A. Dilatometric analysis

The variations in the relative change of length as a function of temperature, f(T) = ΔL/L ₀, and its corresponding differential thermal expansion coefficient, α_T = d(ΔL/L ₀)/dT,^{Reference James, Spittle, Brown and Evans5} are shown in the dilatation curves in Fig. 2, and these reveal the contraction and expansion that the steel undergoes with a ferrite and pearlite initial microstructure at a continuous heating and cooling rate of 15 °C/min.

FIG. 2.

(a) ΔL/L ₀ versus T on heating, (b) ΔL/L ₀ versus T on cooling, (c) α_T versus T on heating, and (d) α_T versus T on cooling and with a test rate of temperature change of 15 °C/min.

With a change in temperature, the total strain (ε_total) of steel includes the following three components^{Reference James, Spittle, Brown and Evans5,Reference San Martín, Rivera Díaz del Castillo and de Andrés9} : (i) the thermal strain of each phase with the temperature change; (ii) the difference in specific volume between the phases involved in the phase transformation; and (iii) the variations in the specific volume of austenite due to carbon enrichment or depletion. Therefore, the total strain of steel can be expressed as,

(1)$${\varepsilon _{{\rm{total}}}} = {\varepsilon _{{\rm{trans}}}} + {\varepsilon _{\rm{T}}}\quad ,$$

and

(2)$${\varepsilon _{\rm{T}}} = \left\{ \matrix{ {\varepsilon _{\rm{F}}} + {\varepsilon _{\rm{P}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T \;\le\; {A_{{\rm{c}}1}}\left( {{A_{{\rm{rp}}}}} \right) \hfill \cr {\varepsilon _{\rm{A}}} + {\varepsilon _{\rm{P}}} + {\varepsilon _{\rm{F}}}\,\,\,\,{A_{{\rm{c}}1}}\left( {{A_{{\rm{rp}}}}} \right) \;\lt\; T \;\le\; {A_{{\rm{cp}}}}\left( {{A_{{\rm{r}}1}}} \right){\rm{\ }} \hfill \cr {\varepsilon _{\rm{F}}} + {\varepsilon _{\rm{A}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{A_{{\rm{cp}}}}\left( {{A_{{\rm{r}}1}}} \right) \;\lt\; T \;\le\; {A_{{\rm{c}}3}}\left( {{A_{{\rm{r3}}}}} \right) \hfill \cr {\varepsilon _{\rm{A}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T > {A_{{\rm{c}}3}}\left( {{A_{{\rm{r3}}}}} \right)\quad , \hfill \cr} \right.$$

where ε_trans and ε_T are the phase transition strain (lattice change and the variation in the specific volume of austenite due to carbon enrichment or depletion) and the thermal strain. ε_A, ε_F, and ε_P are the thermal strains of austenite, ferrite, and pearlite, respectively. A _c1, A _cp, and A _c3 are the temperatures at which pearlite dissolution starts, pearlite dissolution ends, and austenization finishes on heating, while A _r3, A _r1, and A _rp are the temperatures at which austenite decomposition begins, the eutectoid reaction starts, and the eutectoid reaction finishes on cooling.

The phase transformation process of hypoeutectoid steel during continuous heating/cooling includes the pearlite ↔ austenite and the ferrite ↔ austenite transformations. Generally, those two phase transformations will overlap at a small temperature range. Some investigations^{Reference Oliveira, Andrade and Cota17,Reference Martín, Cock, García Junceda, Caballero, Capdevila and de Andrés20,Reference de Andrés, Caballero and Capdevila21} have experimentally shown that these two phase transformations can be made a separation successive steps. By comparing Figs. 2(a) and 2(c) at heating and Figs. 2(b) and 2(d) at cooling, the phase transformation temperature range can be divided into stage I (I′) (pearlite ↔ austenite transformation) and stage II (II′) (ferrite ↔ austenite transformation). Using the heating process as an example, as can be seen from Fig. 2(a), the first inflection point A (the ε_total deviates from linearity) is seen as the temperature rises, which indicates the start of the α + Fe₃C → γ reaction (pearlite dissolution).^{Reference De Andres, Caballero, Capdevila and Alvarez7} Because ferrite α has a body-centered cubic structure, it will transform into austenite γ with a face-centered cubic structure. This phase transformation will cause a volume contraction of approximately 1.6%^{Reference Kop, Sietsma and Van Der Zwaag10} and then ε_trans < 0. At the same time, each phase of steel expands as the temperature continues to rise, │ε_T = ε_A + ε_F + ε_P│ > │ε_trans│, and ε_total will increase along with the temperature [AB and A′B′ range in Figs. 2(a) and 2(c)]. When│ε_trans│ = │ε_T = ε_A + ε_F + ε_P│, the second inflection point B is seen, and the differential thermal expansion coefficient α_T = 0 [point B′ in Fig. 2(c)]. As the temperature continues to rise, │ε_trans│ > │ε_T = ε_A + ε_F + ε_P│, │ε_total│ will decrease, and negative thermal expansion (ε_total < 0) then occurs [BC range in Fig. 2(a) and B′C′ range in Fig. 2(c)], with the pearlite to austenite phase transition happening more rapidly. Thereafter, with the continued dissolution of pearlite, ε_trans decreases while the amount of austenite increases, and thus the thermal strain │ε_T = ε_A + ε_F + ε_P│ continues to rise. When the absolute value of ε_trans equals ε_T (│ε_trans│ = │ε_T = ε_A + ε_F + ε_P│) once again, the third inflection point C occurs. In fact, at the late stage of the eutectoid reaction, │ε_trans│ < │ε_T = ε_A + ε_F + ε_P│, ε_total > 0. When the eutectoid reaction ends, the ferrite to austenite transformation (α → γ) begins, and this causes the fourth inflection point D. But this inflection point D is less evident to see in Fig. 2(a) but easier to determine from the first derivative, α_T = d(ΔL/L ₀)/dT, as shown in Fig. 2(c).^{Reference Pawłowski16}

As the ferrite gradually transforms into austenite, the carbon in the carbon-rich austenite (with a maximum value of about 0.77C, wt%^{Reference Howell22}) will diffuse into the ferrite, and thus the volume of austenite is decreased. Both the lattice change (α → γ) and the decreasing carbon content of the austenite cause a contraction of the sample. At the start of the α → γ transformation [DE and D′E′ ranges in Figs. 2(a) and 2(c)], │ε_trans│ < │ε_T = ε_A + ε_FD│, and ε_total increases along with the temperature. When the temperature reaches the point E, │ε_trans│ = │ε_T = ε_A + ε_F│, ε_total = 0, and the differential thermal expansion coefficient α_T = 0 [point E′ in Fig. 2(c)]. As the temperature continues to rise, │ε_trans│ > │ε_T = ε_A + ε_F│, │ε_total│ will decrease, and the steel shows negative thermal expansion (ε_total < 0) [EF or E′F′ ranges in Figs. 2(a) and 2(c)]. At late stage of the austenization, │ε_trans│ < │ε_T = ε_A + ε_F + ε_P│, ε_total > 0 [FG and F′G′ ranges in Figs. 2(a) and 2(c)]. When austenization finishes, ε_total has a linear relationship with temperature.

B. Effect of heating/cooling rate on the critical temperature of phase transformation

The dilatation curves at various heating or cooling rates are shown in Fig. 3. The characteristic temperatures of phase transition, including A _c1, A _cp, A _c3, A _r3, A _r1, and A _rp, are shown in Table II. It can be seen that the characteristic temperatures of A _c1, A _cp, and A _c3 increased along with the heating rates, while A _r3, A _r1, and A _rp decreased as the cooling rates rose. Because the characteristic temperature of phase transformation depends on the nucleation rate of the new phase (austenite or ferrite), which is controlled by the nucleation time, nucleation site, temperature gradient, and nucleation style (e.g., heterogeneous nucleation). At higher heating or cooling rates, the effective holding times at each temperature are shorter.^{Reference Martín, Cock, García Junceda, Caballero, Capdevila and de Andrés20} Also, the nucleation rate is believed to be controlled by either the volume diffusion of carbon or by boundary diffusion of substitutional alloying elements, like Mn and Si.^{Reference Oliveira, Andrade and Cota17} When the heating or cooling rate increases, the amount of diffusion of elements, such as C, Mn, and Si, will decrease per unit of time, which is the result of a fall in the nucleation rate. Under these conditions, the new phase has less time to nucleate, which can result in that the heating/cooling rates increased. Consequently, the characteristic temperature of phase transformation is increased/decreased.

FIG. 3.

The dilatation curves at various rates of temperature change during heating and cooling: (a) V _c = 5 °C/min, (b) V _c = 10 °C/min, (c) V _c = 15 °C/min, and (d) V _c = 20 °C/min.

TABLE II.

The critical temperature for phase transition (in °C).

Despite this, it can be seen in Table II that the increasing or decreasing gradient of the characteristic temperatures decreased with the heating or cooling rates increased. This is due to that the greater heating or cooling rate is, the higher the superheat or subcooling per unit time is. So, the radius of critical nucleus will be decreased, which can aid the nucleation of the new phase during phase transformation at higher (heating) or lower (cooling) temperature. As a result, the gradient of those characteristic temperatures decreases. Similar effects of heating or cooling rates on critical temperatures of low-carbon microalloyed steel were also been previously reported in Refs. Reference San Martín, Rivera Díaz del Castillo and de Andrés9, Reference Martín, Cock, García Junceda, Caballero, Capdevila and de Andrés20, and Reference Clarke, Van Tyne, Vigil and Hackenberg23.

In addition, it can be seen in Table II that the phase transition temperature range (A _c3–A _c1) expanded as V _c increased at heating process, while the range for A _r3–A _rp contracted at cooling process. This phenomenon was also observed by Oliveira et al.^{Reference Oliveira, Andrade and Cota17} in studying the kinetics of austenite formation during continuous heating in a low-carbon steel. An investigation of the temperature range of α ↔ γ during the heating and cooling processes showed that the temperature range (A _c3–A _cp) of α → γ increased as V _c rose in the heating process, while the range of A _r3–A _r1 decreased. It means that the rates of nucleation and growth of austenite at heating are lower than that of ferrite at cooling. Moreover, while the temperature difference (A _c1–A _rp) increased as V _c rose, the gradient of this temperature difference then decreased as V _c continued to rise. The same phenomena also occurred for (A _c3–A _r3).

C. Effect of heating/cooling rate on thermal expansion coefficient

Figures 4(a) and 4(b) show the dilatation curves and differential thermal expansion coefficients as a function of temperature with a heating or cooling rate of 15 °C/min. It can be seen that α_T of the same temperature point was inconsistent in the heating and cooling process. In addition, the maximum │α_T│ was significantly different in the heating and cooling process, and this difference is defined as │Δα_T│, as shown in Fig. 4(b).

FIG. 4.

(a) ΔL/L ₀ versus T, (b) α_T versus T, and (c) $\overline \alpha$ vs. T on heating/cooling and with a test rate of temperature change of 15 °C/min.

To clarify the causes of this difference, the phase transformation rate of austenite K _γ is defined as,

(3)$${K_\gamma } = {{{\rm{d}}{f_\gamma }} \mathord{\left/ {\vphantom {{{\rm{d}}{f_\gamma }} {{\rm{d}}T}}} \right. \kern-\nulldelimiterspace} {{\rm{d}}T}}\quad ,$$

where df _γ is the increase in the austenite volume fraction as a function of the temperature change dT. f _γ is determined by a semiempirical method of dilatometric analysis, as described in more details below.

The atomic volume (V) of the sample is determined by a linear combination of the atomic volumes of each phase, which can be expressed as,

(4)$$V = {f_\alpha }{V_\alpha } + {f_\gamma }{V_\gamma } + {f_{\rm{P}}}{V_{\rm{P}}}\quad ,$$

where f _α, f _γ, and f _P are the volume fractions of ferrite, austenite, and pearlite, while V _α, V _γ, and V _P are the atomic volumes of ferrite, austenite, and pearlite, respectively, which are related to the lattice parameters by $V_\alpha = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}a_\alpha ^3$; $V_\gamma = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}a_\gamma ^3$; V ^P = (1 − ρ)V ^α + ρV ^θ, with V ^θ = 1/12(a _θb _θc _θ), and ρ is the cementite fraction in the pearlite.

The lattice parameters of ferrite, cementite, and austenite are given by^{Reference Lee, Clarke and Van Tyne6,Reference De Andres, Caballero, Capdevila and Alvarez7,Reference Onink, Brakman, Tichelaar, Mittemeijer, Van der Zwaag, Root and Konyer24}

(5)$${a_\alpha } = \left( {2.8863 + \mathop\sum\limits_i {{k_{i,\alpha }}{A_i}} } \right)\left[ {1 + 1.75 \times {{10}^{ - 5}}\left( {T - 800} \right)} \right]\quad ,$$

(6)$${a_\gamma } = \left( {3.6306 + 0.78 \times {\chi _\gamma } + \sum\limits_i {{k_{i,\gamma }}{A_i}} } \right)\left[ {1 + \left( {24.9 - 50 \times {\chi _\gamma }} \right) \times {{10}^{ - 6}}\left( {T - 1000} \right)} \right]\quad ,$$

(7)$${a_\theta } = 4.5234\left[ {1 + \left( {5.311 \times {{10}^{ - 6}} - 1.942 \times {{10}^{ - 9}} \times T + 9.655 \times {{10}^{ - 12}} \times {T^2}} \right) \times \left( {T - 293} \right)} \right]\quad ,$$

(8)$${b_\theta } = 5.0883\left[ {1 + \left( {5.311 \times {{10}^{ - 6}} - 1.942 \times {{10}^{ - 9}} \times T + 9.655 \times {{10}^{ - 12}} \times {T^2}} \right) \times \left( {T - 293} \right)} \right]\quad ,$$

(9)$${c_\theta } = 6.7426\left[ {1 + \left( {5.311 \times {{10}^{ - 6}} - 1.942 \times {{10}^{ - 9}} \times T + 9.655 \times {{10}^{ - 12}} \times {T^2}} \right) \times \left( {T - 293} \right)} \right]\quad ,$$

where k _i,α and k _i,γ are the coefficients for the effects of alloying element i on the ferrite and austenite lattice parameters in at.%, as listed in Table III.^{Reference Lee, Clarke and Van Tyne6}A _i is the at.% of alloy element i. χ^γ is the at.% of carbon in austenite.

TABLE III.

Coefficients for the effects of alloying elements on the lattice parameter (in nm).

Considering that the carbon content of the steel is 0.12 wt%, the carbon concentration in austenite w ^γ during this transformation can be estimated from the mass conservation as follows:

(10)$${w_\gamma } = {{\left( {0.12 - {f_\alpha }{w_\alpha }} \right)} \mathord{\left/ {\vphantom {{\left( {0.12 - {f_\alpha }{w_\alpha }} \right)} {{f_\gamma }}}} \right. \kern-\nulldelimiterspace} {{f_\gamma }}}\quad .$$

From Eq. (10), a value of w ^γ = 0.72 wt% at the critical temperature of A _cp(A _rp) can be obtained from f _α = 0.86 and f _γ = 0.14, and by assuming a constant carbon content in ferrite of w _α = 0.0218 wt%.

The at.% of C, Si, and Mn in austenite can be calculated as follows,

(11)$${\chi _\gamma } = {{{w_\gamma }} \over {12}}{\left[ {\sum\limits_i {{{{w_i}} \mathord{\left/ {\vphantom {{{w_i}} {{\omega _i}}}} \right. \kern-\nulldelimiterspace} {{\omega _i}}}} + {{\left( {100 - {w_\gamma } - \sum\limits_i {{w_i}} } \right)} \mathord{\left/ {\vphantom {{\left( {100 - {w_\gamma } - \sum\limits_i {{w_i}} } \right)} {56}}} \right. \kern-\nulldelimiterspace} {56}}} \right]^{ - 1}}\quad ,$$

(12)$${A_i} = {{{w_i}} \over {{\omega _i}}}{\left[ {{{{w_\gamma }} \over {12}} + \sum\limits_i {{{{w_i}} \mathord{\left/ {\vphantom {{{w_i}} {{\omega _i}}}} \right. \kern-\nulldelimiterspace} {{\omega _i}}}} + {{\left( {100 - {w_\gamma } - \sum\limits_i {{w_i}} } \right)} \mathord{\left/ {\vphantom {{\left( {100 - {w_\gamma } - \sum\limits_i {{w_i}} } \right)} {56}}} \right. \kern-\nulldelimiterspace} {56}}} \right]^{ - 1}}\quad ,$$

where w _i and ω_i are the weight percent and atomic weight of alloy element i, respectively.

The volume fractions of ferrite and cementite in pearlite are based on a Fe–Fe₃C phase diagram and the densities of the two phases at room temperature. The mass fractions for the two phases, F(%) and Fe₃C(%), are calculated from the phase diagram, as follows:

(13)$${\rm{F}}\left( \% \right) = {{\left( {6.69 - {\rm{C}}\% } \right)} \mathord{\left/ {\vphantom {{\left( {6.69 - {\rm{C}}\% } \right)} {6.69}}} \right. \kern-\nulldelimiterspace} {6.69}}\quad ,$$

(14)$${\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( \% \right) = {{{\rm{C}}\% } \mathord{\left/ {\vphantom {{{\rm{C}}\% } {6.69}}} \right. \kern-\nulldelimiterspace} {6.69}}\quad ,$$

where C% is the weight percentage of carbon in the steel, and the C% in ferrite is assumed to be zero at room temperature.

The volume fractions of ferrite and cementite in pearlite are then equal to

(15)$${f_{\alpha {\rm{P}}}} = {{{{{\rm{F}}\left( \% \right)} \mathord{\left/ {\vphantom {{{\rm{F}}\left( \% \right)} {{\rho _\alpha }}}} \right. \kern-\nulldelimiterspace} {{\rho _\alpha }}}} \over {{{{\rm{F}}\left( \% \right)} \mathord{\left/ {\vphantom {{{\rm{F}}\left( \% \right)} {{\rho _\alpha }}}} \right. \kern-\nulldelimiterspace} {{\rho _\alpha }}} + {{{\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( {\rm{\% }} \right)} \mathord{\left/ {\vphantom {{{\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( {\rm{\% }} \right)} {{\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}}}} \right. \kern-\nulldelimiterspace} {{\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}}}}}\quad ,$$

(16)$${f_{\theta {\rm{P}}}} = {{{{{\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( {\rm{\% }} \right)} \mathord{\left/ {\vphantom {{{\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( {\rm{\% }} \right)} {{\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}}}} \right. \kern-\nulldelimiterspace} {{\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}}}} \over {{{{\rm{F}}\left( \% \right)} \mathord{\left/ {\vphantom {{{\rm{F}}\left( \% \right)} {{\rho _{\rm{\alpha }}}}}} \right. \kern-\nulldelimiterspace} {{\rho _{\rm{\alpha }}}}} + {{{\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( {\rm{\% }} \right)} \mathord{\left/ {\vphantom {{{\rm{F}}{{\rm{e}}_3}{\rm{C}}\left( {\rm{\% }} \right)} {{\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}}}} \right. \kern-\nulldelimiterspace} {{\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}}}}}\quad ,$$

where ${\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}}$ and ρ_α are the mass densities of cementite and ferrite, as follows,^{Reference De Andres, Caballero, Capdevila and Alvarez7}

(17)$${\rho _{{\rm{F}}{{\rm{e}}_3}{\rm{C}}}} = 7686.45 - 6.63 \times {10^{ - 2}} \times T - 3.12 \times {10^{ - 4}} \times {T^2}\quad ,$$

(18)$$\!{\rho _\alpha } = 7875.95 - 0.2975 \times T - 6.62 \times {10^{ - 5}} \times {T^2}\quad .$$

Assuming that the pearlite ↔ austenite and ferrite ↔ austenite transformations are carried out with successive steps of heating and cooling, the volume fractions of ferrite (f _α) and pearlite (f _P) during the transformations are given by^{Reference Suh, Oh, Han and Kim8}

(19)$${f_\alpha } = {{V - {V_\gamma }} \over {{V_\alpha } - {V_\gamma }}}\;{\rm{and}}\;\left( {{f_\alpha } + {f_\gamma } = 1} \right)\quad ,$$

(20)$${f_{\rm{P}}} = {{V - {V_\gamma } + {f_\alpha }\left( {{V_{\rm{P}}} - {V_\alpha }} \right)} \over {{V_{\rm{P}}} - {V_\alpha }}}\;{\rm{and}}\;\left( {{f_\alpha } + {f_\gamma } + {f_{\rm{P}}} = 1} \right)\quad .$$

If the volume change associated with the phase transformation is isotropic and relatively small, the change in the atomic volume of the sample is related to the change in length, as measured by dilatometry, as follows,^{Reference Suh, Oh, Han and Kim8}

(21)$${{\Delta V} \over {{V_0}}} = {{V - {V_0}} \over {{V_0}}} = {V \over {{V_0}}} - 1 = {{3\Delta L} \over {{L_0}}}\quad ,$$

and

(22)$$V = k{V_0}\left( {{{3\Delta L} \over {{L_0}}} + 1} \right)\quad ,$$

where V ₀ and ΔV are the initial atomic volume and change in atomic volume, respectively, and k is a scaling factor. Ideally the value of this scaling factor is 1, but a number of issues could cause it to be different, such as inaccuracies in the measuring system, or nonisotropic contraction/expansion in the sample due to the presence of texture and transformation induced plasticity effects.

Eqs. (19) and (20) were numerically solved for f _α, f _γ, and f _P in temperature steps of 1 °C. Figure 5 shows the evolution of the fractions of each phase during heating and cooling at a temperature change rate of 15 °C/min.

FIG. 5.

The evolution of the fractions of each phase: (a) heating and (b) cooling.

The phase transformation rate K _γ of austenite is calculated using Eq. (3), as shown in Fig. 6. It can be seen that the phase transformation speeds (K _γ) of ferrite ↔ austenite are different in the heating and cooling process. The K _γ value on heating decreases and reaches the minimum value at 885 °C (corresponding to peak 2 in Fig. 4), and then subsequently increases along with the temperature on further heating. However, the phase transformation speed of austenite (K _γ) to ferrite increases and reaches the maximum value at 810 °C (corresponding to peak 2′ in Fig. 4), and then decreases as the temperature falls on cooling. The phase fraction of austenite [f _γ = 0.705, in Fig. 5(a)] on heating is smaller than that on cooling [f _γ = 0.802, in Fig. 5(b)]. It means that the transformation rate difference resulted in the thermal expansion coefficient difference.

FIG. 6.

df _γ/dT versus T on heating and cooling.

The evolution of the C, Si, and Mn contents in austenite are calculated using Eqs. (10)–(12), as shown in Fig. 7. It can be seen in Fig. 7(a) that the carbon content in austenite rapidly decreases in the initial phase transformation stage and then slowly declines in the later stage on heating. At the same time, the Si and Mn contents in austenite rapidly increase at the initial phase transformation stage and then slowly increase in the later stage. Figure 7(b) shows that the carbon content in austenite increases slowly in the initial phase transformation stage and then rapidly increases as the temperature declines, with the Si and Mn contents in austenite slowly decreasing in the initial phase transformation stage and then rapidly decreasing in the later late stage.

FIG. 7.

The evolution of the C, Si, and Mn contents in austenite: (a) heating and (b) cooling.

In addition, the average linear expansion coefficient, $\overline \alpha = {{\left[ {{{\left( {L - L_0 } \right)} \mathord{\left/ {\vphantom {{\left( {L - L_0 } \right)} {\left( {T - T_0 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T - T_0 } \right)}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\left( {L - L_0 } \right)} \mathord{\left/ {\vphantom {{\left( {L - L_0 } \right)} {\left( {T - T_0 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T - T_0 } \right)}}} \right]} {L_0 }}} \right. \kern-\nulldelimiterspace} {L_0 }}$,^{Reference James, Spittle, Brown and Evans5} is shown in Fig. 4(c), with the temperature changing at a rate of 15 °C/min. By comparing Figs. 4(b) and 4(c), it can be seen that the expansion coefficients are significantly different between $\overline \alpha$ and α_T. This difference is because the increase in length ΔL has been averaged based on the average linear expansion coefficient, $\overline \alpha = {{\left( {{{\Delta L} \mathord{\left/ {\vphantom {{\Delta L} {\Delta T}}} \right. \kern-\nulldelimiterspace} {\Delta T}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Delta L} \mathord{\left/ {\vphantom {{\Delta L} {\Delta T}}} \right. \kern-\nulldelimiterspace} {\Delta T}}} \right)} {L_0 }}} \right. \kern-\nulldelimiterspace} {L_0 }}$, which obscures the phase transformation and expansion characteristics at some of the temperature range. By contrast, the differential thermal expansion coefficient α_T presents the amount of expansion between the two test temperature points, more accurately than the average linear expansion coefficient $\overline \alpha$ is able to. Therefore, if the average linear expansion coefficient $\overline \alpha$ is used to simulate the stress/strain, then this will produce misleading results.

Figure 8 shows the differential thermal expansion coefficient as a function of temperature. There is a linear relationship between the thermal expansion coefficient and temperature during the phase reign of pearlite + ferrite (below the A _c1 and A _rp) and single austenite (above the A _c3 and A _r3), and the heating or cooling rate has little effect on the expansion coefficient. The thermal expansion coefficients (α_T) at the phase reign of pearlite + ferrite and single austenite were about 1.61 × 10⁻⁵ and 2.25 × 10⁻⁵ °C⁻¹, respectively. This α_T of single austenite measured in this study agrees well with those reported in the previous works, such as 2.09 × 10⁻⁵,^{Reference Onsøien, M'hamdi and Mo25} 2.31 × 10⁻⁵,^{Reference Petrov, Kestens and Houbaert26} 2.065 × 10⁻⁵,^{Reference Caballero, Capdevila and García De Andrés27} and 2.1 × 10⁻⁵ °C⁻¹.^{Reference Long, Dong, Chen, Zhang and Zhang18} As can be seen in Fig. 8, when A _c1 (A _rp) < T < A _c3 (A _r3), the relationship between the thermal expansion coefficient and temperature is significantly nonlinear, and the value of α_T is between −6.31–2.25 × 10⁻⁵ °C⁻¹ for the different rates of temperature change. More specifically, the α_T of the pearlite dissolution process (stage I) is between −0.90–1.61 × 10⁻⁵ °C⁻¹ during the heating process, and that of the austenization process (stage II) is between −2.49–2.25 × 10⁻⁵ °C⁻¹. At the same time, the α_T value of the eutectoid reaction process (stage I) is between 0.52–1.61 × 10⁻⁵ °C⁻¹ during the cooling process, and that of the austenite decomposition process (stage II) is between −6.31–2.25 × 10⁻⁵ °C⁻¹. The peak α_T values for the pearlite dissolution process (heating) or eutectoid reaction (cooling) are significantly smaller than those for the austenization process (heating) or ferrite precipitate process (cooling).

FIG. 8.

Differential thermal expansion coefficients as a function of temperature: (a) heating and (b) cooling.

By comparing Figs. 8(a) and 8(b), it can be seen that the α_T values of the same temperature point were inconsistent in the heating and cooling processes. The difference (│Δα_T│) of the maximum │α_T│ between the heating and cooling processes increases along with rate of temperature change. Specifically, with the V _c of 5, 10 , 15, and 20 °C/min, the values of │Δα_T│ are 0.70 × 10⁻⁵ °C ⁻¹, 1.43 × 10⁻⁵, 3.82 × 10⁻⁵, and 4.17 × 10⁻⁵ °C ⁻¹, respectively.