Characterization of internal lattice strain

5.3.3.1 Stress portioning between ferrite and cementite

Figure 5-6 shows the evolution of the ferrite and cementite phases during RT tensile deformation in the axial (tensile) and transverse direction. Phase strains corresponding to phase stresses were determined from the relative change in the lattice parameter according to eq. (11). The phase strains for ferrite and cementite increase linearly with increasing applied true stress. In spite of variation in the specimen, cementite’s Young's modulus is quite similar with that of the ferrite, agrees well with previous work [53]. Beyond the σ_0.2, it is worth noting that phase strain of ferrite ceases to increase, while the increasing rate of cementite phase strain becomes larger in both the axial and the transverse direction, accompanying high work hardening in the flow stress curve. This transition behavior occurs at a higher external stress in the fine lamellar specimen than in the coarse one, exhibiting effective coincidence of yield

Fig. 5-5 Neutron diffraction profile obtained from the tensile direction magnified around the ferrite 110 peak for fine pearlite to show cementite peak, where low-intensity peaks were indexed as cementite for the unloaded

profile.

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strength difference. These results indicate that the yield strength of ferrite is higher for the fine lamellar specimen than for the coarse one, whereas the work hardening due to stress partitioning; that is, overall load transfer from ferrite to cementite, which agrees well with previous works by Morooka et al. [166] and Shinozaki et al. [167]. The stress partitioning is determined in the transverse direction results (see Fig. 5-6), although the changes are approximately 0.3 of those in the axial direction, due to the Poisson’s effect. Unloading then results into residual strains, which are tensile in the axial and compressive in the transverse direction. The residual phase strain of cementite is larger than that of ferrite and of opposing, for instance, the ferrite is in axial compression and the cementite in axial tension. This is a typical behavior during tensile deformation at RT of a two-phase steel.

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5.3.3.2 Stress portioning between different crystallographic orientation of the grains

The intergranular strain was calculated from the change in lattice spacing for

<hkl>-oriented grain families along the axial direction, according to the following eq. (12). Figure 5-7 illustrates the evolution of the intergranular strains of ferrite (110), (200), and (211), as well as cementite (122), (301), and (212), for the fine pearlite specimen, as a function of true stress in both the axial and transverse directions. The different slopes in the elastic deformation region between <hkl>

grains, so-called “diffraction Young’s modulus”, are due to elastic anisotropy: the Fig. 5-6 Phase strain with applied true stress for each constituent phases of

fine and coarse pearlite. The error bars for phase strain, which represent fitting error, are too small to be observed.

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diffraction Young’s modulus determined for ferrite <110>, <200> and <211> and cementite <122>, <301> and <212> are 229.8GPa, 172.5GPa, 170.2GPa, 171.1GPa, 235.3GPa and 215.6GPa, respectively, showing good agreement with previous reports [77,78,168]. Consequently, some grain orientations take more load than others. The difference in lattice strain among <hkl> grain families continues to appear in the elasto-plastic region, which suggests that the plastic flow differs among individual grains.

Following unloading, all lattice strain in ferrite <hkl> grain families are compressive, whereas those of cementite are tensile, showing strong agreement

Fig. 5-7 Intergranular strain with applied true stress for various

<hkl>-oriented grain families of fine pearlite in the axial and transverse directions.

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with previous reports [54,163]. Oliver et al. discussed the superposition of phase stresses and intergranular stresses in plastically deformed carbon steel (volume fraction of cementite is 34%), with spherical cementite particles using neutron diffraction and XRD experiments, coupled with calculations using the elasto-plastic self-consistent (EPSC) model or crystal plasticity finite element method (CPFEM) [85]. The current results also reveal the plastic anisotropic behavior features both in ferrite and cementite phases.

5.3.3.3 Stress portioning between different lamellar orientation of the colonies

As typical examples, ferrite (200) diffraction peaks are illustrated in Fig. 5-8.

The diffraction profiles in the axial direction, obtained in unloaded stage (0%

strain) and following 1.5% tension (a), 1.5% compression (b), and 4.7% tension (c) stages, are compared. The peak shift, diffraction intensity change, and profile shape change are observed. In order to compare the profile shape for clarity, these two profiles are normalized to be of identical heights (see the embedded figure). Then, the result suggests that the broadening on the right-hand side (see arrow) is larger than that on the left in upon tensile deformation, which indicates asymmetry. Meanwhile, it shows opposite trend during compressive deformation.

On the contrary, the diffraction profiles of the deformed specimen in the transverse direction in (a’-c’) exhibit slightly larger broadening on the left-hand side during the tensile case; that is, the larger broadening side is opposite in the axial and transverse directions. Such asymmetry becomes more obvious with increasing plastic deformation. The asymmetric diffraction line profile of plastically deformed metal was first discussed by Ungár et al. using XRD for a Cu single crystal [169] and by Jakobsen et al. for a Cu polycrystal [170]. These authors proposed a composite model composed of a hard region of dislocation

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cell walls and a soft region of the cell interior. These hard and soft regions bear different stresses, resulting in double peaks with varying lattice spacing for the X-ray diffraction profile. A similar asymmetric profile line broadening was observed recently by means of neutron diffraction for a quenched lath martensite steel [171,172]. Lath martensite exhibits a hierarchical topology, consisting of lath, block, packet, and prior austenite grain. The packet is composed of several blocks with the same habit plane; therefore, slip deformation is strongly influenced by the block’s alignment direction, leading to the separation of hard and soft packets with tensile deformation. Therefore, diffraction profiles of deformed lath martensite steel consist of double peaks associated with soft and hard blocks, which is similar to the asymmetric features in Fig. 5-8. In the case of pearlite steel, hard and soft regions with the same crystallographic orientation must correspond to soft and hard colonies, as claimed by the studies on surface observations [86,87].

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Further, the diffraction profiles were fitted according to double peaks with different lattice spacings. The results for the axial and transverse directions are presented in Fig. 5-9, respectively. The hard colony (HC) peak (blue curve) shifts Fig. 5-8 Examples of ferrite 200 diffraction peaks before and after deformation in axial (a-c) and transverse (a’-c’) direction. The top, second and third rows show

the 1.5% tensile, 1.5% compressive and 4.7% tensile test, respectively.

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towards wider lattice spacing, while the soft colony (SC) peak (green curve) is narrower, and the magnitude of SC is lower than that of HC in the axial direction.

Transverse direction exhibits opposite trend and the amount of shift is three times lower than that in the axial direction, due to the Poisson’s effect. The strains of SC and HC (ε_SC and ε_HC) were determined by eq. (22) and (23) according to eq. (12).

where d_hkl^H , d_hkl^S and d_hkl⁰ refer to the hkl plane spacing for HC, SC and stress-free conditions, respectively, Here, ferrite (110), (200), and (211) diffraction peaks in axial direction were employed for double peak fitting.

Figure 5-10 compares the HC and SC results for the fine pearlite specimen. In addition, the results for the coarse pearlite will be shown in Appendix. As can be seen, the lattice strain for all grain families progresses linearly with the applied true stress. At the yielding point, HC and SC start to diverge from the hitherto linear behavior, and subsequently ε_HC increases rapidly, while ε_SC ceases to increase, retaining hkl dependence. Recently, micro-pillar deformation tests have been performed on lath martensite steel [173] and pearlite steel [174].

These results claim that flow stress is greatly influenced by packet or colony alignment in terms of the loading direction: the flow stress of a specimen with an alignment of 0° or 90° is much higher than that at 45°. Hence, in addition to stress partitioning behaviors of phase and intergranular stresses, colony stress is overlapped.

ε_HC=(d_hkl^H −d_hkl⁰ )/d_hkl⁰ (22) ε_SC=(d_hkl^S −d_hkl⁰ )/d_hkl⁰ (23)

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After unloading, the residual lattice strains are tensile in HC, but compressive in SC. In general, residual strains or stresses are compressive for the ductile phase and tensile for the brittle phase in the loading direction [54,78]. However, both HC and SC exhibit compressive residual lattice strain in coarse pearlite.

This result infers the HC in fine pearlite relatively takes a larger portion of the load than that in coarse pearlite. It is suggested that load transfer occurs more effectively with decreased lamellar spacing. The effects of morphology and volume fraction of cementite on the amount of accumulated residual strain in ferrite have been discussed [85,168]. Unfortunately, it remains lack of convincing explanation. Thus, the evolution of the residual lattice strain during plastic deformation is open for future discussion.

Fig. 5-9 Examples of double peaks fitting for ferrite (200) diffraction peaks of fine pearlite. These double peaks corresponding to the HC (blue curve) and

SC (green curve) in the axial and transverse directions.

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The FWHM value, computed as the average of the FWHMs for (110), (200), and (211) reflections, is shown in Fig. 5-11. It is worth noting that FWHM apparently increases in HC, but not in SC, upon yielding. In this case, FWHM is believed to mainly depend on dislocation density. It is speculated that the pile-up of dislocations at the ferrite/cementite interface is affected by lamellar alignment:

in the case of ~45° in terms of the tensile direction (SC), the free path for dislocation movement is large, and hence work hardening must be low, due to poor accumulation and easy dislocation annihilation. On the other hand, the increment of HC in fine pearlite is significantly larger than that in coarse pearlite, indicating that the responses of colony stress relate to the lamellar spacing.

Fig. 5-10 Colony strain evolution during tensile deformation of fine pearlite in the axial direction.

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Fig. 5-11 Change of FWHM with tensile deformation for fine and coarse pearlite.