Numerical Investigation on the Strain Evolution of Ti-6Al-4V Alloy during Multi-directional Forging at Elevated Temperatures

Processing parameters

It is widely accepted that the optimized processing parameters are determined by processing map in hot forging process [24, 25]. With the help of processing map, it is possible to obtain the optimal parameters for multi-directional forging. Isothermal compression tests were carried out in the strain rate range of 0.0005 to 1 s^‒1 and the temperature range of 1073 to 1323 K by a Gleeble-3500 simulator to plot the processing map, as shown in reference [26]. It can be determined that the optimized zone for hot forging is in the temperature range of 1160 to 1190 K, strain rate range less than 10^‒3 s^‒1 and strain less than 0.5 [26]. The power dissipation efficiency is maximum in this zone. In addition, plasticity instability domain is not in this zone. Consequently, deformation temperature of 1173 K, strain rate of 10^‒3 s^‒1 and strain per pass of 0.3, 0.4, and 0.5 was chosen as optimized multi-forging processing parameters for simulation.

Constitutive modeling

Constitutive models are mathematical equations that describe the relationship between flow stresses and deformation parameters such as strains, strain rates and temperatures. Ti-6Al-4V alloy, as the most common titanium alloy, is used as billet material for FEM simulation.

Three typical models as Arrhenius-type model [27, 28], Johnson Cook model [29, 30] and Zerilli-Armstrong model [28, 29]. Arrhenius-type model is one of the most widely applied constitutive model [27]. Johnson Cook model is a simple form model with only five material constants [31]. Additionally, Zerilli-Armstrong model is also a widely applied model which has been utilized to analyze various kinds of material over a wide range of temperature [28].

Generally, an ideal constitutive model should meet a number of factors. Firstly, it is able to represent the flow behavior of the material over a definite processing range. Secondly, the accuracy and reliability of the constitutive model is adequate. Thirdly, the constitutive model requires fewer materials constants thus the time required for evaluating these material constants is shorter.

Arrhenius-type constitutive model for Ti-6Al-4V alloy can be described as follows [32, 33]:

where A and α are material constants, σ is the flow stress, n is the stress exponent, Q is the activation energy, R is the gas constant, and T is the deformation temperature.

Johnson Cook model is expressed as follows [31]:

where σ is flow stress (Mpa), ε is the true strain, ε˙∗=ε˙/ε˙0 is the dimensionless strain rate with ε˙ being the strain rate (s^‒1) and ε˙0 the reference strain rate (s^‒1), A is the yield stress at reference temperature and strain rate, B is the coefficient of strain hardening, n is the exponent of strain hardening and T∗ is homologous temperature and expressed as eq. (3):

where T is current temperature (K), T_m is the melting temperature (1903 K for Ti-6Al-4V alloy) and T_ref is the reference temperature. C and m are the material constants that represent the coefficient of strain rate hardening and thermal softening exponent, respectively.

Zerilli-Armstrong constitutive model is expressed as follows [29]:

where ε˙∗=ε˙/ε˙0 is the dimensionless strain rate, ε˙0 the reference strain rate (s^‒1), T∗=T−Tref and Tref are the current and reference temperature (K), respectively. C₁, C₂, C₃, C₄, C₅, C₆ and n are the materials constants.

The results from reference [34] show that Johnson Cook model is inadequate to predict high-temperature flow behavior over the entire range while Zerilli-Armstrong and Arrhenius-type could predict flow behavior well (In reference [34], The deform temperature ranges from 1073 to 1223 K, strain rate ranges from 10^‒3 to 1 s^‒1, strain ranges from 0 to 0.5, which is suitable to the processing parameters in part 3.1. After quantifying the predictability of the constitutive models in terms of Correlation Coefficient and Average Absolute Relative Error, it is indicated that Zerilli-Armstrong constitutive model could represent the elevated temperature flow behavior more accurately (Correlation Coefficient=0.994, Average Absolute Relative Error=8.85 % [34]. Johnson Cook and Arrhenius-type constitutive models involve seven and sixteen material constants, respectively, the time required for evaluating these material constants involved in Johnson Cook model is much shorter than that of Arrhenius-type model [34].

Consequently, Zerilli-Armstrong model is ideal constitutive model in the temperature range of 1073 to 1223 K, strain range of 10^‒3 to 1 s^‒1 and strain ranges of 0 to 0.5. The optimized multi-forging processing parameters (deformation temperature of 1170 K, strain rate of 10^‒3 s^‒1 and strain per pass less than 0.5) belong to the effective domain of Zerilli-Armstrong model (temperature range of 1073 to 1223 K, strain range of 10^‒3 to 1 s^‒1 and strain ranges of 0 to 0.5). Therefore, Zerilli-Armstrong model and relevant constants are reliable, accurate and suitable for simulation of multi-directional forging of Ti-6Al-V alloy at deformation temperature of 1170 K, strain rate of 10^‒3 s^‒1 and strain per pass less than 0.5. The material constants of Zerilli-Armstrong model derived from reference [34] is listed in Table 1.

Friction modeling

The classical constant shear friction model is used for MDF deformation. The friction stress in this model is defined as follows [35]:

where fs denotes the friction stress, K is the shear stress, and m is the friction coefficient. The equation states that the friction stress is directly proportional to the shear stress of the billet. Friction coefficient is set as 0.3, 0.1 and 0.5 to discuss the influence of friction coefficient on simulation results.

Thermal parameters

The thermal parameters can be derived from reference [36], in which the thermal parameters are based on the temperature of 1173 K. Thermal conductivity, heat transfer coefficient, specific heat, and emissivity are defined as 16 [W/(m^.K)], 2470 [W/(m^2.K)], 1.15 [J/(g^.K)] and 0.6.

Geometrical modeling

The dimensions of the cylindrical billets are 60 mm (diameter) and 90 mm (height). The dies for MDF include the up die and the bottom die. The dimensions of the dies are 150 mm (length), 150 mm (width), and 30 mm (height).

FEM procedure

The first step was heating the billets to the temperature of 1173 K. In the second step, the six-pass MDF was conducted between the up die and the bottom die at an initial billet temperature of 1173 K. The die temperature was equal to the initial billet temperature. In this study, the compression direction during the MDF process was set as the z-axis direction (Figure 1). The compression strain rate was set as 10^‒3 s^‒1. The forging direction was turned by 90° from pass to pass. Figure 1 details the six-pass MDF process for FEM simulation, including the compression direction, rotation axis, and rotation direction. The surfaces in contact with the dies for each pass were surface A, surface C, surface B, surface A, surface C, and surface B. In view of the forging directions of the MDF, the first three passes composed the first cycle, and the last three passes composed the second cycle. The shape of the billet altered from cylinder to cuboid after the six-pass MDF.

Figure 1:

Schematic illustration of the six-pass MDF.

Deformation zoning in MDF

To study the MDF process systematically, the deformation zoning method was utilized to analyze the MDF process. The results were obtained in the case where the compression strain per pass was 0.5 and friction coefficient of 0.3. Figure 2 shows the seven geometrically-symmetrical zones of the billet and the dependence of effective strain in the geometrically-symmetrical zones on time. It is easily seen that the deformation distribution of MDF is inhomogeneous. The coordinates of the central point of geometrically-symmetrical zones were defined based on the three-dimensional Cartesian coordinate system, where the origin was at the volume center of the undeformed billet (Figure 2). In terms of the geometric symmetry of the billet, the coordinates of the central points of the geometrically-symmetrical zones before deformation were defined as follows: Z1(0, 0, 0), Z2(0, 0, ‒45), Z3(0, ‒30, 0), Z4(30, 0, 0), Z5(0, 30, 0), Z6(‒30, 0, 0), and Z7(0, 0, 45). It seems that some geometrically-symmetrical zones such as Zones Z3 and Z4 were strain-equivalent (Figure 2), which necessitated analysis by FEM whether the geometrically-symmetrical zones were strain-equivalent. The dependence of the effective strain in the seven geometrically-symmetrical zones on time is depicted in Figure 2. The definite curves (curves of Zones Z2 and Z7, Zones Z3 and Z5, and Zones Z4 and Z6) almost coincide with each other, indicating that the effective strain evolution characteristics in these geometrically-symmetrical zones are identical. In other words, some geometrically-symmetrical zones are equivalent in terms of the equivalence of effective strain evolution. By eliminating the strain-equivalent zones, four individual strain zones remained, namely, Zone Z1, Zone Z2, Zone Z3, and Zone Z4, whose coordinates before deformation were Z1(0, 0, 0), Z2(0, 0, ‒45), Z3(0, ‒30, 0), and Z4(30, 0, 0).

Figure 2:

Dependence of effective strain in geometrically-symmetrical zones on time.

Figure 3 shows the dependence of the effective strain in the four individual strain zones on time and the space variation of the effective strain in the billet. The curves can be divided into six parts in terms of the non-differentiable points of the curves, which correspond to six passes (two cycles). The effective strain in the four individual strain zones increases with the increase in the plastic deformation amount. However, the strain increment in the four individual strain zones varies from pass to pass.

Figure 3:

Dependence of effective strain in the four individual strain zones on time.

Figure 4 shows the dependence of effective strain increment per pass in the four individual strain zones on the pass number. During the first MDF pass, Zone Z2 contacted with MDF dies with the strain increment of 0.16, whereas Zones Z3 and Z4 were on the side edge of the billet with the strain increment of 0.44. In addition, Zone Z1 was in the core of the billet with the strain increment of 0.70. The maximum strain of the four individual strain zones (in Zone Z1) was 0.70, whereas the minimum strain of the individual strain zones (in Zone Z2) was 0.16 at the end of the first pass. During the second MDF pass, Zones Z2 and Z4 were on the side edge of the billet with the respective strain increment of 0.54 and 0.24, whereas Zone Z3 contacted with MDF dies with the strain increment of 0.29. Additionally, Zone Z1 was in the core of the billet with the strain increment of 0.62. The maximum strain of the individual strain zones (in Zone Z1) was 1.32, whereas the minimum strain of the individual strain zones (in Zone Z4) was 0.68 at the end of the second pass. During the third MDF pass, Zones Z2 and Z3 were on the side edge of the billet with the respective strain increment of 0.31 and 0.43, whereas Zone Z4 contacted with MDF dies with the strain increment of 0.10. Zone Z1 was also in the core of the billet with the strain increment of 0.57. The maximum strain of individual strain zones (in Zone Z1) was 1.88, whereas the minimum strain of individual strain zones (in Zone Z4) was 0.77 at the end of the third pass. During the second cycle, the effective strain evolution process of the billet was similar to that during the first cycle. However, the strain increment in the four individual strain zones was different from that during the first cycle (Figures 3 and 4).

Figure 4:

Dependence of effective strain increment per pass in the four individual strain zones on pass number.

The effective strain increment of the individual strain zones in the core is high (ranging from 0.57 to 0.69, shown in red dotted line in Figure 4) due to the three-direction compressive stress of the zones and the absence of influence from friction force on the zones. By contrast, the increment of the individual strain zones contacting with the MDF dies is low (ranging from 0.10 to 0.29, shown in blue dotted line in Figure 4) because of the influence of the friction force. Additionally, the effective strain increment in the individual strain zones on the side edge of the billet is on a medium level (ranging from 0.31 to 0.54, shown in green dotted line in Figure 4).

Effective strain distribution from the core to the exterior

For further analysis of the effective strain distribution, the billet was split along two orthogonally symmetrical planes and the core was revealed (Figure 5). The results were obtained in the case where the compression strain per pass was 0.5 and friction coefficient was 0.1, 0.3 and 0.5, respectively. The split billet shows that the distribution of the effective strain is inhomogeneous. Figure 5 (a) shows the effective strain distribution in the billet subjected to the third pass (end of the first cycle) MDF with the friction coefficient of 0.1. Moreover, the maximum strain (in the core) reaches 1.72, whereas the minimum strain (in the exterior) reaches 0.69. Figure 5 (b) shows the effective strain distribution in the billet subjected to the sixth pass (end of the second cycle) MDF with the friction coefficient of 0.1. The maximum strain (in the core) reaches 3.30, whereas the minimum strain (in the exterior) reaches 1.70. The deviation between the maximum and minimum strains after the third and sixth passes are 1.60 and 1.03, respectively. Similarly, for billet subjected to the third pass MDF with the friction coefficient of 0.3, the maximum strain reaches 1.92 and the minimum strain reaches 0.68 (Figure 5 (c)); for billet subjected to the sixth pass MDF with the friction coefficient of 0.3, the maximum strain reaches 3.72 and the minimum strain reaches 1.69 (Figure 5 (d)). The deviation between the maximum and minimum strains after the third and sixth passes are 1.24 and 2.03, respectively. Additionally, for billet subjected to the third pass MDF with the friction coefficient of 0.5, the maximum strain reaches 2.06 and the minimum strain reaches 0.63 (Figure 5 (e)); for billet subjected to the sixth pass MDF with the friction coefficient of 0.5, the maximum strain reaches 3.93 and the minimum strain reaches 1.55 (Figure 5 (f)). It can be calculated that the differences between the maximum and minimum strains after the third and sixth passes are 1.43 and 2.38, respectively. The results show that the deviation between the maximum and the minimum strain increases with the increase in passes, irrespective of friction coefficient. The deviation between the maximum and the minimum strain increases with the increase of the friction coefficient, irrespective of pass number.

Figure 5:

Effective strain distribution of the billet: (a)-(f) billet subjected to MDF; (g)-(i) functional relationship of the effective strain vs. distance from the core to the exterior (m means friction coefficient).

Figure 5(g)–(i) depicts the relationship of the effective strain vs. distance from the core to the exterior in the x-axis, y-axis and z-axis directions. The effective strain in all three directions decreases linearly from the core to the exterior. To quantitatively describe the linear distribution of the effective strain, the functional relationship between effective strain and distance from the core to the exterior was established by linear fitting. The linear relationship of the billet subjected to the third pass MDF can be derived as follows:

Similarly, the linear relationship subjected to the sixth pass MDF can be derived as:

The gradient of the effective strain distribution is simply the slope of the effective strain vs. the distance curve. With the increase in passes, the slope of effective strain vs. distance curves significantly increases, which denotes that the gradient of the effective strain increases with the passes in all three directions. With the increase in friction coefficient, the slope of effective strain vs. distance curves significantly increases, which denotes that the gradient of the effective strain increases with the increase in friction coefficient in all three directions. The results are in consistence with the variation of deviation between the maximum and the minimum strain discussed above. The results may be due to the difference of deformability in different individual strain zones. With the increase in passes and the friction coefficient, the difference of deformability in different individual strain zones is increasingly shown, which is reflected by the increase in the gradient of the effective strain.

Strain accumulation effect

In the MDF process, the accumulative compression strain (ACS) is defined as the sum of compression strains from the initial to a certain MDF pass:

where εC1, εC2, …,εCn is the compression strain of the 1^st, 2^nd, …, n^th MDF pass (n≤6 in this paper), and εAC is the accumulative effective strain. For example, for the MDF process with the compression strain per pass of 0.3, the ACS after 1, 2, 3, 4, 5 and 6 passes are 0.3, 0.6, 0.9, 1.2, 1.5 and 1.8, respectively. For the analysis of the influence of ACS on the average effective strain (AES) and maximum effective strain (MES) in the billet, the compression strain per pass is set with different values such as 0.3, 0.4, 0.5, and the friction coefficient is set as 0.1, 0.3 and 0.5. Figure 6 depicts the functional relationship of AES vs. ACS. The AES in the billet linearly increases with the increase in the ACS. Figure 7 shows the functional relationship of MES vs. ACS, which is similar with that of AES vs. ACS. The MES in the billet linearly increases with the increase in the ACS.

Figure 6:

Functional relationship of AES vs. ACS: (a) friction coefficient of 0.1, (b) friction coefficient of 0.3, (c) friction coefficient of 0.5.

Figure 7:

Functional relationship of MES vs. ACS: (a) friction coefficient of 0.1, (b) friction coefficient of 0.3, (c) friction coefficient of 0.5.

Through linear fitting, the functional relationship of AES ε_AE vs. ACS ε_AC is (m means friction coefficient):

The functional relationship of MES ε_ME vs. ACS ε_AC is:

Based on eqs. (16) and (17), the functional relationship of MES ε_ME vs. AES ε_AE is:

The slope of AES-ACS function decreases with the increase of friction coefficient, while the slope of MES-ACS function increases with the increase of friction coefficient. For the definite friction coefficient, the strain accumulative effect of MDF can be described by the fact that the AES and MES are in proportion to the ACS. Hence, for the billet with lower-plasticity, although the compression strain per pass is limited, it can be refined by increasing the passes of MDF. For the billet with higher-plasticity, however, the few-pass MDF process is optimized owing to its advantage in productivity.