Characterization of Hot Deformation Behavior of a New Near-β Titanium Alloy: Ti555211

Flow behavior

The typical stress–strain curves obtained at 750°C, 800°C, 850°C, 900°C, and 950°C and different strain rates are shown in Figure 4. These curves exhibit the following characteristics:

1. Three distinct stages of stress evolution with strain. Work hardening (WH) plays an important role in the deformation in the first stage, and the flow stress increases rapidly to a critical value [5]. The curve has a very large slope, and the material deforms mainly by WH and dynamic recovery (DRV).

2. In the second stage, the increase in the flow stress slows until a peak value or an inflection of work-hardening rate is reached, which indicates that the thermal softening due to DRX and DRV exceeds WH [5].

3. The curves are gradually stabilized in the third stage and exhibit flow softening phenomena and steadily decreasing flow characteristics.

4. High strain rates result in discontinuous yielding at some of the experimental temperatures. The severity of this yielding increases with increasing deformation temperature.

Figure 4:

True stress–true strain curves of Ti555211 alloy: (a) 750°C; and (b) 800°C; (c) 850°C; (d) 900°C; (e) 950°C.

The phenomenon of discontinuous yielding

In the initial stages of deformation, the material deforms primarily by the cross slip of dislocations, which is the main softening mechanism operating during the high-temperature plastic deformation of metals. The hardening that results from increasing dislocation density is substantially larger than the dynamic softening stemming from cross slip and therefore, the flow stress increases significantly with increasing strain. When the flow stress reaches the peak stress, the vacancy concentration of the material increases with increasing strain, and subsequent softening results from dislocation climb in the mid and late stages of deformation. Moreover, the steady-flow stage begins when a dynamic equilibrium between hardening and softening mechanisms is reached [23–27].

The steady-state deformation mechanism encompasses the proliferation of dislocation and the interaction of different numbers of dislocation. The dynamic theories can be used to explain the phenomenon of discontinuous yielding.

A high density of mobile dislocations is generated in the grain boundaries during deformation. However, these dislocations are hindered by the boundaries, thereby resulting in a sharp increase in the flow stress. Subsequent dislocation climb into the interior of the grain boundaries results in significant dislocation annihilation, which in turn leads to discontinuous yielding when some critical dislocation density is reached [24–27]. Therefore, pronounced discontinuous yielding occurs at high strain rates. Discontinuous yielding of Ti555211 titanium alloy during high-temperature deformation is well-explained by the dislocation dynamics theory.

Weiss and Semiatin [28] reported that discontinuous yielding during the hot deformation of Ti-15v-3Al-3Cr-3Sn alloy resulted from the multiplication of mobile dislocations. Philinpart observed similar trends via TEM analysis of the microstructural evolution of Ti-6.5Mo-4.5Fe-1.5Al alloy. As Figure 5 shows, the grain boundaries are hindered by the high density of dislocations generated therein. Moreover, the amount of DRV in the β phase increases significantly when the dislocation density reaches some critical value, thereby resulting in a significant decrease in the flow stress; this decrease confirms the occurrence of discontinuous yielding.

Figure 5:

Micrograph showing grain boundary dislocation generation in Ti555211 alloy deformed at 950°C and 1 s⁻¹ to a true strain of 0.02.

Constitutive relationship model considering the effect of strain

The classical relationship can be generally expressed as an Arrhenius equation. Furthermore, the effect of the strain rate and temperature on the deformation behavior can be expressed as a Zener–Hollomon parameter exponent-type equation. The Arrhenius hyperbolic equation provides a good fit for the flow stress and Zener–Hollomon parameter.

where A, α, and n are the material constants, α=β/n; σ, Q, R, T, Z, and ε˙ are the true stress, activation energy of hot deformation (kJ mol⁻¹), universal gas constant (8.31 J mol⁻¹K⁻¹), absolute temperature (K), Zener–Hollomon parameter, and strain rate (s⁻¹), respectively.

Determination of material constants for Arrhenius-type constitutive relationship

The influence of the strain on the stress is typically not considered in eqs (1) and (2). Therefore, an Arrhenius-type constitutive relationship that includes strain compensation is established in order to determine the effect of strain on the flow stress. The corresponding material constants can be evaluated from the stress–strain curves resulting from the compression tests. The material constants are determined at a deformation strain of 0.1 as follows.

At low and high stress levels, the power law and exponential law are substituted for F(σ) in eq. (2) and the strain rates are then obtained from

(4)ε˙=Bσn1

(5)ε˙=Cexp(βσ)

where B and C are the material constants, which are independent of the deformation temperature.

Taking the logarithm of both sides of eqs (4) and (5) gives

(6)ln(σ)=1n1ln(ε˙)−1n1ln(B)

(7)σ=1βln(ε˙)−1βln(C)

However, the final microstructure after deformation at temperatures of 875°C or higher consisted of only β grains. Therefore, the relevant material constants should be calculated separately for the α+β phase and the single β phase.

The values of β and n₁ can be determined from the slope of the lines in the plot of σ−lnε˙ and lnσ−lnε˙. The slope of the lines in Figures 6(a), 6(b), 7(a), and 7(b) changed only slightly. At a deformation strain of 0.1, β and n₁ were computed as 0.035509 (α+β phase) and 3.12467 (α+β phase)/0.0670552 (β phase) and 2.85447 (β phase) in Figures 6 and 7.

Figure 6:

Relationship between (a) ln σ and ln ε˙; (b) σ and ln ε˙; (c) ln[sinh(ασ)] and ln ε˙; (d) ln[sinh(ασ)] and T⁻¹/10⁻³K⁻¹ for α+β phase.

Figure 7:

Relationship between (a) ln σ and ln ε˙; (b) σ and ln ε˙; (c) ln[sinh(ασ)] and ln ε˙; (d) ln[sinh(ασ)] and T⁻¹/10⁻³K⁻¹ for β phase.

Therefore, α=β/n₁=0.00797 MPa⁻¹ (α+β phase)/0.0171605 MPa⁻¹ (β phase).

Equation (2) can then be rewritten as

(8)ε˙=A[sinh(ασ)]nexp−QTR

Taking the logarithm of both sides of eq. (8) gives

(9)ln[sinh(ασ)]=lnε˙n+QnTR−lnAn

(10)Q=Rndlnsinhασd1/T

(11)Z=Asinhασn

Taking the logarithm of both sides of eq. (11) gives:

(12)lnZ=lnA+nlnsinhασ

The flow stress can then be expressed as a function of the Z parameter, i.e.,

(13)σ=1αlnZA1/n+ZA2/n+11/2

The material constants ln A, n, and Q at a deformation strain of 0.1 can be determined by eqs (9–12). In addition, the regression analysis of eq. (12) and Figure 8 yields ln A=48.84(α+β phase), ln A=18.23 (β phase), Q_ave=476.557 kJ mol⁻¹ (α+β phase), Q_ave=222.173 kJ mol⁻¹ (β phase), n=2.99194 (α+β phase), and n=2.82151 (β phase).

Figure 8:

Relationship between lnZ and ln[sinh(ασ)] for (a) α+β phase; (b) β phase.

The material constants (α, n, Q, and A) are calculated from eqs (9–13), and the flow stress is determined from eq. (13). However, the effect of strain is neglected in this equation. In true stress–true strain curves, the flow stress exhibits a significant dependence on the strain, which is manifested as strain hardening and dynamic softening [29, 30]. Therefore, in order to accurately predict the flow stress, the effect of strain should be considered. The material constants (a, n, Q, and A) in eq. (14) are typically considered strain dependent. Therefore, in this work, the material constants (i.e., α, A, n, Q) are calculated in 0.05 intervals for strains ranging from 0.1 to 0.7. As shown in the following equation, these constants can be fitted by a polynomial, in which the strain is a variable, i.e.,

Q=D6ε6+D5ε5+D4ε4+D3ε3+D2ε2+D1ε+D0

(14)α=F6ε6+F5ε5+F4ε4+F3ε3+F2ε2+F1ε+F0

A=eG6ε6+G5ε5+G4ε4+G3ε3+G2ε2+G1ε+G0

n=H6ε6+H5ε5+H4ε4+H3ε3+H2ε2+H1ε+H0

where D₀–D₆, F₀–F₆, G₀–G₆, and H₀–H₆ are the coefficients of the polynomial.

A sixth-order polynomial accurately represents the influence of strain on the material constants (Figures 9 and 10). Some of the material constants (n_{(α+β phase)}; Q_{(β phase)}; n_{(β phase);} ln A_{(β phase)}) vary in a complex manner. Higher-order (i.e., >6) polynomials resulted in an over-fit, and hence the influence of strain was not accurately represented. However, lower-order (i.e., <6) polynomials exhibited insufficient fitting capacity and reflected the nature of the variables only somewhat. The results of the polynomial fit of α, n, Q, and ln A of Ti555211 alloy are listed in Table 2 (α+β phase) and Table 3 (β phase).

Figure 9:

Polynomial fit of relationships: (a) Q; (b) n; (c) α; (d) ln A; with strain for α+β phase.

Figure 10:

Polynomial fit of relationships: (a) Q; (b) n; (c) α; (d) ln A; with strain for β phase.

Table 2:

Results of the polynomial fit of Q, α, ln A, and n for the α+β phase.

Q	α	ln A	n
D0=553.76948	F0=0.00784	G0=57.29243	H0=3.15221
D1=−930.91058	F1=−0.000872446	G1=−101.77048	H1=−3.08687
D2=1,862.55132	F2=0.02996	G2=202.5848	H2=19.66898
D3=−2,918.766	F3=−0.10252	G3=−329.01037	H3=−57.49236
D4=2,418.89475	F4=0.19676	G4=319.97452	H4=87.20392
D5=−667.34872	F5=−0.19849	G5=−165.26221	H5=−68.96821
D6=−80.48794	F6=0.0829	G6=41.6677	H6=22.91159

Table 3:

Results of the polynomial fit of Q, α, ln A, and n for the β phase.

Q	α	ln A	n
D0=202.15525	F0=0.01736	G0=13.95315	H0=2.79185
D1=375.32167	F1=−0.00964	G1=89.53075	H1=0.75068
D2=−2,371.40657	F2=0.11637	G2=−672.13406	H2=−6.619
D3=8,003.89099	F3=−0.49253	G3=2,520.29115	H3=25.31954
D4=−16,123.01192	F4=1.05187	G4=−5,054.43695	H4=−49.26467
D5=17,449.52169	F5=−1.09863	G5=5,118.44525	H5=48.19971
D6=−7,630.74978	F6=0.44757	G6=−2,049.88235	H6=−19.08359

The constitutive model considering the effect of strain is developed and the flow stress for all conditions can be calculated once the strain-compensated material constants are evaluated.

The constitutive equations describing the dependence of the flow stress of Ti555211 alloy on the strain rate and deformation temperature can be written as

(15)ε˙=Aα+βsinh(αα+βσ)nα+βexp−Qα+βTR(α+β phase)

(16)ε˙=Aβsinh(αβσ)nβexp−QβTR(β phase)

where α, n, Q, and A of Ti555211 alloy can be calculated by sixth-order polynomial (eq. (14)), and D₀–D₆, F₀–F₆, G₀–G₆, and H₀–H₆ listed in Table 2 (α+β phase) and Table 3 (β phase) are the coefficients of the polynomial.

Verification of the accuracy of the constitutive model considering the effect of strain

As Figure 11 shows, the flow behavior of this material is accurately described by the predicted flow stress data for all experimental conditions except for 750°C, 1 s⁻¹ and 850°C, 1 s⁻¹; significant deviations occur between the experimental and predicted data for experimental conditions of 750°C, 1 s⁻¹ and 850°C, 1 s⁻¹. Phenomenological models based on physical theories of plastic deformation, although less rigorous than the polynomial method, make more accurate predictions for limited processing conditions where specific deformation mechanisms operate. In addition, DRX typically occurs at low strain rates. The extent of this recrystallization increases with decreasing strain rate. In contrast, bands of flow localization form and longitudinal cracking occurs at high strain rates (≥1 s⁻¹). The difference in the deformation mechanisms operating at low and high strain rates may therefore lead to significant divergence between the predicted and experimental values, as observed in the case of 750°C, 1 s⁻¹ and 850°C, 1 s⁻¹. Ji et al. [29] used a strain-compensated constitutive model to predict the high-temperature deformation behavior of AerMet100 steel; significant divergence occurred due to the difference in the physical mechanisms (cracking, shear bands, and twin kink bands vs. DRV and recrystallization, respectively) operating in the instability and stability regimes.

(17)R=∑i=1N(Ei−Eˉ)(Pi−Pˉ)∑i=1N(Ei−Eˉ)2∑i=1N(Pi−Pˉ)2

(18)AARE(%)=1N∑i=1NEi−PiEi×100

Figure 11:

Comparison between the experimental and predicted flow stress at different temperatures: (a) 750°C; (b) 800°C; (c) 850°C;(d) 900°C (e) 950°C.

The correlation coefficient (R) is used to evaluate the accuracy of the predictions of the constitutive equation [30]. An R of 0.99084, as calculated from eq. (17) using the data plotted in Figure 12, reflects the excellent predictive capability of the developed constitutive equation. An AARE of 6.914% for the Arrhenius-type constitutive model is calculated from eq. (18).

Figure 12:

Comparison of the predicted data obtained from the Arrhenius-type constitutive model (taking the compensation of strain into account) and experimentally determined data.

Microstructure evolution

Figure 13 shows the microstructures resulting from isothermal compression (ε=50%, ε˙=0.001 s⁻¹) of Ti555211 alloy at different deformation temperatures. As Figure 13(a) and 13(b) shows, the primary α phase becomes increasingly equiaxed but its volume fraction increases only slightly with increasing deformation temperature of 750–800°C. A portion of the secondary α phase is also dissolved and becomes globular. In addition, as Figure 13(c) shows, the morphology of the primary α phase is insensitive to the deformation temperature. However, the corresponding grain size and volume fraction decrease significantly. This decrease stems from partial transformation of the primary α phase to the β phase, and the almost complete dissolution of the secondary α phase. At a deformation temperature of 900°C (Figure 13(d)), the primary α phase transformed completely to the β phase, and only the β grain boundaries are observed.

Figure 13:

Microstructure of Ti555211 alloy at the strain rate of 0001 s⁻¹: (a) 750°C; (b) 800°C;(c) 850°C; (d) 900°C; (e) 950°C.

The deformation process (Figure 13(a)) is longer than all of the other processes and the elevated deformation temperatures significantly enhance the diffusion process. As such, the primary α phase consumes the nearby secondary α phase, thereby leading to the impingement of neighboring α grains [31–33]. Furthermore, the grain size is refined due to the transformation of primary and secondary α to the β phase. The two mechanisms occur simultaneously, in general, but the mechanism governing the phase transition plays the dominant role when the deformation temperature is increased to a certain level [34].

Due to the small strain rate, the entire deformation process is longer than the other processes. Figure 13(d) shows that small DRX grains nucleated on the β grain boundaries, and many elongated β grains are present in the deformed microstructures; the new DRX grains are easily distinguished from the other microstructural features [35]. When the deformation temperature was increased to 950°C (Figure 13(e)), the β grains grew with serrated boundaries and fewer DRX β grains were observed, compared to those at 900°C. Moreover, the large β grains consumed the neighboring small β grains.

The deformation temperature has a significant effect on the morphology and volume fraction of the secondary α phase. For example, the volume fraction of the long rod-like secondary α phase decreases substantially when the deformation temperature is increased from 750°C to 800°C, where this phase exists mainly as short rod-like features. At 850°C, the secondary α phase is no longer observed due to the phase transformation. Furthermore, for a given strain rate, the α phase is refined with increasing deformation temperature.

At a strain rate of 1 s⁻¹, the microstructure (Figure 14) exhibits features which are typical of DRV. The volume fractions of primary and secondary α phase decrease due to the heat generated during deformation [36]. In addition, the temperature increase is strain rate dependent and hence, larger increases occur at high strain rates than at low ones. Crystal defects, such as dislocations, phase boundaries, and grain boundaries, constitute potential heterogeneous nucleation sites for the secondary α phase [37, 38]. The density of these defects increases significantly with increasing strain rate, leading to precipitation of the secondary α phase [38]. The TEM images in Figure 15 reveal the large number of dislocations and dislocation cells, which are generated during hot deformation of Ti555211 alloy; the interaction between dislocations also leads to a ridged structure (Figure 15(a), 15(b)). These dislocation substructures interact with and consume other dislocations continuously, thereby resulting in an increase in the misorientation of the subgrain boundary. Furthermore, subgrain boundaries are continuously transformed to HAGBs and form new subgrains (Figure 15(c)). Elongated β grains with irregular grain boundaries are a typical feature of DRV. However, high strain rates hinder DRX since there is not enough time for the dislocations to be continuously consumed or generated.

Figure 14:

Microstructure of Ti555211 alloy at the strain rate of 1 s⁻¹: (a) 750°C; (b) 800°C; (c) 850°C; (d) 900°C; (e) 950°C.

Figure 15:

TEM observation of the deformation microstructure: (a) and (b) dislocation glide; (c) subgrain.

Figures 16–18 compare, via grain boundary orientation maps, the misorientation angle distributions of the grain boundaries, under various hot deformation conditions. As the figures show, compared to that of the initial microstructure, the fraction of subgrain boundaries decreases gradually increasing deformation. Moreover, the fraction of subgrain boundaries and LAGBs decreases gradually with increasing temperature and decreasing strain rate. Some of the subgrain boundaries and LAGBs also became HAGBs.

Figure 16:

The grain boundary orientation map of Ti555211 alloy after hot deformation: (a) 750°C, 0.001 s⁻¹; (b) 800°C, 1 s⁻¹; (c) 800°C, 0.001 s⁻¹; (d) 950°C, 0.001 s⁻¹.

Figure 17:

Distribution of grain boundary misorientation angles (α phase): (a) 750°C, 0.001 s⁻¹; (b) 800°C, 0.001 s⁻¹; (c) 800°C, 1 s⁻¹.

Figure 18:

Distribution of grain boundary misorientation angles (β phase): (a) 750°C, 0.001 s⁻¹; (b) 800°C, 0.001 s⁻¹; (c) 800°C, 1 s⁻¹ (d) 950°C, 0.001 s⁻¹.

Figures 17 and 18 show the misorientation angle distributions of the grain boundaries in the α and β phase, respectively. Deformation conditions of 750°C, 0.001 s⁻¹ (Figures 17(a) and 18(a)) and 800°C, 0.001 s⁻¹ (Figures 17(b) and 18(b)) resulted in a decrease (from 29.5% to 15.7%) in the fractions of HAGBs in the α phase; the fractions of HAGBs in the β phase decreased significantly due to the partial dissolution of the secondary α phase and the growth of the β grains.

As Figures 17(b) and 18(b) (800°C, 0.001 s⁻¹) and Figures 17(c) and 18(c) (800°C, 1 s⁻¹) show, the fractions of HAGBs in the α and β phases decrease from 15.7% to 4.8% and change only slightly, respectively. This decrease results mainly from the subgrain boundaries in the α phase not having sufficient time, at a strain rate of 1 s⁻¹, to absorb dislocations, for transformation to HAGBs. The fraction of HAGBs increased substantially for deformation conditions of 950°C, 0.001 s⁻¹ (Figure 18(d)), thereby resulting in an increased number of nucleation sites; this increased number of nucleation sites resulted in intense DRX.