Effect of isothermal transformation temperature on the microstructure, precipitation behavior, and mechanical properties of anti-seismic rebar

2.1 Materials prepared

The raw material for the experimental steel consists of 500 MPa grade high-strength anti-seismic rebars. Niobium-iron alloy and titanium-iron alloy were added to the medium-frequency induction heating furnace and melted into 15 kg ingots. The chemical composition of the ingots was analyzed using various instruments including a carbon and sulfur analyzer (CS, LECO CS844, USA), a nitrogen–hydrogen–oxygen analyzer (ONH, LECO ONH836, USA), and an Inductively Coupled Plasma Emission Spectrometer (ICP, PerkinElmer 8300, USA). The results are presented in Table 1.

2.2 Thermo-mechanical processing

The ingot of steel was processed into rectangular specimens of dimensions 120 mm × 20 mm × 20 mm, and flat compression experiments were conducted on a Gleeble-3800 thermal simulation machine. According to the solubility theory and relevant studies [24–27], a thermal simulation process was designed for the experimental steel: the specimen was heated to 1,200°C at a rate of 20°C·s⁻¹ and held for 10 min to ensure full homogenization of alloying elements. Subsequently, the specimen was cooled to 1,100°C at a rate of 5°C·s⁻¹ for the first deformation stage, with a true strain of 0.3 and a strain rate of 2 s⁻¹. It was then cooled to 1,000°C at the same rate for the second deformation stage, with a true strain of 0.2 and a strain rate of 2 s⁻¹. Finally, it was cooled to 900°C at the same rate for the third deformation stage, with a true strain of 0.2 and a strain rate of 2 s⁻¹. The specimen was then cooled at a rate of 15°C·s⁻¹ to the isothermal transformation temperature and held for 30 s. The isothermal transformation temperatures were 700, 650, and 600°C, respectively, followed by air cooling to room temperature. The thermal simulation process is illustrated in Figure 1.

Figure 1

Thermal simulation process of the experimental steel.

2.3 Characterization methods

The microstructure of the experimental steels was characterized using a Z380T metallographic microscope (MM) and a SUPRA40 scanning electron microscope (SEM). The average ferrite grain size, ILS, ferrite content, and pearlite content were determined separately using the linear intercept method. A Tecnai G2 F20 S-TWIN transmission electron microscope (TEM) equipped with an energy spectrometer (EDS) was employed to analyze the structure, morphology, and composition of the precipitates. The original high-resolution images of the precipitates (HRTEM) were processed using the HRTEM filter function of the GMS-3. Electron diffraction spots and lattice stripes of the matrix and precipitates were obtained using the fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT). Electron backscatter diffraction (EBSD) analysis was conducted using Oxford Nordlys max3, with the sample rolling direction aligned with the physical x-axis of the SEM. The EBSD data were analyzed using Channel5. Plate tensile specimens were prepared for the samples under different isothermal transformation temperatures and stretched at a strain rate of 10⁻² s⁻¹ using an MTS810 universal tensile tester. Based on the true stress–strain equation, the stage of uniform plastic deformation was identified, and redundant data points were eliminated using the three-strip sample interpolation method. The work hardening indices at different isothermal transformation temperatures were obtained through differentiation and linear fitting, ensuring an R ² value greater than 0.9.

3.1 Microstructure

Figure 2 shows the microstructure of the experimental steel at different isothermal transformation temperatures.

Figure 2

MM, SEM, and TEM of microstructure under the conditions of different isothermal transformation temperatures: (a1) and (a2) 700°C; (b1) and (b2) 650°C; (c1) and (c2) 600°C (yellow dash line is the interface).

Figure 2 shows the microstructure of the steel at different isothermal transformation temperatures, revealing the presence of polygonal ferrite (PF) and pearlite. The PF appears as the white area, while the black area represents pearlite. The distribution of PF and pearlite is relatively uniform, with a higher proportion of ferrite grains being observed. We observe numerous dislocations between the cementite, with a higher accumulation of dislocations at the boundaries. Additionally, noticeable dislocation entanglements within the ferrite grains form a network that may consist of dislocation cells and walls. Moreover, precipitates with sizes ranging from 3 to 5 nm are present within the lamellae, as depicted in the enlarged image of the red area (a2–c2). Figure 3 illustrates the different isothermal transformation temperatures, along with the inverse pole figure (IPF), grain boundary (GB), and geometric dislocation density (GND) of the experimental steel.

Figure 3

IPF, GB, and GND of simples under the conditions of different isothermal transformation temperatures: (a1)–(a3) 700°C, (b1)–(b3) 650°C, and (c1)–(c3) 600°C.

The analysis of grain boundaries reveals that the content of high angle grain boundary (HAGB) in the experimental steel increases to 82.3%, while the content of low angle grain boundary gradually decreases to 17.7% as the isothermal transformation temperature decreases to 600°C. The GND in the experimental steel gradually diffuses from within the grain interior toward the grain boundaries as the temperature decreases, resulting in a more uniform distribution of GND. Figure 4 presents the orientation distribution function (ODF) and the fiber strengths of 〈110〉//RD, <111>//ND, and 〈011〉//TD in the texture of the experimental steel with a φ ₂ angle of 45°.

Figure 4

(a)–(c) ODF at φ ₂ = 45° section: (a) 700°C, (b) 650°C, (c) 600°C. (d)–(f) Skeleton plots from at φ ₂ = 45° section: (d) α-fiber//RD, (e) γ-fiber//ND, (f) ε-fiber//TD.

In Figure 4(a)–(c), the lines of different colors represent the grain strength (indicating the quantity of grains) for a specific orientation. The red, green, blue, and black lines correspond to values of 2.5, 2.0, 1.5, and 1.0, respectively. Figure 4(a)–(c) shows that there are textures in the experimental steel. Figure 4(e) demonstrates that {111}<110> and {111}<143> exhibit the highest strength in the γ-fiber. In Figure 4(d), as the isothermal transformation temperature decreases, it is evident that the strength of {115}<110>, {111}<110>, and {112}<110> in the experimental steel gradually decreases. It is noteworthy that {112}<110> originates from the Cu-{112}<111> texture in the austenitic parent phase, which significantly reduces the impact toughness of RD [28]. In Figure 4(e), {111}<110> and {111}<143> exhibit the highest strength in the γ-fiber, with {111}<110> component found to enhance elongation and deep drawability. As the isothermal transformation temperature decreased, the strength of {100}<110> gradually decreased, resulting in improved toughness and reduced delamination in the experimental steel. In Figure 4(f), the main components in the ε-fiber was observed to be copper-{121}<111>, {111}<112>, {332}<113>, and Goss-{110}<001>. The static recrystallization of ferrite facilitates the transformation of {111}<110> to {111}<112> and the transformation of brass-{110}<112>_γ into {332}<113>_α. The transformation of these textures follows: {332}<113>_α → {554}<225>_α → {111}<112>_α. At 600°C, {332}<113>_α exhibits the highest strength, while {111}<112>_α has the lowest strength. {332}<113> is an ideal texture that can greatly enhance the plasticity and yield strength of the experimental steel [29]. Figure 5 presents the average ferrite grain size, ILS, microstructure content, and GND for the experimental steels.

Figure 5

(a) Grain size, ILS, and GND. (b) Microstructure content.

With the isothermal transformation temperature decreasing from 700 to 600°C, the average ferrite grain size decreased by 4.73 μm, the ILS decreased by 0.08 μm, the proportion of ferrite decreased by 17.14%, the proportion of pearlite increased by 5.85% and the GND increased by 0.19 × 10¹⁴ m⁻². As the ferrite grain size and the ILS were decreased, the blocking effect on dislocation slip becomes more pronounced, facilitating the accumulation of dislocations between the lamellae and resulting in a higher dislocation density. The existence of a significant angular difference between the HAGB and the slip surface can effectively impede dislocation slip, thereby enhancing dislocation pile-up at the grain boundary. Additionally, TEM observations indicate that a portion of the cementite exhibits continuous curvature. Previous studies [30,31] have shown that the behavior of cementites under shear strain differs depending on their thickness. Specifically, when the thickness of the cementite is reduced below 200 nm, it tends to bend or rotate rather than fracture.

3.2 Precipitates

The Nb/Ti precipitates primary strengthens microalloyed rebars. The composition, size, and interface of the precipitates significantly influence the mechanical properties of microalloyed steels. Figure 6 presents the TEM results of precipitation.

Figure 6

TEM of precipitate: (a) (Nb, Ti)C, (a1) EDS and (b) (Ti, Nb)C, (b1) EDS.

Figure 6 shows that the precipitates appear as circular or rectangular shape, with an average size of ∼50 nm. These precipitates primarily consist of (Nb, Ti)C composite precipitates. Diffusion of solute atoms within the crystal lattice leads to local concentration variations, resulting in the formation of rounded precipitates, whereas solute atom clustering at specific crystal lattice sites leads to the formation of square precipitates [5,16]. NbC predominantly adopts a rounded morphology, while TiC tends to exhibit a square morphology. Research indicates [32,33] that NbC and TiC can make close spatial contact, co-precipitating as a TiC–NbC core–shell structure through epitaxial growth. Depending on the composition of the precipitate shell, precipitates can exhibit either rounded or square shapes. From Figure 6 it can be observed that the precipitates are formed on dislocations. According to the nucleation theory, the formation of precipitated phases requires a certain amount of energy [5]. Due to the significant lattice mismatch between Nb, Ti, and Fe, Nb/Ti is highly prone to biased precipitation at grain or phase boundaries. This effectively reduce the interfacial energy [34,35]. The intense deformation during rolling accelerates the γ → α phase transition, generating a large number of randomly dispersed supersaturated carbide particles within the ferrite matrix [36]. The high-density dislocations within the pearlite lamellae have a remarkable ability to capture carbon atoms, leading to the preferential nucleation of Nb/Ti precipitates at these locations [37,38]. The microstructure exhibits a significant presence of numerous high-density dislocations and subgrain boundaries, serving as nucleation sites for the precipitates [39,40]. Furthermore, the high-density dislocations in the steel synergize with the fine nanoscale Nb/Ti carbides, resulting in a high work hardening capability [41]. During the γ → α phase transformation, the fine precipitates effectively anchor the grain boundaries, thus impeding the migration of austenite along the grain boundaries and refining the ferrite grains. Wang et al. [42] demonstrated that the crystal structure of (Nb, Ti)C precipitated phases in microalloyed steels was a NaCl-type structure, with a specific OR between the crystal structure of the precipitated phase and the ferrite matrix. The precipitated phase tightly binds to the ferrite matrix, nucleating and growing on it. Furthermore, the chemical composition of the precipitates formed at lower austenite temperatures or during the transformation to ferrite is predominantly composed of niobium and carbon [15]. Figure 7 shows the HRTEM of the (Nb, Ti)C precipitates with the FFT and IFFT of the corresponding regions.

Figure 7

HRTEM, FFT, and IFFT of precipitate: (a) (Nb, Ti)C and (b) (Ti, Nb)C (red dash line indicates matrix, the solid blue line indicates precipitates, yellow ⊥ indicates dislocations).

In Figure 7(a), it can be observed that the matrix of the precipitates consists of PF, within crystal planes spacings that are 1.8456 and 2.2197 nm, corresponding to crystal planes of (001) and (011), respectively. The precipitates exhibit a single-crystal structure of (Nb, Ti)C. The crystal zone axis is [ 0 1 ̅ 1 ] with crystal planes ( 0 2 ̅ 2 ̅ ) and ( 1 ̅ 1 ̅ 1 ̅ ) , with an angle of 69.20° between the crystal planes. Figure 7(b) shows that the matrix of the square-shaped precipitates is likely Fe₃C in the orthorhombic system. It has a crystal zone axis of [ 0 2 ̅ 1 ̅ ] and a crystal plane spacing of 0.2108 nm, corresponding to a crystal plane of ( 0 1 2 ̅ ) . The precipitate is (Nb, Ti)C, with a crystal zone axis of [ 5 ̅ 6 1 ̅ ] . Its crystal plane spacings are 0.2079 and 0.2117 nm, corresponding to crystal planes of (111) and ( 1 0 5 ̅ ) , respectively. The angle between the crystal planes is measured at 110.9°. In Figure 7(b), IFFT reveals the presence of dislocations within the matrix, which may have been generated during the rolling process. Additionally, the presence of diffraction rings at the center of the FFT of the precipitates suggests that the precipitates are likely composed of polycrystalline material. Figure 8 shows the HRTEM images of the matrix/precipitate interface, along with the FFT and IFFT of the corresponding regions.

Figure 8

HRTEM, FFT, and IFFT of interface of matrix/precipitate: (1) matrix, (2) precipitate, and (3) interface (red dash line indicates matrix, the solid blue line indicates precipitate, yellow dash circle indicates unknown spot, yellow ⊥ denotes dislocation, and red dots denote atoms in the corresponding positions).

Figure 8(b) shows that the matrix is likely α-Fe in the BCC system, with a crystal zone axis of [ 100 ] and a crystal plane spacing of 0.2057 nm, corresponding to a crystal plane of ( 110 ) . Additionally, a small amount of dislocation is observed within the matrix. The precipitates exhibit a single-crystal structure of (Nb, Ti)C, with a crystal zone axis of [ 1 ¯ 10 ] and crystal planes ( 111 ) and ( 200 ) , with an angle of 17.3° between the crystal planes. The IFFT at the interface in Figure 8 reveals a twisting and distortion of approximately 2.07° in the regular atomic arrangement. The length of the interface distortion measures approximately 4.6059 nm, indicating a probable semi-coherent interface. According to Bramfitt’s two-dimensional mismatch theory [43], the mismatch between the different crystal planes of the precipitates and the matrix was calculated, and the results are shown in Table 2.

The interface between the precipitates and the matrix exhibited a mismatch of 14.24 and 15.65%, respectively. The two crystal planes were unable to form a co-lattice or semi-coherent interface with the matrix, indicating a non-co-lattice interface (12%). This finding contradicts the experimental observations in this study and is also inconsistent with reports from other researchers [44]. Consequently, the growth of precipitates may not occur directly on the substrate, and the introduction of dislocations becomes necessary to alleviate the interfacial strain energy and promote the formation of a semi-coherent interface. Based on these observations, the following conjectures have been proposed:

The IFFT of the interface of Figure 8 reveals the atomic arrangement with a 2.07° twist, indicating the formation of a spiral dislocation during the deformation process. When a spiral dislocation occurs at a smooth interface, it gives rise to a helical step on the surface perpendicular to the dislocation line. The helical step assists in the nucleation and growth of the Nb/Ti precipitates, as shown in Figure 9.

Figure 9

Nucleation of Nb/Ti precipitates with the help of screw dislocation.

This could explain why the precipitates are able to nucleate and grow despite a matrix/precipitate mismatch of more than 12%. Comparing the positions of diffraction spots in Figure 8 from 1, 2, and 3, besides the diffraction spots originating from the matrix and precipitates, there are a few diffraction spots from unknown crystal planes in the interface FFT (yellow dash circles in the 3-FFT). From the analysis above, it is presumed that the spiral dislocation leads to the deviation of atoms from the original Bravais dot matrix, resulting in the formation of unknown diffraction spots. Wang et al. [45] observed that, owing to the lattice distortion within the ferrite matrix, vanadium (V) forms a co-lattice structure with the ferrite matrix, facilitating the nucleation of V(C, N) on the ferrite matrix. This observation strongly aligns with the hypothesis of this article.

3.3 Mechanical properties

The stress–strain of the experimental steel under different isothermal transformation temperature conditions is shown in Figure 10. The stress–strain data statistics are shown in Table 3.

Figure 10

Stress–strain of the steel under the conditions of different isothermal transformation temperatures (700–600°C).

When the isothermal transformation temperature is decreased from 700 to 600°C, the yield strength and tensile strength increased by 76 and 120 MPa, respectively. Notably, for an isothermal transformation temperature of 600°C, the steel achieved its highest values for yield strength (592 MPa) and tensile strength (734 MPa). The reduction rates of the work hardening exponent and the work hardening exponent at different stages of the experimental steel are illustrated in Figure 11 and summarized in Table 4, respectively.

Figure 11

Work hardening exponent of experimental steel under the conditions of different isothermal transformation temperatures: (a)–(c) Ln(strain)–Ln(stress) ((a) 600°C, (b) 650°C, (c) 700°C) and (d) work hardening exponent.

Figure 11(a)–(c) shows three distinct stages of work hardening in the experimental steel at different isothermal transformation temperatures. The work hardening exponent (n) of the experimental steels decreases from 0.3983 to 0.1838 at 600°C, from 0.3576 to 0.1830 at 650°C, and from 0.2879 to 0.1758 at 700°C. The work hardening exponent of the experimental steel gradually decreases, indicating a reduction in the plastic deformation capability. Figure 11(d) shows that the work hardening exponent of Stage I gradually increases as the isothermal transformation temperature decreases. The work hardening exponent indices of Stages II and III remain similar and show no significant changes. The work hardening exponent is an important parameter to measure the work hardening ability, it is related to the dislocation density in the microstructure. Dislocations undergo slip, intersecting with each other and forming a dislocation network within the microstructure, thereby increasing the dislocation density. The higher the dislocation density, the stronger the work hardening effect becomes. The greater the dislocation density, the more pronounced the work hardening effect becomes. Based on the statistical results presented in Figure 4, the experimental steel at 600°C exhibits the lowest average ferrite grain size and ILS. The microstructure of the experimental steel demonstrates the strongest hindrance to dislocation slip and the highest potential for increasing dislocation density at Stage I, thereby showcasing the greatest work-hardening ability. At Stage II and Stage III, the large number of dislocations undergo cross-slip during Stage I, leading to difficulties in further slip as dislocations intersect with each other. Similar work-hardening behavior in microalloyed steel has also been reported by Ghazani et al. [46]. Table 4 shows that as the isothermal transformation temperature decreases, the reduction rate of the work hardening exponent from Stage I to Stage II gradually increases (21.67% → 40.47%). The reduction rate of the work hardening exponent from Stage II to Stage III gradually decreases. The reduction rate of Stage I → Stage II is consistently greater than Stage II → Stage III.

4.1 Effect of isothermal transformation temperature on microstructure

Based on the characterization of microstructure and mechanical properties, it is evident that ferrite grain size and ILS both play crucial roles in determining mechanical properties. Moreover, the growth of both ferrite and pearlite is influenced by the diffusion of carbon atoms. By the kinetics of pearlite growth (equation (1)) [47], it becomes possible to establish a relationship between the growth rate of pearlite and the diffusion of carbon:

where D _C is the diffusion coefficient of carbon atoms, λ ^α and λ ^θ are the thickness of ferrite and cementite, respectively, λ is the interlayer spacing of pearlite, λ _C is the theoretical minimum spacing, C eq γα and C eq γθ indicate the carbon concentration in austenite before ferrite and cementite, respectively, while C eq α and C eq θ are the equilibrium carbon concentration in ferrite and cementite, respectively, and k is a geometrical constant related to the volume diffusion mechanism.

As shown by the pearlite growth kinetics, the pearlite growth rate G is proportional to the carbon atom diffusion rate. According to Fick’s law, it is known that the carbon diffusion rate is related to the carbon diffusion coefficient, and equation (2) is the equation of the carbon diffusion coefficient with temperature [48]:

where D ₀ is the carbon diffusion coefficient in austenite, R is the gas constant, and T is the transition temperature. D ₀, Q _i , and R are 0.23 cm²·s⁻¹, 138 kJ·mol⁻¹, and 8.31 J·(mol⁻¹·K⁻¹) in austenite, respectively [22,49].

Considering that the Nb/Ti atoms in the steel are strongly carbonized elements that can impede the migration of carbon atoms and reduce the diffusion coefficient of carbon atoms. Lee et al. [49] provided the following equation between Nb and carbon diffusion coefficient:

where X _Nb is the molar fraction of Nb and D C 0 denotes the diffusion coefficient of carbon atoms without the addition of Nb and Ti.

Since Ti atom is also a strong carbonizing element, X _Nb is corrected to X _Nb+Ti, and the corrected carbon diffusion coefficient is calculated as a function of temperature. Figure 12 shows TTT and diffusion coefficient of carbon.

Figure 12

(a) TTT and (b) diffusion coefficient of carbon.

In Figure 12(a), it can be observed that as the isothermal transformation temperature is decreased from 700 to 600°C, the starting time of ferrite transformation in the experimental steel first decreases and then increases, while the starting time of pearlite transformation gradually decreases. However, 600°C is not the fastest transformation time for ferrite, but it is considered the fastest transformation time for pearlite due to the presence of pearlite organization in the steel. As a result, isothermal transformation at 600°C enables an increase in pearlite content while simultaneously controlling the ferrite grain size. The data presented in Figure 4 also confirms that the experimental steel exhibits the highest pearlite content and the smallest ferrite grain size at 600°C. Clearly, in Figure 12(b), the diffusion coefficient of carbon atoms gradually decreases with decreasing temperature, leading to a decrease in diffusion ability. This trend holds true regardless of the presence of the Nb/Ti dragging effect. Adding Nb/Ti elements significantly reduces the diffusion coefficient of carbon atoms compared to that of the rebars without Nb/Ti. Moreover, the reduction rate of the diffusion coefficient of carbon atoms is much higher in the rebars containing Nb/Ti compared to the rebars without the Nb/Ti dragging effect. Since the microstructure of anti-seismic rebars typically consists of ferrite and pearlite, pearlite exhibits a size effect as the strengthening phase. The finer ILS and cementite thickness during deformation hinders effectively dislocations and inhibits the formation of dislocation cells [28]. Moreover, the decreased diffusion coefficient of carbon atoms contributes to a slower growth rate of pearlite and leads to the refinement of cementite.

4.2 Effect of isothermal transformation temperature on the precipitation behavior of Nb/Ti

Based on the analysis of precipitates in Section 3.2, the size of Nb/Ti precipitates is approximately 50 nm. During the γ → α phase transition, the dislocations and fine precipitates at grain boundaries effectively restrict grain boundary migration and inhibit grain growth. Additionally, the high density of dislocations synergizes with the fine nanoscale Nb/Ti carbides, resulting in a high continuous work hardening capability [41]. According to equation (4) [5], the average grain diameter is directly proportional to the size of the precipitated phase when the volume fraction of the precipitated is constant. Consequently, the smaller the size of the precipitated phase, the smaller the ultimate average grain size.

where D ̅ lim is the limiting grain average size, φ is the volume fraction of precipitated phase per unit volume of alloy, and r is the size of precipitated phase.

Based on the nucleation growth theory and Avrami’s equation [50,51], Yong [16] derived the precipitation kinetic model (equations (5)–(7)) and calculated the difference in Nb/Ti precipitation time as a function of temperature variations:

where t _0a is the temperature-independent parameter (s), t _0.05a is the precipitation fraction of 5% (s), d* is the critical nucleation radius, ΔG* is the critical nucleation work (J), k is the Boltzmann constant, and T is the temperature (K). When the precipitated phase is n ₁ = n ₂ = 1, α = A ₁ for grain boundary nucleation, n ₁ = 2/3, n ₂ = 2.5, α = (1 + β)^3/2 for dislocation nucleation, β is a coefficient associated with dislocation line nucleation, A is the core energy per unit blade dislocation line (J·m⁻¹), ΔG _V is the volume free energy (J·m⁻³), σ is the interface energy between precipitates and matrix (J·m⁻²), and Q is the atomic activation energy (J·mol⁻¹).

PTT of the precipitates were calculated as shown in Figure 13.

Figure 13

PTT of NbC and TiC: (a) grain boundary nucleation and (b) dislocation nucleation.

Figure 13 shows that TiC precipitates precede NbC in both grain boundary nucleation and dislocation nucleation. Moreover, the temperature for NbC grain boundary nucleation is higher than that for dislocation nucleation. In Figure 13(b), TiC precipitates first at 1,200°C. At 1,100°C, NbC begins to precipitate and combines with TiC to form a composite precipitates, such as (Nb, Ti)C or (Ti, Nb)C. Remaining NbC in the experimental steel continues to precipitate even after the pearlite transformation takes place. As the temperature decreases, the precipitation time also decreases. The isothermal transformation temperature of 600°C is closest to the nasal tip precipitation temperature (618.7°C), with the precipitation rate being the fastest and the precipitate content the highest.

4.3 Strengthening mechanism

Four strengthening mechanisms are known to enhance the performance of steel: solid solution strengthening (σ _SS), fine grain strengthening (σ _GB), precipitation strengthening (σ _PR), and dislocation strengthening (σ _Dis). The relationship between yield strength and strengthening mechanism is shown in the following equation:

where σ ₀ is the friction stress, which has a value of 53.9 MPa [7]. The four strengthening effects were calculated as follows [16,35,52]:

where [Mn], [Si], [P], and [Cr] are the mass fractions (wt%) of the corresponding elements in the steel. σ _f and σ _p are the contributions of ferrite and pearlite to strength, V _f and V _p are the percentages of ferrite and pearlite in the experimental steel [37], d _PR is the average precipitate size (∼50 nm), f is the volume fraction of Nb/Ti precipitates in the ferrite (%), α is the constant 0.24 [53], M is the deformation-induced orientation factor of about 2 [37], G is the shear modulus (81,600 MPa for steel) [54], b is the Burgers vector (0.248 nm) [54], and ρ is the dislocation density in the steel.

The strengthening of ferrite and pearlite can be calculated by the following equation [55]:

where k _HP is the proportionality constant and takes the value of 17.4 MPa·mm^0.5 [56], d is the average ferrite grain size (nm), and ILS (nm).

The volume fraction of (Nb, Ti)C in the ferrite can be calculated by the following equation [57]:

where M_i(M_i = Nb, Ti) is the content of microalloying elements in steel (wt%), [M_i] is the solid solution amount of microalloying elements in ferrite, C is the content of element C in steel (wt%), [C] is the solid solution amount of C in ferrite, ρ _Fe and ρ _MC are the densities of Fe and MC (MC = (Nb, Ti)C), i.e., ρ _Fe=7.8 g·cm⁻² and ρ _{(Nb, Ti)C} = 6.385 g·cm⁻².

For solid solution strengthening, the solid solution of Nb/Ti elements is almost zero as the temperature decreases [16]. Therefore, the contribution of solid solution strengthening mainly comes from elements such as Mn, Si, and P. Based on the theory of “grain boundary dislocation emission source,” Yong [16] combined the dislocation strengthening and fine-grain strengthening mechanisms by using the root mean square method, considering that grain boundaries can be regarded as dislocations of total length of L. Some researchers also adopted this theory and achieved better results [8]. The following equation shows the modified strengthening mechanism:

Figure 14 illustrates the increment of four strengthening mechanisms on the yield strength.

Figure 14

(a) Strengthen increment of σ _YS and (b) strengthen increment of σ _PR, σ _GB, and σ _Dis.

Figure 14(a) shows that the vector superposition of the strengthening mechanism aligns more closely with the experimental yield strength, albeit slightly lower than the theoretical yield strength. This discrepancy may arise from the overestimation of solid solution strengthening and friction stresses in theoretical calculations compared to actual values. Figure 14(b) shows that grain boundary strengthening emerges as the primary contributor to yield strength. As the isothermal transformation temperature decreases, the contributions of precipitation strengthening and dislocation strengthening gradually increase. However, the increase in grain boundary strengthening is the most significant. As discussed in Sections 3.1 and 4.1, the presence of numerous sub-grains and defects within the grains of the rolled experimental steel provides nucleation sites for new grains during the γ → α phase transition [21,58]. Subsequent temperature rise prompts the migration of ferrite grain boundaries and accelerated diffusion of carbon atoms, leading to the coarsening of ferrite grains and cementite. In some cases, abnormal growth may occur, resulting in the diminishing effectiveness of fine grain strengthening. As described in Section 4.2, with increasing temperature, the diffusion rate of Nb/Ti atoms also increases. This leads to the growth of precipitates, causing an increase in precipitates size and a subsequent decrease in the strengthening effect of precipitation [51].