult to separate each strengthening contribution specifically by this approach. In order to study mechanical properties at a local level in direct relation with the intragranular microstructure, which in the dimensions of micro-meter and nano-meter, contact mechanics techniques, and more specifically nanoindentation have been proposed in this study.
The technique of contact mechanics has been widely used to estimate the strength of material [102-104]. Compared to the tensile tests, it gives an efficient way to reveal
154
some of the mechanical properties of material by testing the specimen in a smaller scale.
The results depend on the type of indenter used during the indentation process. Vickers [105], Spherical, Conical, and Berkovich are commonly used in contact mechanics. One procedure to estimate the strength of material is to simply measure the diagonal length on the specimen after indentation, which is generally used in Vickers hardness tests [106]. This procedure inevitably leads to errors as the size of sample decreases. For the small-scaled indentation, nanoindentation has been developed to solve this problem.
The method of nanoindentation to determine mechanical properties of material had been discussed elsewhere [107-109] and been introduced by Oliver and Pharr since 1992 [110]. The principal goal of nanoindentation test is to extract elastic modulus and hardness of the specimen from experimental depth-penetrating curve. During the test, the indenter moves down to the surface of the material and the force and the depth of penetration beneath the specimen surface are recorded and presented. The main outputs of the depth-penetrating curve are:
a. The maximum load,
b. The maximum penetration depth, c. The Stiffness,
d. The residual penetration depth as the indenter is removed,
Figure 7-1 illustrated these four quantities in a typical depth-penetrating curve. The output depends on the type of indenter, the yield stress, the strain hardening properties, and the elastic modulus of the materials. The depth of penetration together with the known geometry of the indenter provides an indirect measure of the area of contact at
Pmax
hmax
S
hr
155
full load, from which the hardness can be estimated by the mean contact force dividing by the projected contact area.
Figure 7-1 A schematically illustration showing the main quantities presented in the depth-penetrating curve
The elastic moduli of materials are determined from the slope of the unloading curve, which is formally called the indentation modulus of the specimen. The results also further provide information on the elastic modulus, on the hardness, on the strain hardening. Recent works have shown that some information can be obtained concerning cracking, plasticity induced phase transformation, and energy absorption [111-113]. All the above information requires a specific analysis of the depth indentation versus applied force curve.
Since the depth measured during the indentation includes both plastic and elastic displacement, the elastic contribution must be subtracted from the data to obtain hardness. It is worth highlighting that when load is removed from the indenter, the material attempts to regain its original shape, but plastic deformation in the specimen
156
after indentation results in a permanent deformation. These descriptions are presented in Figure 7-2. An analysis of the initial portion of this elastic unloading response gives an estimate of the elastic modulus. The portion of elastic and plastic deformation consequently determines the shape of depth-penetrating curve after each indentation.
For a purely elastic case, the unloading curve is identical to the loading curve. For a plastically deformable material, the unloading curve will be a straight line and the initial slope gives access to the elastic indentation modulus.
Figure 7-2 The ideal condition of the deformation and recovery of specimen during the indentation test.
The instrumented nanoindentation can provide more detailed and rich information regarding the mechanical behavior compared to the conventional Vickers hardness. As the indentations made by a nanoindentation instrument are very small, the price to pay to use this technique is that specimens must have a smoothly polished surface [102], around 1 µm roughness. .
Dao et al. [108] have presented a series of dimensionless functions to process the raw data from nanoindentation in order to obtain the Young’s modules, E, the representative stress, (which is related to the yield strength in a non-trivial manner) representative strain (known as to relate to the type of indenter), , and strain
σr
er
157
hardening exponent, n. The relations between this “apparent quantities, and the standardised tensile test results, depend of the indenter geometry. Bucaille et al. [104]
then proposed another approach to improve the precision in the analysis of strain hardening by using two different indenters. In the present work, we have decided to use Bucaille’s approach rather than Dao’s one because it has been shown elsewhere that the former approach is capable of providing more reasonable predictions [104].
7.2 The conversion of the depth-penetrating curve into mechanical properties 7.2.1 Bucaille’s approach
For metallic materials, Dao et al. had proposed several dimensionless equations to process the raw data of nanoindentation. These equations provide an efficient way to obtain Young’s modulus, strain hardening exponent, yield strength of material. The general procedures to have these quantities are described as follows. The detailed derivations and discussions of these equations can be found in [104, 108].
The first step is to estimate the reduced Young’s modulus, E*, by considering that
(7-1)
where , , and are given from the individual load-penetration curve shown in Figure 7-1. The obtained reduced modulus, , then is introduced in equation 7-2
(7-2)
, is directly measured during the test, and the other term in the r.h.s of eq.2 is obtained by derivation of the loading curve. stands for the elasto-plastic deformation prior to the unloading. It is depending on the geometry of the indenter and on the extent of deformation. Table 7-1 gives the values of for different deformations and a variety of indenter shapes.
Table 7-1 List of given in Dao’s study [108]
Berkovich 1.167 1.2370
Vickers 1.142 1.2105
The reduced modulus obtained from equation (7-1) allows to determine the Young’s modulus of material, E as shown in eq.7-3 By giving the Young’s modulus, , and Poisson’s ratio, , of indenter material (diamond), the value of E can be determined from
(7-3)
From the curvature C of the loading curve, one can get the representative stress via eq.
7-4
(7-4)
Here is the representative stress associated with Berkovich indenter, for a representative strain of 0.033.. Either Dao or Bucaille follow the same procedures presented above to determine , , E, and , but they did not use the same method to determine strain hardening exponent.
It has been shown that the equation (7-4) actually varies with the choice of representative strain (which is somehow arbitrary and may be different from 0.033). The variation of representative strain with different indenter geometries are summarized in Table 7-2. A general relation of the representative strain with the half tip-angle of the indenter was subsequently proposed as
(7-5)
Table 7-2 The representative strains associated with different included angle [104]
Inclined angle of indenter Representative strain
70.3 0.033
This finding motivated them to consider another way to determine the strain hardening exponent. By extending Dao’s approach, Bucaille et al. proposed additional three dimensionless equations based on different representative strain
(7-6)
and a general form is given by
(7-7)
In Bucaille’s approach, the same specimen is indented by two different indenters of different shapes (one of them has to be Berkovich indenter). Then, the yield strength and strain hardening exponent are obtained s by solving
3 2
* * *
0.0537 0.0537 0.0537 0.0537
0.06463 ln 2.2102 ln 21.589 ln 28.5741
C E E E
0.02937 ln 0.9324 ln 8.4034 ln 7.532
C E E E
0.02842 ln 0.648 ln 4.9036 ln 3.806
C E E E
tan2 0.02552 ln 0.72526 ln 6.34493 ln 6.47458
r r r r
(7-8)
The calculated flow strength and strain hardening exponent can be used further to describe the elasto-plastic behavior of material, which can be described by using the following relation (Ludwig’s law).:
, K =Enσ1y−n (7-9)
7.2.2 The reliability of the processed data
It has to notice that the above procedures to process the raw data of nanoindentation are all based on idealized assumptions: perfect geometry of the indenter, absence of friction between the indenter and the tested material. In addition the validity of the results for hardness and modulus is very sensitive to thermal drifts and mechanical vibration. In addition, intrinsic problems with the indentation test have to be considered,
It has to notice that the above procedures to process the raw data of nanoindentation are all based on idealized assumptions: perfect geometry of the indenter, absence of friction between the indenter and the tested material. In addition the validity of the results for hardness and modulus is very sensitive to thermal drifts and mechanical vibration. In addition, intrinsic problems with the indentation test have to be considered,