Crystal Plasticity Analysis of Microscopic Deformation Mechanisms and GN Dislocation Accumulation Depending on Vanadium Content in β Phase of Two-Phase Ti Alloy

Abstract

Inhomogeneous deformation of a single α-β colony in a Ti–6Al–4V alloy under uniaxial tensile conditions was numerically simulated using a crystal plasticity finite element (CPFE) method, and we predicted density changes in geometrically necessary dislocations (GNDs) depending on the vanadium concentration in the β phase (V_β). The geometric model for the CPFE analysis was obtained by converting data from electron back-scatter diffraction patterns into data for the geometric model for CPFE analysis, using a data conversion procedure previously developed by the authors. The results of the image-based crystal plasticity analysis indicated that smaller V_β induced greater stress in the α phase and smaller stress in the β phase close to the α-β interfaces in the initial stages of deformation because of the elastically softer β phase with lower V_β. This resulted in greater strain gradients and greater GND density close to the interfaces in the initial stages of deformation within the single α-β colony when the β phase plastically does not deform.

1. Introduction

Titanium (Ti) alloys have been widely used for over 50 years, typically in the areas of aerospace, household goods, and medical applications, because of their significant specific strength, fracture toughness, and good corrosion resistance, among other advantages.¹⁾ While Ti–6Al–4V alloys are the most commonly used Ti alloys,²⁾ their mechanical properties and deformation mechanisms are not fully understood. Therefore, the investigation of these alloys will lead to an enhancement of the reliability and safety of numerous products.

The microstructures of Ti–6Al–4V alloys change depending on the processing route, such as heat treatment,^2,3) including fully equiaxed α, fully lamellar α, and bi-modal. The fully lamellar and bi-modal microstructures comprise hexagonal close-packed (hcp) α phase and body-centered cubic (bcc) β phase, and lamellar structures with α and β phases existing in both microstructures. The typical lamellar structure is shown in Fig. 1. The Ti alloys containing such lamellar structures typically exhibit greater strength and better fatigue strength than those with fully equiaxed α^3,4) because of the smaller crystal grains of the α phase divided by the β phase. The sizes and slip lengths of α grains decrease with the division by the β phase, called the alloy element partitioning effect.³⁾ As discussed above, two-phase Ti alloys exhibit better mechanical properties. However, the deformation mechanism within the α-β colonies are not well understood.

Fig. 1

Back scattered electron image (a) and schematic illustration (b) of a typical lamellar structure of a two-phase Ti–6Al–4V alloy.

Inhomogeneous deformation occurs within the α-β colonies when they deform. One of the reasons for this is the difference in mechanical properties between the α and β phases. The β phase is typically softer than the α phase in the elastic range, and is easily deformed compared to the α phase under slight deformation, while the dislocation movements in the α phase are prevented by the β phase in the plastic ranges.^5–7) In other words, the difference between the mechanical properties in both phases in the elasto-plastic ranges significantly affects the deformation mechanism within the α-β colonies.

The difference in the mechanical properties between the α and β phases generates a strain gradient close to the interface under plastic deformation. Geometrically necessary dislocations (GNDs) are accumulated close to the interface because of the strain gradients, and the accumulation of GNDs results in changes in the work-hardening rate within the α-β colonies.^5,6) The relationship between the interaction of the α and β phases, the deformation mechanism, and the accumulation of GNDs has been studied by experimental^5,8) and numerical methods;⁹⁾ however, studies that focus on the effect of the mechanical properties of the β phase, depending on the chemical composition of the deformation mechanism in the two-phase alloy, have not been conducted.

Aluminum (Al) and vanadium (V) are strong stabilizers for the α phase and the β phase, respectively, and each element is concentrated within the individual phase. The volume fraction levels of the β phase in slow-cooled Ti–6Al–4V alloys are approximately 10 vol%.²⁾ The volume fractions change depending on the processing history, such as heat treatments. The concentration of V in the β phase (V_β) at smaller volume fractions is easier to change with the volume fraction; the mechanical properties of the β phase also change with the V_β. However, the effects of the V_β on the deformation mechanism in the α-β colonies are unclear.

Numerical analysis is typically utilized to investigate the activity of slip systems and the process of dislocation accumulations in the scale of crystal grains.^10–17) The numerical methods employed in the investigations can be classified broadly into discrete and continuous methods represented, respectively, by molecular dynamics (MD)^15,16) and crystal plasticity analysis.^10,13) While the crystal plasticity analysis does not track the movement of atoms in detail, it is better suited for investigation of the activities of slip systems in crystal grains in wider visual fields than the discrete methods.

The authors have developed a procedure to convert the data of microstructural maps obtained by electron back scatter diffraction patterns (EBSD) into geometric models for crystal plasticity finite element (CPFE) analysis.¹⁸⁾ The procedure solves the operation complexity in the process of the data conversion, and we can easily obtain geometrical models for CPFE analysis using the procedure. In this study, we adapt the procedure to a microstructure of two-phase titanium alloy, and investigate the distributions of slip strains and density of GNDs in a single α-β colony depending on the elastic constants as a function of the V_β, using a crystal plasticity finite element (CPFE) method. First, the maps of crystal orientations and phases of a single α-β colony in a two-phase Ti–6Al–4V alloy are obtained by electron back-scatter diffraction (EBSD) patterns. Second, the data of the map are converted into data of a geometric model for CPFE analysis. Third, uniaxial tensile deformations of the α-β colony are represented by the CPFE method employing a dislocation-based constitutive equation and, finally, the relationship between the density of GNDs and V_β is investigated.

2. Materials and Geometric Model for the Analysis

The sample material used in this study is a Ti–6Al–4V alloy with the chemical composition presented in Table 1. The specimen is cut from the regions at a distance from the surfaces of the material ingot with dimensions ϕ245 mm × 360 mm, and the heat treatment shown in Fig. 2(a) is conducted for the purpose of homogenization of the microstructure. An SEM-backscattered electron image of the specimen is shown in Fig. 2(b). The EBSD measurements are then conducted at 0.1 µm intervals. The data obtained by the measurements are cleaned and the phases and crystal orientations in their measurement points are distinguished by TSL OIM software.

Table 1 Chemical composition of Ti–6Al–4V alloy employed in this study [wt.%].

Fig. 2

Condition of heat treatment (a), and backscattered electron image of sample material (b). Bright and dark contrasts correspond to α and β phases in Fig. (b), respectively.

3. Conversion into Geometric Model for CPFE Analysis

The data obtained by EBSD measurements are converted into geometric model data for CPFE analysis using the data conversion procedure developed by Kawano et al.¹⁸⁾ The maps, comprising hexagonal pixels typically employed in the visualization of EBSD measurement data, are converted into geometric models with cuboid elements. The geometric model obtained is shown in Fig. 3. The thickness of the geometric model is assumed to be 0.3 µm, although this is not observed in the EBSD measurement data, and the size was 15 µm × 15 µm × 0.3 µm. The distribution of crystal orientations within each phase is almost homogeneous, and the crystal orientations are shown in Fig. 3. The misorientations are 1.5° between ⟨0001⟩_α and ⟨110⟩_β and 3.4° between $\langle 11\bar{2}0\rangle _{\alpha }$ and $\langle 1\bar{1}0\rangle _{\beta }$; in other words, the Burgers orientation relationship (BOR)¹⁹⁾ is approximately satisfied in the geometric model.

Fig. 3

Geometric model for the CPFE analysis and the loading condition. The plane strain condition is established on planes normal to the Z-direction. A_Y-B_Y is the central cross-sectional line to observe the distributions of stress, strain, and GND density. The volume fraction of the β phase is approximately 10%. The hexagonal prism and the cube in the figure show crystal orientations of α and β phases.

4. Methods and Conditions for Crystal Plasticity Analysis

4.1 Numerical procedure for crystal plasticity analysis

The clp, that is a code for three-dimensional CPFE analysis developed by Ohashi,^10,20–23) is employed in this study. The clp was employed for the analysis of the deformation of materials with face-centered cubic structures,^23–27) bcc structures,^28,29) and hcp structures.^30,31) The constitutive equations employed in this study are shown below.

It is assumed that Schmid’s law is proven, and an elasto-plastic constitutive equation for slip deformation can be given as follows:³²⁾   

\begin{equation} \dot{\varepsilon}_{ij} = \left[S_{ijkl}^{e} + \sum_{n}\sum_{m}\{h^{(\textit{nm})}\}^{-1}P_{ij}^{(n)}P_{kl}^{(m)}\right]\dot{\sigma}_{kl}, \end{equation} (1)

where $\dot{\varepsilon }_{ij}$, $\dot{\sigma }_{kl}$, $S_{ijkl}^{e}$, $P_{ij}$, and h denote the increment of the strain, the increment of the Cauchy stress tensor, the elastic compliance, the Schmid tensor, and the coefficient of work hardening, respectively, and the superscripts m and n represent slip systems. The critical resolved shear stress (CRSS) is calculated by a modified Bailey–Hirsh model as follows:²²⁾   

\begin{equation} \theta^{(n)} = \theta_{0}(T) + \sum_{m}a\mu\tilde{b}\Omega^{(\textit{nm})}\sqrt{\rho_{S}^{(m)}}, \end{equation} (2)

where θ₀, a, μ, $\tilde{b}$, Ω, and ρ_S denote the lattice friction stress depending on temperature T, a numerical factor of the order of 0.1, the elastic shear modulus, the magnitude of Burgers vector, the interaction matrix between slip systems, and the density of statically stored dislocations (SSDs), respectively. The increment in the SSD density is given as follows:²⁰⁾   

\begin{equation} \dot{\rho}_{S}^{(m)} = \frac{c\dot{\gamma}^{(m)}}{\tilde{b}L^{(m)}}, \end{equation} (3)

where c, $\dot{\gamma }$, and L denote a numerical coefficient of order 1, the increment of the plastic strain, and the mean free path of mobile dislocations, respectively. L is given as follows:²¹⁾   

\begin{equation} L^{(m)} = \frac{c^{*}}{\sqrt{\displaystyle\sum_{k}\omega^{(\textit{km})}(\rho_{S}^{(k)} + \|\rho_{G}^{(k)}\|)}}, \end{equation} (4)

where $c^{*}$, ω, and ρ_G denote a numerical constant to represent the ease with which dislocation accumulation occurs, a weight matrix to reproduce dislocation interactions, and the GND density, respectively. The norm of the GND density is defined as follows:¹⁰⁾   

\begin{equation} \|\rho_{G}^{(m)}\| = \sqrt{(\rho_{G,\textit{edge}}^{(m)})^{2} + (\rho_{G,\textit{screw}}^{(m)})^{2}}, \end{equation} (5)

  

\begin{equation} \rho_{G,\textit{edge}}^{(m)} = -\frac{1}{\tilde{b}}\frac{\partial\gamma^{(m)}}{\partial\xi^{(m)}}, \end{equation} (6)

  

\begin{equation} \rho_{G,\textit{screw}}^{(m)} = \frac{1}{\tilde{b}}\frac{\partial\gamma^{(m)}}{\partial\zeta^{(m)}}, \end{equation} (7)

where ρ_G,edge and ρ_G,screw denote the densities of the edge and screw components of GNDs, respectively, and ξ ζ represent directions parallel and perpendicular to the slip direction on the slip plane, respectively. The increment in CRSS is given as follows:   

\begin{equation} \dot{\theta}^{(n)} = h^{(\textit{nm})}\dot{\gamma}^{(m)} \end{equation} (8)

The strain-hardening coefficient h is obtained from eqs. (2)–(4) and eq. (8) as follows:   

\begin{equation} h^{(\textit{nm})} = \frac{ac\mu\Omega^{(\textit{nm})}}{2L^{(m)}\sqrt{\rho_{S}^{(m)}}}. \end{equation} (9)

4.2 Conditions of CPFE analysis

4.2.1 Elastic compliances of individual phase

In slow-cooled Ti–6Al–4V alloys, the volume fraction of the α phase increases to approximately 90%.²⁾ In other words, the volume fraction of the α phase is typically significantly greater than that of the β phase. In this study, therefore, changes in the concentration of Al in the α phase with the volume fraction are ignored, and the elastic constants of the α phase are assumed to remain unchanged. The elastic compliances of pure Ti presented in Table 2 are employed as approximately those of the α phase; however, the elastic constants of Ti–7Al alloys with an hcp structure are similar to those of pure Ti.³³⁾

Table 2 Elastic compliance of pure titanium [(TPa)⁻¹].

By contrast, the V_β is changed depending on the processing, such as heat treatments, because of the changes in its volume fraction.³⁴⁾ Therefore, the elastic constants in the β phase also change with the V_β. Figure 4 shows the elastic compliances of single crystals of Ti–V alloys with a bcc structure from plots obtained by Alers³⁵⁾ and Katahara et al.,³⁶⁾ and the regression lines are obtained using the least-squares method. The regression lines can be expressed as functions of V_β as follows:   

\begin{align} S_{11} & = -0.1391 \times V_{\beta}(\text{wt.%}) + 20.189\\ S_{12} & = 0.0640 \times V_{\beta}(\text{wt.%}) - 8.4624 \\ S_{44} & = -0.0419 \times V_{\beta}(\text{wt.%}) + 26.417. \end{align} (10)

The β phase is stable at room temperature only if it is enriched with more than 15 wt.% V.²⁾ Three conditions (15, 30, and 50 wt.%) of V_β are prepared for this analysis. The elastic compliances for the three conditions are presented in Table 3.

Fig. 4

Elastic compliance of Ti–V alloys.

Table 3 Elastic compliances of β phase depending on V_β. A₁ is the indicator of elastic anisotropy calculated from the elastic stiffness (A₁ = 2C₄₄/(C₁₁ − C₁₂)).

4.2.2 CRSS for individual phase

Active slip systems of the α phase in Ti–6Al–4V alloys are basal ⟨a⟩, prismatic ⟨a⟩, pyramidal ⟨a⟩, and pyramidal ⟨c + a⟩.¹²⁾ The CRSS for each slip system exhibited variety in previous reports,^37–41) and the dependency of the strain rates on CRSS are also different among the slip systems. In this study, a constant strain rate is assumed and the CRSS is determined referencing Bridier et al.,⁴²⁾ as presented in Table 4. In the α phase, it is commonly believed that activity of twinning under the deformation decreases with increasing the Al content.⁴³⁾ Twin deformation does not appear when the Al concentration is 6 wt.%, although it is possible that twinning activity is increased with increasing Al content at smaller Al levels.⁴⁴⁾ Therefore, twining is not taken into account in this simulation.

Table 4 CRSS and Schmid factor of each slip system.

In experimental observations, Chan et al.⁴⁵⁾ and Suri⁵⁾ reported that the flow stress level of α-β colonies increased when the movement of dislocations in the α phase were inhibited by β laths. Ashton et al.⁹⁾ incorporated this phenomenon into the simulation, and successfully reproduced changes in the flow stress level depending on the conditions, whereby dislocations move either through or in the direction parallel to the β lath.

The role of the β phase within single α-β colonies under deformation becomes clear by comparing the results obtained under the conditions where the β phase obstructs movements of dislocations with the results obtained under the conditions where the β phase does not obstruct them. Three CRSS sets for the β phase (CRSS_β) are presented in Table 4 and are assumed to represent the conditions under which the β phase is easily deformed (CRSS_β = 370 MPa; Condition 1), the β phase is possible but difficult to deform (CRSS_β = 500 MPa; Condition 2), and the β phase completely prevents movement of dislocations (i.e., the β phase does not plastically deform at all) (CRSS_β = ∞; Condition 3). Typical slip systems of β phase in two-phase titanium alloys are $\{ 110\} \langle 1\bar{1}1\rangle $ and $\{ 121\} \langle 1\bar{1}1\rangle $.⁴⁶⁾ Deformation twinning may occur in the β phase while the activity decreases with higher V_β.⁴⁷⁾ In this study, the twinning is ignored, and we assume that only the slip systems of $\{ 110\} \langle 1\bar{1}1\rangle $ and $\{ 121\} \langle 1\bar{1}1\rangle $ are possible to activate in the β phase.

4.2.3 Geometry of the analysis

Figure 3 shows the loading condition applied to the geometric model for the CPFE analysis. The Y-directional tensile loading is applied by the forced displacement, the plane-displacement condition is assumed for the planes perpendicular to the Z-axis, and zero stress is applied to the planes perpendicular to the Y-axis. The Schmid factors for each slip system are presented in Table 4.

5. Results

Figure 5 shows the results obtained under the conditions where V_β = 15 wt.%V and CRSS_β = 500 MPa in the plastic range (nominal strain (ε_n) is 1.0%). Figure 5(a) shows the spatial distribution of norm of GND density; greater GND density exists close to the α-β interfaces. Such phenomena can be also observed for the case of actual two-phase Ti alloys.^5,6) In this section, we investigate the reason why such distribution is obtained, and validate the results.

Fig. 5

Distributions of the norm of GND density (a), profiles of σ_YY (b), and γ of the α phase (c) along the cross-sectional line A_Y-B_Y when V_β = 15 wt.%V and CRSS_β = 500 MPa. ε_n = 1.0%.

Figure 5(b) and (c) show the distribution of normal stress in the tensile direction (σ_YY) and slip strain (γ) in the α phase in the central cross-sectional line A_Y-B_Y, respectively. Lower stress within the β phase and greater stress in the α phase close to the α-β interfaces exist in the distribution (Fig. 5(b)). This is because the α phase close to the interfaces is required to support the force instead of the elastically softer β phase. The stress concentration in the α phase close to the interfaces leads to the activation of slip systems, and locally greater slip strain occurs in the α phase close to the interfaces (Fig. 5(c)). The slip system activated is the basal one, and the GNDs accumulated are also by the basal one. The reasons for the activation of the basal slip system are its high Schmid factor compared with the other slip systems, and relatively low CRSS.

The dislocation accumulation close to the α-β interfaces occurs in two-phase Ti alloys when the β phase inhibits the movement of dislocations.^5,6,9) β laths can act as a filter for the slip glide,⁴⁸⁾ which means that the difficulty in dislocation movements through the α-β interfaces differs among the slip systems. This could cause the basal slip operation in lamellar colonies even if the prismatic ⟨a⟩ slip system is the primary one in globular α grains. This is because the CRSS for a basal ⟨a⟩ slip system could be smaller than that for prismatic ⟨a⟩ slip systems because of BOR.^48,49) The activation of pyramidal ⟨a⟩ slip systems in the α phase is observed^37,39) and the CRSS for the slip systems is estimated.³⁷⁾ The CRSS for the pyramidal ⟨a⟩ slip systems can be seen as the same level with the prismatic ⟨a⟩ slip systems;³⁷⁾ however, the shear stress passing through the α-β interface remains unknown in the pyramidal ⟨a⟩ slip systems. In the current simulation, pyramidal ⟨a⟩ slip systems did not operate due to the low Schmid factor; the result is reasonable when the α-β interfaces are assumed to work as a resistance of dislocations moving through.

It is predicted that four of the twelve pyramidal ⟨c + a⟩ slip systems can pass more easily through the α-β interfaces than those of the other pyramidal ⟨c + a⟩ slip systems when BOR is satisfied in the α-β colonies.⁵⁰⁾ However, the CRSS for the ⟨c + a⟩ slip systems and defect energies for the ⟨c + a⟩ dislocations are significant compared with those of the other slip systems. Littlewood et al.⁸⁾ reported that ⟨a⟩-type GNDs are dominantly accumulated rather than ⟨c + a⟩ types in a two-phase Ti–6Al–4V alloy. These previous studies indicate that activities of ⟨a⟩-type slip systems are greater than those of pyramidal ⟨c + a⟩ slip systems. The pyramidal ⟨c + a⟩ slip systems did not activate in the current simulation due to their high CRSS.

The α phase yielded earlier than the β phase which does not almost plastically deform in the current simulation. We discuss the validity of this phenomenon as follows. If we assume that there is no dislocation source possible to activate within the β phase under low stress levels, and there is a greater work-hardening rate because of the size effect, it appears to be reasonable that we employ a model to represent the β phase in which the deformation is developed by the movement of dislocations derived from, and pathing through, the α phase. Actually, Jun et al.⁴⁹⁾ reported that the slip seems to start in the α phase and progresses through the β phase when micropillar compression tests of two-phase Ti alloy were conducted. Therefore, the simulation results obtained by the current models, where the β phase starts to plastically deform after the yielding of the α phase, or the β phase cannot plastically deform in the early stages of deformation, represents one of the realistic phenomena in the initial stage of the deformation.

6. Discussion

6.1 Controlling factor for localized slip strain in α-β interfaces

We successfully reproduced the accumulation of GNDs close to the α-β interfaces in the current numerical simulation (Fig. 5(a)). The existence of GNDs requires a gradient of slip strain. The difference of deformability between the α and β phases induces inhomogeneous deformation, which results in strain gradient; this is a candidate for the cause of GND accumulation. On the basis of the condition employed in the simulation of Fig. 6, we conduct several simulations with changing CRSS_β and V_β, and investigate their effects on GND accumulation. The GND accumulation is directly relevant to the formation of strain gradient; we firstly examine the changes in distributions of stress and strain depending on CRSS_β, secondly, their dependency on V_β is investigated and, finally, we reveal the effects of V_β and CRSS_β on GND accumulation.

Fig. 6

Profile showing the relationship between $\bar{\sigma }_{YY}$ and ε_n depending on the CRSS_β or V_β. In the symbols of the figure, 15, 30, and 50 wt.%V are conditions of V_β.

6.2 Distributions of stress and strain depending on V_β

In this section we describe the effects of changes in the elastic constants that depend on V_β and CRSS_β of the stress and strain in the specimen. Figure 6 shows the relationship between the average normal stress in the loading direction ($\bar{\sigma }_{YY}$), and nominal strain (ε_n). Under the current simulation conditions, the flow stress levels are greater when CRSS_β is greater, while the effect of changes in the elastic compliances of the β phase as a function of V_β has a negligible impact on the stress-strain curves.

In the microscopic viewpoint, however, distributions of stress and strain show the dependency of V_β as well as of CRSS_β. Figure 7 shows the distributions of stress and strain in the elastic or plastic range along the central cross-sectional line A_Y-B_Y. Profiles of Figs. 7(a) and (b) show the dependency of the distributions of σ_YY and normal strain in the tensile direction (ε_YY) on CRSS_β in the elastic range (ε_n = 1.0%). The β phase is elastically softer than the α phase in the current simulation conditions, and lower stress and greater strain exist in the β phase than for the α phase. The tendency of the distribution of stress and strain is more pronounced with lower V_β. This is because the β phase is elastically softer with lower V_β due to the dependency of elastic compliances on V_β (Fig. 4). As shown in Fig. 7(a), the stress concentration in the α phase close to the α-β interfaces occurs when the β phase is easier to elastically deform than the α phase, and this tendency is stronger with lower V_β.

Fig. 7

The profiles showing distributions of (a) σ_YY and (b) ε_YY along the central cross-sectional line A_Y-B_Y, depending on V_β in the elastic range (ε_n = 0.1%).

The significant stress in the α phase close to the interfaces induces the activation of slip systems in these regions. That is, γ in the α-β interfaces also shows the dependency of the stress concentration on V_β. Figures 8(a) and (b) show the distributions of σ_YY and γ of the α phase in the central cross-sectional line A_Y-B_Y depending on V_β when the ε_n is 1.0% (plastic range). The stress concentration close to the α-β interfaces also occurs in the plastic range, and the tendency is stronger with lower V_β, which is the same as for the case of the elastic range. The greater stress with lower V_β induces the activation of slip systems, and the γ in the α phase close to the α-β interfaces increases with lower V_β. The active slip system in the α phase was basal one in all conditions of V_β. From the above results, we notice that stress, elastic strain, and γ tend to be greater in the α phase close to the α-β interfaces when the β phase is elastically softer (i.e., V_β is lower).

Fig. 8

The profiles showing distributions of σ_YY and γ of the α phase along the central cross-sectional line A_Y-B_Y depending on V_β ((a), (b)) and CRSS_β ((c), (d)) in the plastic range (ε_n = 1.0%).

6.3 Distributions of stress and strain depending on CRSS_β

In this section, we focus on dependency of the distribution of stress and strain on CRSS_β. The three conditions of CRSS_β described in Table 4 are employed to compare the results obtained with those shown in Figs. 8(c) and (d). The condition CRSS_β = 370 MPa (Condition 1) is that the β phase is easier to plastically deform than the α phase; we choose lower CRSS_β than that of the condition in Fig. 5, and we assume that both $\{ 110\} \langle 1\bar{1}1\rangle $ and $\{ 121\} \langle 1\bar{1}1\rangle $ are possible to activate virtually to express the plastic softness of the β phase. Under this condition, the β phase starts to yield earlier than the α phase due to its lower CRSS and higher Schmid factor than that of the α phase. This represents the condition where the β phase contains enough dislocation sources for it to be possible to emit dislocations under lower stress. The condition CRSS_β = 500 MPa (Condition 2) is the same condition with the one used in the simulation of Fig. 5. The condition CRSS_β = ∞ (Condition 3) is the condition where the β phase does not plastically deform at all; this represents the condition where there are no dislocation sources to emit dislocations under lower stress, and slip transfers between phases are impossible.

Figures 8(c) and (d) are the distributions of σ_YY and γ in the α phase in the central cross-sectional line A_Y-B_Y depending on CRSS_β when ε_n is 1.0% (plastic range). Figure 8(c) shows the stress concentration in the α-β interfaces for any values of CRSS_β. That is, the stress concentration occurs close to the α-β interfaces even when the β phase is plastically softer than the α phase, and the γ is also greater in the α phase close to the interfaces in all CRSS_β conditions, reflecting the distribution of stress. In addition, the results obtained for the condition CRSS_β = ∞ are almost the same as those of the condition CRSS_β = 500 MPa; this is because the β phase is not almost deformed at such low ε_n when CRSS_β = 500 MPa.

The above results indicate that stress concentration occurs in the α phase close to the α-β interfaces regardless of the plastic deformability of the β phase when the β phase is elastically softer than the α phase, and it results in greater γ close to the interfaces.

6.4 Distributions of GND density depending on V_β and CRSS_β

Local increases of γ generate its spatial gradient, and the existence of GNDs is required to satisfy the spatial compatibility within such regions. Thus, the γ concentration close to the interface observed in Fig. 8 leads to the accumulation of GNDs. Figure 9 shows the dependency of distributions of the norm of GND density on V_β and CRSS_β. Figures 9(a) and (b) are distributions of the norm of GND density along A_Y-B_Y depending on V_β when CRSS_β = 500 MPa and 370 MPa, respectively. In addition, the distribution of the norm of GNDs on the condition that CRSS_β = ∞ is almost the same as the result for the condition that CRSS_β = 500 MPa, and is left out. Figure 9(c) shows the dependency of distributions of the norm of GND density on CRSS_β when V_β = 15 wt.%V. A greater norm of GND density close to the α-β interfaces exists for any conditions (Fig. 9) reflecting gradients of γ (Fig. 8), and the magnitude changes depending on both CRSS_β and V_β.

Fig. 9

Distributions of the norm of GND density along the cross-sectional line A_Y-B_Y. (a) and (b) show the dependency of V_β when CRSS_β = 500 and 370 MPa, respectively. (c) shows the dependency of CRSS_β when V_β = 15 wt.%. ε_n is 1.0%.

The maximum GND density in the distribution depending on V_β becomes greater with lower V_β when the β phase is difficult to plastically deform (Fig. 9(a)). This is because, as stated above, the elastically softer β phase induces stress and strain concentration in the α phase close to the β phase. On the other hand, such a relationship between GND density and V_β is not established when the β phase plastically deforms easily (Fig. 9(b)); the GND density is greater in the order of the conditions V_β = 30 wt.%V, 15 wt.%V, and 50 wt.%V. The cause for this is relevant to plastic as well as elastic deformability of the β phase.

The stress in the α phase close to the α-β interface is greater with an elastically softer β phase (Fig. 7(a)); however, the β phase tends to be more difficult to plastically deform with lower V_β due to lower stress in the β phase (Fig. 7(a)). Figure 10 shows the changes in the profiles of average normal strain in the tensile direction ($\bar{\varepsilon }_{YY}$) within the β phase as a function of ε_n, and the yielding times are earlier with higher V_β. That is, the plastic strain in the β phase is larger with higher V_β. A plastically softer β phase leads to steeper gradients of γ (Fig. 8(d)), and the GND density tends to become greater. Actually, profiles of GND density as a function of V_β (Fig. 9(c)) show greater density of GNDs with lower V_β. This means that both the elastic and plastic softness of the β phase affect the density of GNDs close to the α-β interfaces. However, it is noted that the conditions that the β phase yields earlier than α phase are satisfied only when there are mobile dislocations within the β phase under a lower stress level.

Fig. 10

The profiles showing $\bar{\varepsilon }_{YY}$ within the β phase changes with ε_n. The strain in the β phase rapidly increases after the yielding when CRSS_β = 370 MPa while the β phase does not almost plastically deform on the condition of CRSS_β = 500 MPa. Δε_β-yield is the difference of yielding timings between the conditions of V_β = 15 wt.%V and V_β = 50 wt.%V.

From the above results, it is revealed that GNDs tend to accumulate close to the α-β interfaces regardless of the plastic deformability, and this tendency is stronger with lower V_β when β phase is not plastically deformed. However, an unclear point still remains; the plastic as well as elastic deformability of the β phase affects the GND density close to the α-β interfaces, while the plasticity of the β phase within α-β colonies is unknown in detail. Deformability of the β phase such as yielding conditions are affected by V_β⁴⁷⁾ when the β laths within α-β colonies are possible to plastically deform by dislocation glide or twining. If there are not any mobile dislocations and twining does not occur within the β phase, plastic deformation within the β phase progresses by dislocation transmission through the α-β interfaces sourced from the α phase;^5–7,51) the difficulty of transmission changes depending on the crystal orientation relationship between the α and β phases. Even if the angles of slip planes are different between α and β phases, dislocations may be transmitted from incoming to outgoing interfaces by the indirect or the direct dislocation transmission mechanism under the condition of dislocation pilling up at the interfaces or stronger external stress.⁵¹⁾ Morphology also affects the deformability of the β phase within α-β colonies.⁵²⁾ Further study is required to reveal the effects of plasticity of the β phase on the accumulation of GNDs.

7. Conclusion

In this study, a microstructural map of an α-β colony in Ti–6Al–4V obtained by EBSD patterns was converted into a geometric model for CPFE analysis using a data conversion procedure that converts EBSD measurement data to geometric data for CPFE analyses. The CPFE analysis was performed under a number of conditions of V concentration in the β phase (V_β). Uniaxial tensile loading was applied to the geometric model of the α-β colony by forced displacement. We investigated the mechanisms of the deformation and accumulation of the GNDs depending on V_β. The results can be summarized as follows:

1. (1)    The absolute values of the elastic compliances of the β phase increase linearly with decreasing V_β; therefore, the β phase is easier to deform under lower V_β in the elastic range, and stress concentration occurs in the α phase close to the α-β interfaces with lower V_β. This results in greater γ and formation of steeper strain gradients, and GNDs are accumulated in the α phase close to the interfaces. In other words, greater GND density exists close to the interfaces with lower V_β under a low strain level.
2. (2)    The accumulation of GNDs close to the α-β interfaces occurs regardless of the yielding conditions of the β phase. This is because the strain gradients are formed close to the interfaces for both conditions, such that the β phase yields earlier and later than the α phase. However, the phenomenon of greater density of GNDs with lower V_β is not always observed for the condition where the β phase is easily deformed. This is because not only elastic but also plastic deformability of the β phase contributes to the GND accumulation close to the interfaces.

Acknowledgement

This work was partly supported by the Council for Science, Technology and Innovation (CSTI), the Cross-ministerial Strategic Innovation Promotion Program (SIP), “Fundamental Research Focusing on Interface for Overcoming Unsolved Issues in Structural Materials” (Founding agency: JST). This work was also supported by the Amada Foundation.

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