Effects of strain state and slip mode on the texture evolution of a near-α TA15 titanium alloy during hot deformation based on crystal plasticity method

1. Introduction

Near-α titanium alloys are widely used in aerospace industry due to their outstanding properties, such as superior specific strength and excellent corrosion resistance [1,2]. However, their mechanical properties are strongly dependent on the material texture owing to the hexagonal close-packed (HCP) crystal structure of α phase in titanium alloys. Due to the brittle nature of HCP metals in low temperature, most of titanium alloys are processed by means of thermo-mechanical processing [3], which determines the microstructure, texture and mechanical properties to a large extent [4]. Therefore, the analysis of the texture evolution of near-α titanium alloys during the hot metal forming can help obtain an optimal crystallographic texture and material performance.

The texture evolution of titanium alloys has been studied extensively based on the EBSD technique. During the hot compression of commercially pure (CP) titanium, the basal planes tended to rotate to an inclination of 45° angle to the compression axis [5]. During the hot tension of a near-α titanium alloys, a new texture component of (01$\bar{1}$0)[0001] generated due to the dynamic recrystallization [6]. In addition to the uniaxial loading process, during the KoBo type extrusion of CP titanium, the (0001) basal plane rotated to the extrusion direction [7]. During the spinning forming of TA15 titanium alloys, the non-standard (0001) texture gradually formed resulting from the activated basal slip system under the compression in the normal direction [8]. During the biaxial tensile testing of CP titanium, the new texture component [11$\bar{2}$0] developed due to the operation of the specific type of twinning systems [9]. During the hot rolling, all the samples of CP titanium developed a dominant basal fiber texture irrespective of the reduction percentages or the mode of rolling [10]. As stated above, the strain state has a remarkable effect on the texture evolution of titanium alloys during the hot metal forming. Li et al. [11] fabricated the titanium tubes with the desired radial texture by allocating the plastic flow during the multi-pass pilgering. Therefore, the relationship between strain state and texture evolution can help control the crystallographic texture and mechanical properties of the titanium alloys component during the hot metal forming. However, the effect of strain state on the texture evolution of titanium alloys during the hot metal forming has not been thoroughly investigated.

The deformation modes realized by different slip and twinning systems are considered to be the decisive causes of the texture evolution of HCP metals [12,13], although grains cannot rotate freely due to the interaction of adjacent grains [14]. The numerical methods, for example the crystal plasticity finite element method (CPFEM), are the ideal tools to study the polycrystalline deformation modes and the texture evolution in-depth in micro-scale [[15], [16], [17]]. No deformation twinning is observed during the hot deformation of near-α TA15 titanium alloys, which may be ascribed to the high Al content and the elevated process temperature [18]. Therefore, the slip mode has a decisive effect on the texture evolution of near-α TA15 titanium alloys during the hot metal forming. The activated slip mode would rotate the grain around the corresponding Taylor axis [19]. The relationship of slip mode and texture evolution of near-α TA15 titanium alloys during the hot metal forming has not been well established.

The present work emphasizes the effects of strain state and slip mode on the texture evolution of the near-α TA15 titanium alloy sheets during the hot deformation process. Firstly, the texture evolution during the hot uniaxial tension was studied by a combination of CPFEM simulation and EBSD observation to analyze the slip mode and its effect on the texture evolution. Then, the relationship between the texture evolution and strain state during the hot deformation process is established based on the present CPFEM.

2. Materials and methods

2.1. Material characterization and uniaxial tensile tests

Near-α TA15 titanium alloy sheets with a chemical composition of Ti-6.5Al-2Zr-1Mo-1 V (in wt%) were utilized in this work. The α→β transformation temperature of the present alloy is ∼980 °C [20]. The microstructure evolution of both initial and deformed samples was characterized by EBSD performed on a FEG LEO 1530 SEM with a scan size of 0.3 μm. The tensile samples with a gauge length of 20 mm and a width of 4 mm were machined from the sheet along the rolling direction (RD) and transverse direction (TD), respectively. The interrupted tensile tests of TA15 samples were carried out on an INSTRON 5500R test machine in air condition at 750 °C with an initial strain rate of 0.005 s^-1. It is a varying-strain-rate deformation condition. After the tensile process, the samples were water quenched immediately to preserve the deformed microstructure.

The inverse pole figures (IPFs) of the undeformed sheet with an average α grain size of 9.4 μm are depicted in Fig. 1(a), in which the color code represents the α grain orientation, and the black region represents the β phase (∼5%). The low fraction of the low angle grain boundaries (∼14.3%) indicated that the initial sheet contained little deformation structures. Fig. 1(b) shows the pole figures (PFs) of the initial sheet, which exhibits four orientations components: A(0°, 90°, 30°), B(180°, 60°, 30°), C(45°, 30°, 0°) and D(90°, 150°, 30°).

2.2. Model description

According to the previous work [18], the β phase has almost no effect on the overall deformation behavior due to its limited fraction (∼5%). In addition, the recrystallized grains, deformation twinning and grain boundary sliding were not observed during the hot tension of the utilized TA15 sheet at 750 °C with an initial strain rate of 0.005 s^-1. Therefore, the dislocation slipping serves as the sole deformation mechanism in this paper. In the α phase of titanium alloys with HCP lattice, the five operating slip systems include basal (0001)[11$\bar{2}$0], prismatic {10$\bar{1}$0}[11$\bar{2}$0], pyramidal {10$\bar{1}$1}[11$\bar{2}$0], pyramidal-1 {10$\bar{1}$1}[11$\bar{2}$3] and pyramidal-2 slip systems {11$\bar{2}$2}[11$\bar{2}$3] according to the Taylor method [21]. Accordingly, this paper deals with the texture evolution of α grains in the near-α TA15 titanium alloys during the hot deformation process with the dislocation slipping as the sole deformation mechanism.

The same theoretical model with the previous work [18] was utilized in this paper, because the deformation condition and the utilized TA15 samples were consistent with ones in the previous work. So, the kinetic law, hardening behavior and evolutionary behaviors of dislocation density of a slip system were described briefly here, which can effectively characterize the deformation behavior of the utilized TA15 sample during the hot metal forming.

In a rate-dependent crystal plasticity model, the kinetic law of a slip system is directly linked to the resolved shear stress and thermal activation energy required to overcome lattice resistance, which is given by [22]:

$\dot{γ}=\dot{γ}_0^αexp\{-\frac{F_0}{k_Bθ}[1-<\frac{|τ^α|-τ_{cr}^α}{S^α}>^p]^q\}sign(τ^α)$ (1)

where $\dot{γ}_0^α$ is a reference shear strain rate, F₀ is the thermal activation energy, k_B is the Boltzmann constant, θ is the absolute temperature, τ^α is the resolved shear stress, S^α is the overall slip resistance, and $τ_{cr}^α$ is the critical resolved shear stress (CRSS). The overall slip resistance S^α is caused by the total dislocation density including screw and edge dislocation as given by [22]:

$S^α=λμb\sqrt{∑\limits_βh^{αβ}ρ_T^β}$ (2)

$ρ_T^β=ρ_{Se}^β+ρ_{Ssw}^β$ (3)

where λ is the statistical coefficient, μ is the shear modulus at the temperature of interest, b is the magnitude of Burgers vector, and $h^{αβ}$ denotes the strength of dislocation interactions. $ρ_{Se}^β$ and $ρ_{Ssw}^β$ represent the edge and screw dislocation components, respectively. The corresponding evolutionary behaviors follow from the work [23]:

$\dot{ρ}_{Se}^β=\frac{C_{Se}}{b}[K_{Se}\sqrt{∑_βρ_T^β}-2d_{Seρ_{Se}^β}]|\dot{γ}α|$ (4)

$\dot{ρ}_{Ssw}^β=\frac{C_{Ssw}}{b}[K_{Ssw}\sqrt{∑_βρ_{ST}^β}-ρ_{Ssw}^β\{K_{Ssw}πd_{Ssw}^2\sqrt{∑_βρ_T^β}+2d_{Ssw}\}]|\dot{γ}^α|$ (5)

where C_Se and C_Ssw represent the relative contributions to plastic strain produced by edge and screw dislocations, K_Se and K_Ssw are related to the mean free path of the edge and screw dislocations, and d_Se and d_Ssw are the critical annihilation distances of the edge and screw dislocations with the opposite sign.

The crystal plasticity model given above was solved by a user subroutine (UMAT) developed with the general finite element software ABAQUS. A detailed description of theoretical model and parameter verification can refer to the previous work [18]. The parameters were listed in Table 1. It was worth mentioning that the utilized ratio of the values of CRSSs of basal: prismatic: pyramidal: 1ts pyramidal: 2nd pyramidal slip systems was 1:0.7:3:3:3 [6,24,25].

In order to analyze the texture evolution, the simplified polycrystal model [26] was utilized in this paper. As shown in Fig. 2(a), the model contained 512 grains, where one element represented a grain [27]. The mesh size was set to 10 μm according to the grain size. The initial grain orientations were extracted from the EBSD data of the undeformed sample, then, 512 randomly selected initial orientations were distributed into the polycrystal model randomly as shown in Fig. 2(b). The large quantity of orientations and the random distribution can properly characterize the initial texture of the undeformed sample. As for the imposed boundary conditions of the polycrystal model, take the tension along RD as an example as shown in Fig. 2(a), planes of X-Y, Y-Z and X-Z were set to U3 = 0, U1 = 0 and U2 = 0, respectively. The tensile deformation was implemented by setting a constant velocity on the front surface along the X-axis (RD). According to the simulated and experimental results as shown in Fig. 2(c-e), no necking was observed during the hot tensile deformation, which verified the effectiveness of the simulated results. The high formability of the tensile sample was ascribed to the elevated deformation temperature and relatively low strain rate. Therefore, no necking was observed within the true strain of 0.3 during the hot tension. Fig. 2(f) shows the fitting curves of the true stress-strain of TA15 samples during the hot tension along RD (RD-tension) and TD (TD-tension) at 750 °C with an initial strain rate of 0.005 s^-1, in which the solid lines and dot lines represent the experimental and predicted results, respectively. Their good match verified the reliability of the present model.

According to the previous work [18], no dynamic recrystallization was observed under this deformation condition, and dynamic recovery served as the dominant mechanism of microstructure evolution. The temperature rising often takes place during the deformation with high strain rate (>0.1 s^-1), and there was no necking under this deformation condition as shown in Fig. 2. Therefore, the stress variation was related to the dislocation density, grain orientation and strain rate according to the crystal plasticity theory in this paper. The high dislocation density results in the high slip resistance S^α, and the grain orientation affects the resolved shear stress τ^α according to the Schmid law. The slightly increased dislocation density as shown in Fig. 2(g) and the gradually decreased strain rate rendered the nearly constant flow stress during RD-tension. The decreasing stress during TD-tension may be ascribed to the evolution of the Schmid factor (SF) of prismatic slip systems, which were regarded as the dominant slip mode. The small average prismatic SF rendered the higher flow stress at the beginning of TD-tension. With the grains rotation caused by the tensile deformation, the gradually increased average prismatic SF rendered the decreased flow stress as shown in Fig. 2(g).

2.3. Simulated and experimental texture evolution during the hot uniaxial tension

Fig. 3 illustrates the simulated and experimental (0001) and (10$bar{1}$0) PFs during RD- and TD-tension, in which the effective strain was calculated based on the plastic strain-increments theory. The predicted PFs agreed well with the experimental results, which further verified the reliability of the present model. During RD-tension, the (0001) orientations gradually gathered into a narrow band along TD. The initial orientations C and D rotated into two new orientations: E(180°, 120°, 30°) and F(30°, 0°, 0°), respectively. However, the orientations of A and B were preserved. The basal planes of all grains became almost parallel to RD after an effective strain of 0.49. During TD-tension, the intensity of the original orientations A and B demonstrated an obvious attenuation. The orientation C rotated into a new orientation G(90°, 30°, 30°). The basal planes of grains with the orientations D and G were nearly parallel with TD.

3. Results and discussion

3.1. Relationship between slip mode and texture evolution during the hot uniaxial tension

3.1.1. Effect of slip mode on the texture evolution

The slip mode has a decisive effect on the texture evolution of near-α TA15 titanium alloys during the hot metal forming. Generally speaking, once the resolved shear stress on any slip system exceeds its CRSS, the dislocation slipping would take place [18]. The basal, prismatic and pyramidal-2 slip systems are the common slip modes of HCP metals, and they would rotate the grain around the [10$\bar{1}$0], 〈0001〉 and [10$\bar{1}$0] axis, respectively [13]. To better understand the mechanism of the texture evolution, the grains with four initial orientations A, B, C and D were tracked through CPFEM. Fig. 4 shows the simulated texture evolutions and shear strains induced by the dominant slipping of the grains with four initial orientations during RD- and TD-tension.

During RD-tension, in the cases of orientations A and B, the prismatic slipping was the dominant slip mode, and the projection points in PFs were stable. The move of orientations C and D in the (0001) PF was ascribed to the activated basal slipping. The projection points in (10$\bar{1}$0) PF of orientation C rotated clockwise around the normal direction (ND) of the sheet. Here it was interesting to notice that the projection points of orientation D in (10$\bar{1}$0) PF divided into two groups, and rotated clockwise and anticlockwise around ND, respectively.

During TD-tension, in the case of orientation A, as the basal and prismatic slipping could not take place at the beginning of the tensile deformation, the pyramidal-2 slip systems had to be activated. The result is consistent with the previous work on the slip mode, in which once the basal and prismatic slip systems cannot be activated due to the low SFs, the pyramidal-2 slipping operates [18]. The activation of pyramidal-2 slip systems inflicts the stress concentration [19], which would result in the decreased plasticity. With the tensile strain, the lattice rotation around [10$\bar{1}$0] axis caused by the pyramidal-2 slipping inflicted the increased basal SF and the corresponding slight basal slipping. The basal slipping would further rotate the grain around the [10$\bar{1}$0] axes like pyramidal-2 slipping. The projection points of orientation A in (0001) and (10$\bar{1}$0) PFs divided into two groups, and rotated clockwise and anticlockwise around ND, respectively. In the case of orientation B, the basal slipping played a dominant role, however, the prismatic slip systems were slightly activated with the tensile strain. The projection points of orientation B in (0001) PF moved obviously to the RD axis. The move of orientation B in (10$\bar{1}$0) PF should be attributed to the coactivation of basal and prismatic slip systems. In the case of orientation C during TD-tension, the deformation behavior showed a resemblance to the one of orientation C during RD-tension. In the case of orientation D during TD-tension, the deformation behavior showed a resemblance to ones of orientations A and B during RD-tension.

3.1.2. Mechanism of lattice rotation during the hot uniaxial tension

Based on the aforementioned analysis, the basal, prismatic and pyramidal-2 slipping were the dominant slip modes of the utilized TA15 sheet during the hot tension. The basal and pyramidal-2 slipping leaded to the rotation of both (0001) and {10$\bar{1}$0} planes. However, the activated prismatic slip systems only resulted in the rotation of {10$\bar{1}$0} planes. The rotation schematics of (0001) and {10$\bar{1}$0} planes are depicted in Fig. 5.

Angle α was defined as the angle between the tensile direction and the c-axis as shown in Fig. 5(a). Fig. 5(b) describes the relationship between the SFs of basal and pyramidal-2 slipping and angle α. Fig. 5(d) shows the rotation schematic of (0001) plane during the uniaxial tension, in which the red lines show the activated slip systems. When the tensile direction was parallel with the c-axis (α = 0°), the pyramidal-2 slip systems were activated according to the discussion of orientation A during TD-tension. With the rotation around [10$\bar{1}$0] axis caused by pyramidal-2 slipping, the basal SF increased gradually, and reached the maximum value when α was equal to 45°, later decreased to zero when α was equal to 90°. In terms of the utilized approximate ratio of 1:3 of the CRSSs of basal and pyramidal-2 slip systems [6,24,25], the basal slipping overwhelmed pyramidal-2 slipping when α was greater than 7° and less than 80°. With the sequential activations of the pyramidal-2 and basal slip systems, the (0001) plane rotated around [10$\bar{1}$0] axis to the stable orientation (α = 80°). The angle α in a stable orientation depended on the CRSS ratio of the basal slipping to the pyramidal-2 slipping [12]. In reality, when α was equal to or slightly less than 90°, prismatic slipping played the dominant slip mode, while basal and pyramidal-2 slip systems can be barely activated. At this point, the lattice was in a quasi-stable state for (0001) plane. In addition, the sequential pyramidal-2 and basal slipping could rotate the lattice to the stable orientation around [10$\bar{1}$0] axis clockwise or anticlockwise, this was why the projection points of orientation A divided into two groups during TD-tension as shown in Fig. 4(e).

Fig. 5(e) shows the rotation schematic of {10$\bar{1}$0} planes, in which the red lines show the activated slip systems. The main purpose was to discuss the reason why one of the [10$\bar{1}$0] axis was parallel with the tensile direction rather than [11$\bar{2}$0] axis. Angle β was defined as the angle between the tensile direction and one of [11$\bar{2}$0] directions as shown in Fig. 5(a). Fig. 5(c) describes the relationship between the prismatic SFs and angle β. When one of [11$\bar{2}$0] axis was parallel with the tensile direction, the lattice was in a metastable state. As illustrated, the slight clockwise perturbation around 〈0001〉 axis would render the increased SF of P2 and the decreased SF of P1 obviously. On the other hand, P2 and P1 slipping contributed to the clockwise and anticlockwise rotations of HCP lattice around 〈0001〉 axis, respectively. As a result, the slight clockwise rotation would break the balance between P1 and P2, and induce further clockwise rotation around 〈0001〉 axis until P1 was perpendicular to the tensile direction. Similarly, the slight anticlockwise rotation would induce further anticlockwise rotation until the lattice reached to the stable orientation, in which one of [10$\bar{1}$0] axis was parallel with the tensile direction. This was why the projection points of orientation D divided into two groups during RD-tension as shown in Fig. 4(b). Different with the metastable orientation, any rotation perturbations of the stable orientation around the 〈0001〉 axis would induce inverse rotation, which backed the lattice to the stable orientation.

3.2. Texture evolution during hot deformation under different strain states

The good matches of the simulated and experimental strain-stress curves and PFs during the hot uniaxial tension have verified the reliability of the present model. The present CPFEM was also used to predict the texture evolution of TA15 samples with initial random orientations during hot deformation under different strain states at 750 °C with an initial strain rate of 0.005 s^-1.

3.2.1. Texture evolution during the hot uniaxial compression

The imposed stress on X-Y plane along Z-axis during the hot uniaxial compression is shown in Fig. 6(a). Fig. 6(b) and (c) show the simulated (0001) and (10$\bar{1}$0) PFs of TA15 samples with the effective strains of 0.25 and 0.5. With the compressive strain, the (0001) orientations gradually gathered into the center of the PF of X-Y plane. The (10$\bar{1}$0) orientations nearly formed a circle band, and did not manifest an obvious preferential distribution.

Following the similar analysis like the hot uniaxial tension, the rotation schematic of (0001) plane during the hot uniaxial compression is shown in Fig. 6(d), in which the red lines show the activated slip systems. The relationship between the SFs of basal and pyramidal-2 slip systems and angle α can also refer to the above Fig. 5(b). When the compressive direction was perpendicular to the c-axis (α = 90°), the pyramidal-2 slip systems were activated. With the rotation around [10$\bar{1}$0] axis caused by the pyramidal-2 slipping, the basal SF increased gradually, and reached the maximum value when α was equal to 45°, later decreased to zero when α was equal to 0°. In terms of the utilized approximate ratio of 1:3 of the CRSSs of basal and pyramidal-2 slip systems [6,24,25], the basal slipping overwhelmed pyramidal-2 slipping when α was greater than 7° and less than 80°. With the sequential activations of pyramidal-2 and basal slip systems, the (0001) plane rotated around [10$\bar{1}$0] axis to the stable orientation (α = 7°). The angle α in a stable orientation depended on the ratio of CRSSs of the basal slipping and the pyramidal-2 slipping [12]. As stated in Ref [13]. about the texture evolution of AZ31 Mg alloy sheet during the uniaxial tension along RD, the c-axis would rotate to the center of (0001) PF through the basal slipping. One can just ignore the effect of pyramidal-2 slipping. To sum up, the lattice with the c-axis nearly parallel with the compressive direction is in a stable orientation during the hot uniaxial compression.

3.2.2. Consistency of different orientation distributions

The different projection planes of the PF render the different types of the orientation distribution. The simulated (0001) and (10$\bar{1}$0) PFs of TA15 samples with initial random orientations with an effective strain of 0.5 during the hot uniaxial and equi-biaxial compression are shown in Fig. 7. During the hot equi-biaxial compression as shown in Fig. 7(a), the (0001) orientations formed a circumferential band, and the intensity of the (10$\bar{1}$0) orientations was high at the center and in a ring-shaped region around the center in the simulated PF of Y-Z plane as shown in Fig. 7(b). The similar texture development of AZ31 magnesium alloy during the extrusion process under an equi-biaxial compression mode was also experimentally observed [12]. However, with the transformation of projection plane from Y-Z plane to X-Y plane, the (0001) orientations gathered into a narrow band perpendicular to X-axis. The (10$\bar{1}$0) projections gathered to two poles of X-axis and two narrow bands around the two poles. During the hot uniaxial compression as shown in Fig. 7(d), the (0001) orientations rotated into the center of the PF of X-Y plane. However, the (0001) orientations gathered to two poles of Z-axis in the PF of X-Z plane.

As a result, several typical types of orientation distribution in the PFs are essentially consistent. In order to analyze the relationship between the texture evolution and the strain state, all the simulated orientations were projected to the same plane: X-Y plane. X, Y and Z axes represent the directions of the first, second and third principal strain/stress, respectively.

3.2.3. Texture evolution under different strain states

During the metal forming process, the strain states can be divided into three types: tension type, plane state type and compression type. The strain states were divided into five types in this paper: equi-biaxial tension; nonequi-biaxial tension; plane strain; nonequi-biaxial compression; equi-biaxial compression. The simulated (0001) and (10$\bar{1}$0) PFs of TA15 samples with initial random orientations during hot deformation under different strain states with an effective strain of 0.5 are shown in Fig. 8.

The different stress states can result in the same strain states. For example, the stress states of the equi-biaxial tension σ,σ,0 and the uniaxial compression (0,0,-σ) rendered the same equi-biaxial tension strain state (ε,ε,-2ε) based on the plastic strain-increments theory as shown in Fig. 8(a). As a result, it can be concluded that the strain state determined the corresponding texture evolution during the metal forming.

During the forming processes under different strain states, the (0001) orientations formed a band perpendicular to the direction of the first principal strain (X-axis). Under the equi-biaxial tension strain state (ε,ε,-2ε), the lattice with the c-axis nearly parallel with the direction of the third principal strain (Z-axis) is in a stable orientation according to the analysis of the texture evolution during the uniaxial compression. The prismatic planes did not show the preferential distribution in the simulated PFs. Under the equi-biaxial compression strain state (2ε,-ε,-ε), the lattice with one of prismatic planes approximately perpendicular to the direction of the first principal strain (X-axis) is in a stable orientation according to the analysis of the texture evolution during the uniaxial tension. Under the plane strain state (ε,0,-ε), the distribution of orientations was between ones under equi-biaxial tension (ε,ε,-2ε) and equi-biaxial compression strain state (2ε,-ε,-ε).

With the evolution of strain states (ε,ε,-2ε), (5ε,2ε,-7ε), (ε,0,-ε), (7ε,-2ε,-5ε) and (2ε,-ε,-ε), the band of (0001) orientations spread along the direction of the secondary principal strain (Y-axis), and one of [10$\bar{1}$0] axis rotated to the direction of the first principal strain (X-axis) gradually.

3.2.4. Relationship between texture evolution and strain state

The strain state determined the corresponding texture evolution during the metal forming. The first principal strain (ε₁ >0), as the maximal tensile strain, rendered that the (0001) orientations formed a band perpendicular to the direction of the first principal strain (X-axis) as shown in and Fig. 8. The width (d) of the band along the direction of the second principal strain (Y-axis) depended on the tilt angle θ as shown in Fig. 9(a) and Eq. 6.

d=r×tan$\frac{θ}{2}$ (6)

Under the equi-biaxial tension strain state (ε,ε,-2ε), the second principal strain had the same tensile effect on the rotation of the basal plane as the first principal strain, the (0001) orientations gathered into the center of the PF as shown in Fig. 8(a). Under the equi-biaxial compression strain state (2ε,-ε,-ε), the second principal strain had the same compressive effect on the rotation of the basal plane as the third principal strain, the (0001) orientations formed a complete band perpendicular to the X-axis. In order to characterize the compressive and tensile effects of the second principal strain quantitatively, the arbitrary strain state can be decomposed into two parts on the basis of the strain states of equi-biaxial tension or equi-biaxial compression as expressed in Eq. (7) and Eq. (8).

(ε₁,ε₂,ε₃)=(-1/2ε₃,-1/2ε₃,ε₃)+(ε₁+1/2ε₃,ε₂+1/2ε₃,0) (7)

(ε₁,ε₂,ε₃)=(ε₁,-1/2ε₁,-1/2ε₁)+(0,ε₂+1/2ε₁,ε₃+1/2ε₁) (8)

ε₂+1/2ε₃ (<0) and ε₂+1/2ε₁ (>0) represented the compressive and tensile effects of the second principal strain, respectively. The value of the tilt angle θ can be approximately expressed as the ratio of the compressive effect to the tensile effect of the second principal strain as expressed in Eq. (9). Based on the law of volume constancy (ε₁+ε₂+ε₃=0) and the expressions of three principal shear strains (γ1=ε₂-ε₃, γ2=ε₃-ε₁, γ3=ε₁-ε₂), the expression of the tilt angle θ can be given by different forms. The calculated values of the tilt angle θ agreed well with the experimental ones as shown in Fig. 9(c).

$θ=arctan[\frac{-(ε_2+1/2ε_3)}{ε_2+1/2ε_1}]=arctan[\frac{ε_1-ε_2}{ε_2-ε_3}]=arctan[\frac{γ3}{γ1}]$ (9)

Under the equi-biaxial compression strain state (2ε,-ε,-ε) and the equi-biaxial tension strain state (ε,ε,-2ε), the second principal strain had the minimum and maximum tensile effect on the rotation of the prismatic planes, respectively. The volume fraction of the orientation φ₂ as shown in Fig. 9(b) depended on the value of η representing the ratio of the tensile effect to the compressive effect of the second principal strain as expressed in Eq. (10). When the value of η was equal to 0, all the grains manifested as the the orientation φ₁ as shown in Fig. 8(e). When the value of η was equal to +∞, the volume fraction of the orientation φ₂ was equal to the one of orientation φ₁, which rendered a nearly uniform distribution of (10$\bar{1}$0) orientations as shown in Fig. 8(a).

η=$\frac{ε_2+1/2ε_1}{-(ε_2+1/2ε_3)}=\frac{ε_2-ε_3}{ε_1-ε_2}=\frac{γ1}{γ3}$ (10)

3.3. Effect of texture on formability

The relationship between strain state and texture evolution can help control the crystallographic texture and mechanical properties of the titanium alloys component during the hot metal forming. Under a certain strain state, the texture of the component can be predicted based on this relationship. In order to gain a better forming property, the undeformed billet should contain a certain target orientation according to the following deformation state [11]. During the spinning forming [8,28], hot gas forming [1], forging forming [29], the strain state of the billet is about biaxial tension mode, and the preferred texture should be the radial texture as shown in Fig. 8(a) and (b). During the extrusion process, the strain state of the central axis of the billet is approximated by an equi-biaxial compression mode [12], and the preferred texture should be the band texture as shown in Fig. 8(e).

According to this relationship, one can take full advantage of the initial texture of the undeformed billet according to the following deformation state. For example, the (0001) orientations of the used TA15 sheet nearly formed a band perpendicular to RD, just like the texture as shown in Fig. 8(e). Accordingly, in order to obtain a better formability taking advantage of the initial texture, the direction of the first principal strain should be consistent with RD during the forming process.

However, in order to diminish the texture and obtain the nearly isotropic deformation, the complicated non-proportional and non-steady strain state [30] or special heat treatment, such as the high-density electropulsing [31] should be adopted.

4. Conclusions

In this paper, the texture evolution of a near-α TA15 titanium alloy during hot deformation under different strain states was discussed based on the crystal plasticity method. The following conclusions were reached:

(1) During the hot deformation of the TA15 sheet, the basal and prismatic slip systems are the dominant slip modes due to the similar low critical resolved shear stress rotating the lattice around the [10$\bar{1}$0] and 〈0001〉 axis, respectively. Once both of them cannot be activated, the pyramidal-2 slipping occurs rotating the lattice around the [10$\bar{1}$0] axis.

(2) During the hot uniaxial loading process, the angle between the loading direction and the c-axis of the stable orientation is equal to 80° during the uniaxial tension, and 7° during the uniaxial compression, which depends on the ratio of the critical resolved shear stresses of basal and pyramidal-2 slip systems.

(3) The relationship between the texture evolution and strain state is established. All the (0001) orientations formed a band perpendicular to the direction of the first principal strain regardless of the strain state. The tilt angle of the band along the direction of the second principal strain can be expressed as: θ=arctan $[\frac{ε_1-ε_2}{ε_2-ε_3}]$. This relationship can help control the crystallographic texture and mechanical properties of the titanium alloys component during the hot metal forming.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China (No.51401065).

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