Temperature Independences of Fatigue Crack Growth in Ti–0.49 mass%O

Abstract

The temperature dependence of fatigue crack growth in stage IIb was investigated in Ti–0.49 mass%O with an alpha single phase. It was found that the fatigue crack growth rate in stage IIb was temperature independent at temperatures above 300 K. EBSD maps beside the fatigue crack showed that prismatic slips with ⟨a⟩ dislocations were dominantly activated during the fatigue crack growth. The reason why the prismatic slips were dominantly activated is discussed with numerical crystal plasticity analysis. It was shown that the local strain rate at the notch tip was higher than the macroscopic strain rate, which leads to the suppression of non-prismatic slips at the fatigue crack tip.

(a) and (b) are the experimental results. (c), (d) and (e) are the numerical result using crystal plasticity analysis. (a) Orientation map along the wake of a fatigue crack. The crack extended toward the x direction. (b) Colour map showing the deviation of the [0001] direction from the average orientation of [0001] in each grain. The colour key is shown at the middle-right corner. (c) and (d) are maps of the local strain rate at the notch tip. (c) At the nominal strain of 0.00004. (d) At the nominal strain 0.001. (e) Evolution of strain rate at the notch tip versus nominal strain.

1. Introduction

Fatigue is one of the fracture behaviours generated by a crack when the crack propagates under multiple cyclic loads such as far below the yield strength for high cycle fatigue. The crack generation and propagation processes in fatigue are categorised as stage I and stage II, respectively. Stage II is further divided into stage IIa and stage IIb, which are the early stage of the crack propagation and stable crack propagation stage, respectively. The crack growth rate, da/dN, where a is the crack length and N is the number of cycles, in the early stage of fatigue depends strongly on the microstructure in stage IIa, while da/dN in stage IIb does not. Paris and Erdogan¹⁾ suggested the relationship between da/dN and the stress intensity range, ΔK, as follows:   

\begin{equation} \frac{\text{d}a}{\text{d}N} = C\Delta K^{m}, \end{equation} (1)

where C and m are constants within each stage. It has been reported that the crack propagation rate depends on several factors such as residual stress, strain rate, temperature, atmosphere, microstructure, and so on.^2–8) It is essential, therefore, to investigate these dependences from the viewpoint of materials science in order to understand how these factors influence fatigue crack growth.

If the stable crack growth in stage IIb is controlled by plastic deformation, i.e., dislocation activity, the temperature dependence of da/dN in that stage should be influenced by dislocation activities at the fatigue crack tip. There are limited numbers of studies on temperature dependence of fatigue crack growth in stage II.^9–12) Tanaka et al.¹²⁾ pointed out that the fatigue crack growth in steels is one of the thermally activated processes. They also reported the relationship between the activation energy of dislocation glide and that of fatigue crack growth in stage IIb by measuring the activation energy for fatigue crack growth. It is expected from analogy that the fatigue crack growth in Ti is also a thermally activated process controlled by dislocation glide. Therefore, in the present study, the temperature dependence of the fatigue crack growth in α-Ti with oxygen was investigated. The thermally activated process of dislocation glide was also evaluated by measuring the temperature dependence of 0.2% poof stress and activation volume in tensile tests.

2. Experimental Procedure

A casted ingot was forged at 1273 K followed by hot-rolling at 923 K. The rolled plate with a thickness of 1.6 mm was kept at 973 K for 1.8 ks in a salt bath and then air cooled. After the heat treatment, the surfaces were polished to remove the hardened surface layer having been created during the heat treatment with the salt bath. The chemical composition of the materials after the heat treatment is shown in Table 1. The employed materials have alpha single phases, containing 0.49 mass% oxygen as the major impurity after the heat treatment. The oxygen concentration is slightly higher in this study. One of the reasons is to suppress the formation of twinning¹³⁾ during the deformation to investigate only the slip deformation in this study. In addition to that, the concentration of exogen increased from that of preparation as the specimen absorbed oxygens during the hot rolling and the subsequent heat treatment. In order to investigate the temperature dependence of fatigue crack growth in stage IIb, fatigue tests were performed. Figure 1 shows a schematic of the specimen used for the fatigue tests. The specimen thickness was 1.3 mm. A notch was introduced in the as-cut specimen with an electric discharge machine (Mitsubishi Electronic, MV1200R) for the fatigue test. The load direction was parallel to the rolling direction (RD). The maximum stress and a minimum-to-maximum stress ratio, R, were set to be 150 MPa and 0.1, respectively. The tested temperature was between room temperature and 573 K. The crosshead speed was ±3.3 × 10⁻³ m·s⁻¹. The test chamber was vacuumed to an oxygen partial pressure of 3.2 Pa. The crack length was measured using a digital microscope installed in the fatigue test machine (Yonekura, CATY-T3H).

Table 1 Chemical composition of the employed materials.

Fig. 1

Schematic of the specimens for fatigue tests.

To assess the slip systems activated during the fatigue crack growth, the crystal rotations in each grain were investigated by using the Euler angles of EBSD analysis data. First, the average orientation of the [0001] direction in a grain was determined in terms of the specimen coordinates. The angle between the average orientation of the [0001] direction and each measured point in the same grain was measured. The [0001] direction does not deviate only if the prismatic slip with the Burgers vector of a/3 $\langle 11\bar{2}0\rangle $ (the Burgers vector is hereafter denoted as ⟨a⟩ dislocation) is active, because the [0001] direction is the rotation axis for the prismatic slip with ⟨a⟩ dislocation. Therefore, the magnitude of deviation of the [0001] direction in the pole figure can be one of the indexes of how much the slip systems which that are not prismatic with ⟨a⟩ dislocations are activated.

3. Numerical Conditions

The reason why the slip deformation responsible for crack propagation was limited to the prismatic slip systems was investigated by finite element crystal plasticity analysis (CPFEM).^14,15) The relationship between $[10\bar{1}0] - [\bar{1}2\bar{1}0] - [0001]$ (Although a coordinate system in hexagonal lattice is not orthogonal coordinate system, $[10\bar{1}0] - [\bar{1}2\bar{1}0] - [0001]$ is orthogonal coordinate system) in the crystal coordinate system and the x-y-z coordinate system is the equation, as follows.   

\begin{equation} \begin{pmatrix} [10\bar{1}0]\\ [\bar{1}2\bar{1}0]\\ [0001] \end{pmatrix} = \begin{pmatrix} -{\sin\kappa} \cos\theta \sin\varphi + \cos\kappa \cos\varphi & \sin\kappa \cos\theta \cos\varphi + \cos\kappa \sin\varphi & \sin\kappa \sin\theta\\ -{\cos\kappa} \cos\theta \sin\varphi - \sin\kappa \cos\varphi & \cos\kappa \cos\theta \cos\varphi - \sin\kappa \sin\varphi & \cos\kappa \sin\theta\\ \sin\theta \sin\varphi & -{\sin\theta} \cos\varphi & \cos\theta \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix} \end{equation} (2)

Where φ, θ, and κ are the rotation angles with respect to z, y, and z axes. The elastic constants of s₁₁, s₁₂, s₁₃, s₃₃ and s₅₅ were 1.2653, −0.7307, −0.2182, 0.7676 and 2.4155 [×10⁻¹¹ Pa⁻¹], respectively.¹⁶⁾

4. Results and Discussion

4.1 Fatigue tests

Figure 2(a) shows an orientation map of grains on the normal plane. The orientation indicates the crystallographic orientation parallel to RD as observed from ND. The mean grain size was approximately 50 µm, as measured using an intercept method. Figure 2(b) shows inverse pole figures with respect to ND and RD, indicating that the specimens have a texture. The tensile directions of the fatigue tests and tensile tests are parallel to RD; therefore, basal slip is expected to be rare in the specimens employed.

Fig. 2

(a) RD orientation map of grains observed from the normal direction (ND) of the specimen. A colour key is put beside the orientation map. (b) Contour maps of pole density on inverse pole figures for ND and RD, which are taken from the same area with that in (a).

Figure 3(a) shows the crack growth length which was obtained by subtracting the length of the notch from the total length of the crack versus a notch against the number of cycles in specimens tested at 303 K, 373 K, 473 K, and 573 K. The crack growth length at the same number of cycles decreases with temperature because a lower stress was applied to the specimen in fatigue tests at higher temperatures. In order to validate the tests results, fatigue tests were performed twice both at 303 K and 573 K, which are plotted together with other data in the figure. The crack growth length is increased with the number of cycles at all temperatures tested. Figure 3(b) shows da/dN against ΔK when the fatigue tests were performed at 303 K, 373 K, 473 K, and 573 K. In total, 6 fatigue tests were performed, among which twice tests were performed at 303 K and 573 K. da/dN is proportional to ΔK so that Paris low is held at any temperature in this study. da/dN is considered to be independent from temperature since all data are on a single regression line. The slope of the line, that is m in eq. (1), is approximately 2.9. Because the number of samples was limited, fatigue tests were performed twice only both at 303 K and 573 K which is the lowest and the highest temperatures to confirm that the fatigue crack growth rate is temperature independent.

Fig. 3

(a) Crack growth length versus number of cycles. (b) Crack growth rate versus stress intensity factor range taken at several temperatures. (c) Spacing of striations versus stress intensity factor range measured on the fracture surfaces of the specimens in (b).

Figure 4 shows a scanning electron microscopy (SEM) image of the fracture surface at the position where ΔK corresponds to 26 MPa·m^1/2 at 473 K. Almost all the fracture surfaces have striations. The direction of striations changes across grain boundaries, reflecting the change in crystallographic orientations of the grains. Figure 4(b) shows an enlarged image of the striations in the area surrounded by a rectangle in Fig. 4(a). The mean spacing of each striation in a grain was approximately 0.65 µm. The existence of striation suggests that the fatigue crack growth is controlled by plastic deformation, i.e., dislocation activity at the crack tip. Figure 3(c) shows the mean spacing of striations against ΔK, indicating that the striation spacing is in good agreement with da/dN. The mean spacing is also independent from temperature, and is only dependent on ΔK. Because fatigue crack growth is control by slip deformation at the crack tip, the activated slip system at the crack tip was assessed next.

Fig. 4

(a) SEM image of the fracture surface for a ΔK of 26 MPa·m^1/2 at 473 K. (b) Enlarged image from the area surrounded by a rectangle in (a).

The magnitude of deviation of the [0001] direction from the average orientation of the [0001] direction inside the grain was determined along a fatigue crack wake after a fatigue test at 673 K. Figure 5(a) shows an orientation map of the tensile direction (y direction) at the crack wake tested at 673 K. The fatigue crack propagated in the x direction, and the crack surface is normal to the y direction. The stress intensity range at the right end was approximately 10 MPa·m^1/2; therefore, the observed area was included in the stable fatigue crack growth at this temperature, as shown in Fig. 3(b). Figure 5(b) shows a colour map demonstrating the deviation of the [0001] direction from the average [0001] direction in each grain. Figures 5(c1) and (c2) show schematic illustrations of the crystal rotation in the case that a prismatic slip with an ⟨a⟩ dislocation is activated, where the activated prismatic slip planes are coloured grey. If the prismatic slip with ⟨a⟩ dislocations are activated, the rotation axis is parallel to the c-axis. Therefore, the c-axis keeps its initial direction, even if the plastic deformation proceeds in the tensile test. The c-axis does not tilt from the initial direction only in the case that the prismatic slip with the ⟨a⟩ dislocation is activated. Figures 5(c3) and (c4) show schematic demonstrations of the crystal rotation in the case that a pyramidal slip with ⟨a + c⟩ dislocations are activated as an example for the case that the activated slip system is not the prismatic with the ⟨a⟩ dislocation. The activated pyramidal slip planes are coloured grey in the figure. Since the rotation axis is not parallel to the c-axis, the c-axis tilts from the initial direction and the slip proceeds as shown in Fig. 5(c4). The tilt angle is denoted as Φ in the figure. The magnitude of Φ can be used an indicator of how much the slip system which is not the prismatic with ⟨a⟩ dislocation is activated. The colour map of Fig. 5(b) shows the magnitude of Φ. Figure 5(b) shows that the [0001] directions along the crack wake rarely deviate from the average [0001] orientation. This result indicates that the dominant slip planes for the fatigue crack growth are prismatic planes with ⟨a⟩ dislocations even at 673 K.

Fig. 5

(a) Orientation map with respect to the tensile direction (y direction) along the wake of a fatigue crack. The crack extended toward the x direction. (b) Colour map showing the deviation of the [0001] direction from the average orientation of [0001] in each grain. The colour key is shown at the bottom-right corner. (c) Relationship between active dislocations and the rotation of -axis. (c1) Prismatic slip plane with ⟨a⟩ dislocations. (c2) Schematic illustration showing the rotation axis in the case of prismatic slip with ⟨a⟩ dislocations. (c3) Pyramidal slip plane with ⟨a + c⟩ dislocations. (c4) Schematic illustration showing the rotation axis in case of pyramidal slip with the ⟨a + c⟩ dislocations.

We have reported¹⁷⁾ that prismatic slip with ⟨a⟩ dislocation was dominant at low temperatures while other slip systems such as prismatic slips were activated at temperature higher than 673 K in tensile deformation, employing the same materials with the initial strain rate of 4.17 × 10⁻⁴ s⁻¹, where the strain rate is much lower than that corresponds at the crack tip. The critical resolved shear stress of prismatic and other slip systems becomes closer around 700 K,¹⁸⁾ which suggests that the activated slip systems around the crack should be prismatic slip and others. However, Fig. 5 indicated that the dominant slip system in fatigue crack growth is prismatic slip with ⟨a⟩ dislocations even at 673 K. It suggests that the slip systems other than prismatic with ⟨a⟩ dislocations were somehow restricted to be activated at the crack tip. The reason why the prismatic slip with ⟨a⟩ dislocations kept dominant at the crack tip in this study will be discussed in the next section.

4.2 Numerical calculations

The reason why the slip deformation responsible for crack propagation was limited to the prismatic slip systems as shown in Fig. 5 was investigated by CPFEM. Figure 6(a) and (b) show the analytical model of a single crystal model with a U-shaped notch used in this study, and the meshed model, respectively. In order to investigate the relationship between the activated slip systems in the crack propagation process and the crystallographic orientation of the specimen, four different orientations were picked in the CPFEM as shown as grain 1–grain 4 in Fig. 5. The crystal orientation of each grain with respect to the notch is shown in Fig. 6(c) using an HCP unit cell. The Euler angle of each grain is shown in Table 2. Tensile load along the y-direction was applied in the single crystal model with a U-shaped notch under four crystal orientation conditions. Four slip systems were defined for α-Ti: basal, prismatic, 1st order pyramidal and 2nd order pyramidal systems shown in Fig. 7. The value of CRSS in α-Ti is strongly temperature dependent.^18–22) At low temperatures, the value of CRSS of the prismatic slip is much lower than that of the non-prismatic slip system, then the prismatic slip is dominant in the plastic deformation. At high temperatures, the values of CRSS of prismatic and non-prismatic slip systems are close enough for both slip systems to be activated during the plastic deformation. In this study, we focus on the slip deformation at a crack tip in α-Ti at high temperatures, where the values of CRSS are nearly identical.

Fig. 6

(a) Geometry of a single crystal model. The model size is 50 µm × 50 µm × 25 µm. The width and the length of U-shaped notch were 5 µm and 15 µm, respectively. (b) Finite element meshing in the model. (c) The crystal orientation of each model using an HCP unit cell. The coordinate system (x-y-z) is identical with that in (a).

Table 2 Euler angles (φ, θ, κ) of grain 1∼4.

Fig. 7

The four slip systems defined. (a) basal slip systems. (b) prismatic slip systems. (c) 1st order pyramidal slip systems. Slip directions are ⟨a⟩ and ⟨a + c⟩. (d) 2nd order pyramidal slip systems.

Therefore, the lattice friction stresses were set to be 43 MPa,²³⁾ independent of the slip systems. This means that the ease of activity of the slip systems depends only on the Schmid law. Figure 8 shows a contour map of the distribution of Schmid factor in the uniaxial tensile condition. As the analysis model used in this study has a notch, the stress field near the notch is multiaxial. Therefore, it should be noted that the active primary slip system is not determined only by the Schmid factor in the tensile direction. The orientation of the tensile axis (y-axis) in grains 1 to 4 were also plotted in Fig. 8. If we consider that the Schmid factor governs the main active slip system against tensile stress, 2nd order pyramidal slip systems should be activated in grain 1, the prismatic slip systems should be activated in grains 2 and 3, and the basal slip systems should be activated in grain 4. However, since a multiaxial stress field is formed around the notch, there is a possibility that the primary active slip system at the notch tip does not simply depends on the value of Schmid factor. Therefore, the model shown in Fig. 6 was subjected to uniform forced displacement in the y-direction at the upper and lower end faces. The tensile load was applied until the nominal strain in tension in the y-direction reaches 0.001.

Fig. 8

The crystal orientation of each grain in a standard triangle with contour for the highest Schmid factor of slip systems in HCP. The crystal orientation of each grain is for tensile direction (y-axis).

Figure 9 shows the nominal stress-strain curves of the numerical analysis from grain 1 to grain 4. The origin of each curve was offset along the x-axis. The dashed line represents the stress-strain relationship in the elastic regime. The stress-strain curve deviates from the elastic line when the plastic deformation starts even at a small deformation such as the nominal strain of 0.001. The slope of the elastic line for grain 4 is different from those for grains 1–3. This is due to the effect of elastic anisotropy caused by the fact that the crystal orientation of grain 4 differs greatly from those of the other grains. The elastic anisotropy factor of α-Ti is strong. The orthogonal anisotropy factor of the HCP lattice, expressed as s₁₁/s₄₄,²⁴⁾ is 1.53 in this material.

Fig. 9

The nominal stress–strain curves from the numerical analysis obtained by tensile analysis of a single crystal model containing a U-shaped notch. To make each stress-strain curve more visible, the starting position of each curve is offset.

Figure 10 shows the equivalent plastic strain versus the nominal strain. The equivalent plastic strains were calculated separately for the non-prismatic slip system (sum of the base, 1st order pyramidal, and 2nd order pyramidal slip systems) and the prismatic slip systems. In other words, the graph shows which slip system is active against macroscopic deformation with respect to the nominal strain.

Fig. 10

Average equivalent plastic strain versus nominal strain obtained numerical calculation. (a), (b), (c), and (d) are the results from models of grain 1, grain 2, grain 3, and grain 4 respectively.

Figure 10 shows the equivalent plastic strain due to the activity of the prismatic slip system, and that due to the activity of the non-prismatic slip system. For grains 1 to 3 as shown (a)–(d), the average equivalent plastic strain of the prismatic slip is higher than those of the non-prismatic slips in the early stage of deformation. This means that the prismatic slip system is primary activated before the non-prismatic slip system is activated. In the models of grains 1 and 2, the plastic strain of the non-prismatic slip systems is the same level as that of the prismatic slip systems at the nominal strain of 0.001. In the model of grain 3, the average equivalent plastic strain of the prismatic slip system was always higher than that of the non-prismatic slip systems. The average equivalent plastic strain of the non-prismatic slip systems is always higher than that of the prismatic slip system in the model of grain 4. Here, it is to be noted that non-prismatic slip system was activated in any model. In other words, the non-prismatic slip system was activated regardless of the crystal orientation. From the results of the crystal plasticity analysis described above, the following points were clarified. When value of the CRSS of the prismatic and non-prismatic slip systems are the same, i.e., under the assumption that value of the CRSS of all the slip systems are identical, the non-prismatic slip is not suppressed, unlike the experimental results.

Next, the mechanism behind the suppression of the non-prismatic slip systems, which was experimentally obtained in Fig. 5, will be discussed. Figure 11(a) and (b) show the strain rate distribution in the x-y cross section at the centre of the analytical model. Figure 11(a) and (b) show colour maps of the strain rate at the nominal strains of 0.00004 and 0.001, respectively. The former is in the elastic deformation regime while the latter is in the plastic deformation regime. The figures show that the strain rate is locally higher at the notch tip where the stress is also concentrated whereas elastic or plastic regimes. Figure 11(c) shows the strain rate at the notch tip versus the nominal strain. The strain rate at the notch tip is the average of the values of the four elements located at the notch tip. In the elastic deformation range below the nominal strain of 0.0003, the strain rate at the notch tip is nearly 0.003 s⁻¹, which is approximately three times faster than the macroscopic strain rate of 0.001 s⁻¹. At a nominal strain of 0.001, the strain rate at the notch tip is about 0.011 s⁻¹, which is about 10 times faster than the macroscopic strain rate (0.001 s⁻¹).

Fig. 11

(a) and (b) are maps of the local strain rate at the notch tip. (a) At the nominal strain of 0.00004. (b) At the nominal strain 0.001. (c) Evolution of strain rate at the notch tip versus nominal strain.

It is to be noted that the effect of the increase in the strain rate and lowering temperature are equivalent.²⁵⁾ The CRSS of α-Ti also has the temperature dependence. The CRSS of the non-prismatic slip systems is sufficiently higher than that of the prismatic slip systems¹⁸⁾ at low temperatures, which leads to the suppression of the non-prismatic slips.^17,26,27) This suggests that the experimental results which showed that the slip deformation responsible for crack propagation is limited to the prismatic slip systems is due to the increase in the strain rate at the crack tip.

5. Conclusions

The temperature dependence of fatigue crack growth and 0.2% proof stress were investigated in Ti–0.49 mass%O. The following conclusions were obtained.

1. (1)    The fatigue crack growth rate in stage IIb does not depend on temperature above 300 K.
2. (2)    The orientation map of the fatigue crack wake indicates that only prismatic slips were activated during the fatigue crack growth.
3. (3)    CP-FEM analysis indicated that the CRSS of α-Ti at high temperature does not exhibit the suppression of the non-prismatic slip at a notch tip under the assumption that the CRSS is identical among all slip systems.
4. (4)    The local strain rate at the notch tip was approximately three times and ten times higher than the macroscopic strain rate at the elastic regime and the plastic regime, respectively. In such a case, it is expected that the non-prismatic slip systems is suppressed as the plastic deformation at low temperatures.

Acknowledgments

This work is partly supported by JSPS KAKENHI (Grant Number JP19H02462) and Element Strategy Initiative of MEXT (Grant Number JPMXP0112101000).

REFERENCES

• 1)   P.  Paris and  F.  Erdogan: Trans. ASME, J. Basic Eng. 85 (1963) 528–533. doi:10.1115/1.3656900
• 2)   H.  Kitagawa: J. Soc. Mater. Sci. 21 (1972) 710–719. doi:10.2472/jsms.21.710
• 3)   X.H.  Shi,  W.D.  Zeng,  C.L.  Shi,  H.J.  Wang and  Z.Q.  Jia: Mater. Sci. Eng. A 621 (2015) 143–148. doi:10.1016/j.msea.2014.10.022
• 4)   K.  Nagai,  T.  Yuri,  T.  Ogata,  O.  Umezawa,  K.  Ishikawa,  T.  Nishimura,  T.  Mizoguchi and  Y.  Ito: ISIJ Int. 31 (1991) 882–889. doi:10.2355/isijinternational.31.882
• 5)   Y.  Yamazaki,  R.  Sugaya,  U.  Kobayashi and  Y.  Ohta: Mater. Sci. Eng. A 797 (2020) 140248. doi:10.1016/j.msea.2020.140248
• 6)   O.  Umezawa,  M.  Hamada and  T.  Tatsumi: Procedia Mater. Sci. 12 (2016) 48–53. doi:10.1016/j.mspro.2016.03.009
• 7)   Z.  Xu,  A.  Liu and  X.  Wang: Mater. Sci. Eng. A 824 (2021) 141824. doi:10.1016/j.msea.2021.141824
• 8)   B.R.  Anne,  M.  Tanaka,  S.  Yamasaki and  T.  Morikawa: Mater. Trans. 62 (2021) 968–974. doi:10.2320/matertrans.MT-M2020399
• 9)   P.J.  Cotterill and  J.F.  Knott: Acta Metall. Mater. 40 (1992) 2753–2764. doi:10.1016/0956-7151(92)90345-F
• 10)   B.  Lü and  X.  Zheng: Eng. Fract. Mech. 42 (1992) 1001–1009. doi:10.1016/0013-7944(92)90139-6
• 11)   N.-Y.  Tang,  P.  Niessen and  A.  Plumtree: Z. Metallk. 82 (1991) 721–726. doi:10.1515/ijmr-1991-820911
• 12)   M.  Tanaka,  A.  Fujii,  H.  Noguchi and  K.  Higashida: Philos. Mag. Lett. 94 (2014) 95–102. doi:10.1080/09500839.2013.872811
• 13)   S.  Zaefferer: Mater. Sci. Eng. A 344 (2003) 20–30. doi:10.1016/S0921-5093(02)00421-5
• 14)   T.  Ohashi,  M.  Kawamukai and  H.  Zibib: Int. J. Plast. 23 (2007) 897–914. doi:10.1016/j.ijplas.2006.10.002
• 15)   Y.  Okuyama,  M.  Tanaka,  T.  Ohashi and  T.  Morikawa: Mater. Sci. Forum 941 (2018) 1474–1478. doi:10.4028/www.scientific.net/MSF.941.1474
• 16)  G. Simmons and H. Wang: Single Crystal Elastic Constants and Calculated Aggregate Properties, (The MIT Press, Cambridge, 1971) p. 34.
• 17)   M.  Tanaka,  Y.  Hayashi,  Y.  Okuyama,  T.  Morikawa and  K.  Higashida: Mater. Trans. 60 (2019) 80–85. doi:10.2320/matertrans.M2018251
• 18)   A.  Akhtar: Metall. Mater. Trans. A 6 (1975) 1105–1113. doi:10.1007/BF02661366
• 19)   E.D.  Levine: Trans. Metall. Soc. AIME 236 (1966) 1558–1565.
• 20)   T.  Tanaka and  H.  Conrad: Acta Metall. 20 (1972) 1019–1029. doi:10.1016/0001-6160(72)90136-8
• 21)   A.  Akhtar and  E.  Teghtsoonian: Metall. Mater. Trans. A 6 (1975) 2201–2208. doi:10.1007/BF02818644
• 22)   H.  Numakura: Materia Japan 37 (1998) 117–124. doi:10.2320/materia.37.117
• 23)   S.  Naka,  A.  Lasalmonie,  P.  Costa and  L.P.  Kubin: Philos. Mag. A 57 (1988) 717–740. doi:10.1080/01418618808209916
• 24)   D.  Tromans: IJRRAS 6 (2011) 462–483.
• 25)   J.  Harding: Arch. Mech. 27 (1975) 715–732.
• 26)   B.R.  Anne,  Y.  Okuyama,  T.  Morikawa and  M.  Tanaka: Mater. Sci. Eng. A 798 (2020) 140211. doi:10.1016/j.msea.2020.140211
• 27)   B.R.  Anne,  M.  Tanaka and  T.  Morikawa: Mater. Trans. 60 (2019) 1828–1832. doi:10.2320/matertrans.ME201902