Discussion on the microstructural transients during strain reversal based on the effective equivalent strain concept

Abstract

A strain reversal applied for both under hot and cold workings produces a microstructural transient which manifests in different ways and at different scales. The evolution of the dislocation network during strain reversal is the result of the competition between two mechanisms. The first mechanism corresponds to the partial untangling and recovery of previously created dislocation network and the second one is associated to the build-up compatible with the current deformation condition. This study proposes a phenomenological formulation based on an effective equivalent strain concept to describe, in a simple way, the several experimental manifestations of the microstructural transient at grain and intragranular scales and its effect on the static recrystallisation kinetics.

Appendix: Parametrisation study

Present formulation of the effective equivalent strain needs two parameters, as it can be concluded from the whole set of equations from Eqs. (1–7). The first one is the λ parameter while the second one is the proportionality factor A connecting pre-strain and ε₀. Aiming to analyse the effect of these two parameters on the equivalent effective strain predictions, two specific points on the ε_eff–ε_rev plot have been regarded: the minimum of ε _eff (ε ^A_rev , ε ^A_eff ) and the strain required to achieve the same equivalent strain as just before strain reversal ε ^B_rev (ε _eff = ε_pre), see Fig. 12a. The minimum of the equivalent effective strain during strain reversal must satisfy:

Fig. 12

a Typical ε_eff–ε_rev plot wherein the characteristic points (ε_min and ε_rev(ε _eff = ε_pre)) are indicated. Variation of b ε_min and c ε _eff (ε_min) with λ for three different A. d Evolution of ε_rev(ε _eff = ε_pre) for two pre-strains, ε_pre1 and ε_pre2, and two different A (0 and 0.2)

$$ \lambda \left( {\varepsilon_{{\rm{pre}}} - \varepsilon_{0} } \right){\rm{e}}^{{ - \lambda \varepsilon_{\hbox{Min} } }} + \left( {1 - \eta \varepsilon_{\hbox{Min} } } \right){\rm{e}}^{{ - \eta \varepsilon_{\hbox{Min} } }} - 1 = 0 $$

(21)

In Fig. 12b, c, the evolution in the position of the minimum ε ^A_rev and its value ε ^A_eff has been plotted as a function of the λ parameter for three different A. From these plots several conclusions can be extracted. First, ε ^A_rev strongly depends on the λ parameter for low λ values, whereas at larger values, ε_min converges progressively irrespective of A. As A increases, the strain needed to reach the minimum shortens for any λ value. Interestingly, the effective equivalent strain evaluated at the minimum ε ^A_eff decreases monotonically with λ and it increases monotonically with A.

Concerning ε ^B_rev (ε _eff = ε_pre), the variation of this reverse strain with λ for two pre-strains and two A values has been plotted in Fig. 12d. ε ^B_rev (ε _eff = ε_pre) increases monotonically with the pre-strain and it diminishes with A for all λ. It is particularly meaningful that the ε ^B_rev (ε _eff = ε_pre) saturates at lower λ values as the pre-strain is enlarged. The application of different λ–A pairs for describing the experimental data reported in Ref. [6] are shown in Fig. 13. In this particular case, at the low reverse strain region λ_opt = 50 describes more accurately experimental data compared to λ = λ_opt/2 = 25 or λ = 2λ_opt = 100, whereas at large reverse strains, $A \ne 0$ yields larger deviations from experimental data than $A = 0$.

Fig. 13

Application of the different λ–A in ε_eff for measuring dislocation densities through Eq. (10). Assessment of the applicability of the present formulation

In order to generalise this formulation to other hot working conditions, the variation in the pre-strain and/or the changes in the Zener–Hollomon parameter must be taken into account. In this line, further ongoing work is being undertaken.