Modeling of Work Hardening During Hot Rolling of Vanadium and Niobium Microalloyed Steels in the Low Temperature Austenite Region

Abstract

This work extends the application of the well-established Estrin and Mecking (EM) work-hardening model in unstable low temperature austenite region. The interaction between work hardening, recovery and softening attributed to recrystallization and transformation to ferrite under dynamic conditions is considered. Experimental parameters were varied to study the effects of strain, strain rate and temperature during hot rolling in the low temperature austenite region. Hot compression tests were performed two microalloyed steels—one containing V and the other Nb—at strain rates between 0.1 and 10 s⁻¹ over a temperature range of 750-1000 °C. A model is presented that describes the influence of dynamic recovery on flow behavior in the unstable austenite region. The modified work-hardening model incorporates an additional fitting parameter to the EM model and is dependent on the recovery and softening rates. The new model improved prediction in the unstable austenite region, while the original EM model gave better correlation at relatively higher temperatures when dynamic recrystallization is dominant or at relatively lower temperatures when only dynamic recrystallization to ferrite was the softening mechanism.

Appendix: Derivation of the Estrin and Mecking Work-Hardening Model

Estrin and Mecking (Ref 20) modeled the flow curves up to the peak stress based on the evolution of dislocation density from concurrent work hardening and dynamic recovery (DRV) only. In this approach, the evolution of dislocation density with strain was determined as the sum of differential hardening and softening terms as follows (Ref 2, 20):

$$\frac{{\hbox{d}}\rho }{{\hbox{d}}\varepsilon } = \left( {\frac{{\rm{d}}\rho }{{\hbox{d}}\varepsilon }} \right)^{ + } + \left( {\frac{{\hbox{d}}\rho }{{\hbox{d}}\varepsilon }} \right)^{ - } ,$$

where the first term represents the work-hardening part and the second term, the softening due to DRV. The terms in the above equation for dependence of the dislocation density ρ on plastic strain ɛ can be replaced as follows (Ref 20):

$$\frac{{\hbox{d}}\rho }{{\hbox{d}}\varepsilon } = h - r\rho ,$$

where h is the athermal work-hardening rate ($h = r\frac{{\sigma_{\rm sat}^{2} }}{{\left( {\alpha \mu b} \right)^{2} }}$) and r denotes the rate of dynamic recovery at a given temperature and strain rate, independent of the strain.

dɛ can be expressed as: $d\varepsilon = \frac{{\hbox{d}}\rho }{h - r\rho }$

and dρ in turn, can be defined as: $d\rho = \frac{{{\hbox{d}}\left( {h - r\rho } \right)}}{ - r}$

Combining the two equations above leads to: $- rd\varepsilon = \frac{{{\hbox{d}}\left( {h - r\rho } \right)}}{h - r\rho }$

the integration of which gives: ln (h − rρ) = − rɛ + C

The exponential of the above relationship and making dislocation density ρ the subject:

$$\rho = \frac{{h - \exp \left( { - r\varepsilon + C\varepsilon } \right)}}{r} = \frac{h}{r} = \frac{{C_{1} }}{r}\exp \left( { - r\varepsilon } \right)$$

By employing the limiting condition ɛ = 0 and ρ = ρ ₀ (the value at yielding), that of C ₁ can be derived as h − rρ ₀. This leads to the expression:

$$\rho = \frac{h}{r} - \frac{{\left( {h - \rho_{0} r} \right)}}{r}\exp \left( { - r\varepsilon } \right)$$

or

$$\rho = \rho_{0} \exp \left( { - r\varepsilon } \right) + \frac{h}{r}\left[ {1 - \exp \left( { - r\varepsilon } \right)} \right]$$

The dislocation density ρ is converted into σ using the expression $\sigma = M\alpha G_{M} b\sqrt \rho$ so that ρ and ρ ₀ can be replaced by (σ/MG _M b)² and (σ ₀/MG _M b)², respectively. Under these conditions, the flow stress can be given by the following relationship in terms of the plastic strain:

$$\sigma = \left[ {\sigma_{0}^{2} \exp \left( { - r\varepsilon } \right) + \left( {M\alpha G_{M} b} \right)^{2} \frac{h}{r}\left( {1 - \exp \left( { - r\varepsilon } \right)} \right)} \right]^{1/2}$$

where σ ₀ and σ _sat denote the yield (initial) and saturated stress defined as (αMG _M b)² ρ ₀ and (αMG _M b)²(h/r), respectively, α is a shape factor in the order of unity, M is the Taylor factor (3.07 for FCC materials), G _M is the shear modulus, b is the magnitude of Burger’s vector (b was evaluated to be 0.2594 nm for a carbon content of 0.07% and at 1000 °C (Ref 11)), and ρ ₀ is the initial dislocation density.

When ɛ tends to infinity, the above equation can be rewritten as follows:

$$\sigma \approx M\alpha G_{M} b\sqrt {\frac{h}{r}}$$

The expression $M\alpha G_{M} b\sqrt {\frac{h}{r}}$ can be defined as the dynamic recovery saturation stress, σ _sat which allows the work-hardening stress curve to be written in the following formalization (Ref 11, 23):

$$\sigma = \left[ {\sigma_{\rm sat}^{2} - \left( {\sigma_{\rm sat}^{2} - \sigma_{0}^{2} } \right)\exp \left( { - r\left( {\varepsilon - \varepsilon_{0} } \right)} \right)} \right]^{1/2}$$

Further simplification is achieved by the assumption that ρ ₀ ≈ 0 when compared to the stored dislocation density during hot deformation. An assumption is also that it is more pertinent to a low SFE material such as austenite characterized by a sluggish DRV. Neglecting ɛ ₀ in this case yields the following expression for the flow curve:

$$\left( {\sigma_{\rm sat}^{2} - \sigma_{0}^{2} } \right)\exp \left( { - r\varepsilon } \right) = \sigma_{\rm sat}^{2} - \sigma^{2}$$

Considering the plastic flow curve only after yielding (i.e., σ ₀ = 0), the simplified work-hardening curve, therefore, can be rewritten as follows:

$$\sigma = \sigma_{\rm sat} \left( {1 - \exp \left( { - r\varepsilon } \right)} \right)^{0.5}$$

The differentiation of which gives:

$$\frac{{\hbox{d}}\sigma }{{\hbox{d}}\varepsilon } = 0.5\sigma_{\rm sat} r\exp \left( { - r\varepsilon } \right)\left( {1 - \exp \left( { - r\varepsilon } \right)} \right)^{ - 0.5}$$

Substituting 1 − (σ/σ _sat)² for exp (− rɛ), the differential equation can be written as:

$$\theta = \frac{{\hbox{d}}\sigma }{{\hbox{d}}\varepsilon } = 0.5r\left( {\frac{{\sigma_{\rm sat}^{2} - \sigma^{2} }}{\sigma }} \right)\;{\text{or}}\;\sigma \theta = \sigma \frac{{\hbox{d}}\sigma }{{\hbox{d}}\varepsilon } = 0.5r\left( {\sigma_{\rm sat}^{2} - \sigma^{2} } \right)$$

From the above expression, the plots of σθ versus σ ² are linearly fitted from which the slope m and the abscissa and intercept are − 0.5r and − 0.5rσ ²_sat , respectively. Thus, r is the value of the recovery parameter which specifies the “curvature” of the dynamic recovery curve (Ref 11).