Using the Physically Based Constitutive Model and Processing Maps to Understand the Hot Deformation Behavior of 2304 Lean Duplex Stainless Steel

Abstract

The hot deformation behavior of 2304 lean duplex stainless steel was investigated by means of processing maps and physically based constitutive modeling in the temperature range of 850 °C to 1050 °C and strain rate of 0.1 to 15 s⁻¹. For all the processing maps developed for strains of 0.1 to 0.6, dynamic restoration mechanisms were efficient in preventing flow instability at low strain rates (1 to 4 s⁻¹) in the studied temperature range. Hot deformation at medium strain rates of 5 to 7 s⁻¹ results in the risk of flow instability. However, for typical industrial purposes, there is a possibility of hot working at higher strain rates (10 s⁻¹ and above) at lower temperatures without the risk of instability by promoting dynamic recrystallization in the austenitic phase of the steel. The flow behavior of the steel can be accurately modeled by coupling the Estring-Mecking constitutive equation (for the work hardening and recovery region) with the Avrami model, which captures the dynamic recrystallization region.

Appendices

Appendix 1: Derivation of the W.H. relations

According to the modified representation of the Estrin-Mecking model:

$$ \theta \sigma = A - B\sigma^{2} $$

(A1)

Integrating equation results in the equation for the flow stress in the WH + DRV regime according to the Estrin-Mecking modification given by:

$$ \sigma^{{{\text{EM}}}} = \left[ {\sigma_{{{\text{sat}}}}^{2} + \left( {\sigma_{0}^{2} - \sigma_{{{\text{sat}}}}^{2} } \right){\text{exp}}\left( { - 2{\text{B}}\varepsilon } \right)} \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$

(A2)

where

$$ \sigma_{{{\text{sat}}}} = \sqrt{\frac{A}{B}} $$

(A3)

The constants in the above equation can be determined from the plot of product of the work hardening rate and stress, θ_σ, against the square of the stress, σ². From this plot A is the y-intercept and B the slope of the linear approximation of the plot (Figure A1).

Fig. A1

Plot of σθ against σ² to determine the values of A and B at a strain rate of 15 s⁻¹

Appendix 2: Derivation of the constants in the Avrami softening model

$$ X = 1 - \exp \left[ { - r\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{q} } \right] $$

(A4)

where the fractional softening

$$ X_{{\text{s}}} = \left( {\frac{{\sigma_{{{\text{sat}}}} - \sigma }}{{\sigma_{{{\text{sat}}}} - \sigma_{{{\text{ss}}}} }}} \right) $$

(A5)

$$ 1 - X = \exp \left[ { - r\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{q} } \right] $$

(A6)

$$ \ln (1 - X) = - r\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{q} $$

(A7)

$$ - \ln (1 - X) = r\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{q} $$

(A8)

$$ {\text{ln}}\left( { - \ln (1 - X} \right)) = {\text{ln}} r + q {\text{ln}} \left( {\varepsilon - \varepsilon_{p} } \right) $$

(A9)

Thus, the values of the constants r and q will be obtained from the plot of ln(−ln(1−X)) against ln(ε−ε_p) where ln r is the y-intercept and q the slope of the linear approximation of the corresponding plot. From the graph below ln r = 2.1875 and hence r = 8.9 and q = 1.7 (Figure A2).

Fig. A2

Plot of ln(−ln(1−X)) against ln(ε−ε_p) to determine the values of r and q at a strain of 0.8, a strain rate of 15 s⁻¹ and a temperature range of 850 °C to 1050 °C