Work-hardening of ferrite and microstructure-based modelling of its mechanical behaviour under tension

Abstract

Current work-hardening theories have mainly focused on centred cubic structures, e.g. copper and aluminium, and no particular attention has been paid to body-centred cubic materials such as ferrite. This research investigates the work-hardening at room temperature of a set of extra-low and ultra-low carbon steels deformed under tension. Within the homogeneous deformation range, from the end of yield point elongation to the onset of necking, the work-hardening rate of ferrite decreases with stress (strain). A one-parameter microstructure-based model has been applied to predict the mechanical behaviour of ferrite. This model provides a simple tool capable of prediction with only simple input parameters: steel composition and ferrite grain size.

Introduction

The easiest and most extended approach to predict basic tensile properties of steels with ferrite and ferrite–pearlite microstructures entails a number of empirical equations. One of the most popular expresses the yield stress σ_y as a function of different strengthening contributions [1], [2], [3], [4], [5], [6]:

$${\sigma }_{\text{y}}={\sigma }_{\text{f}}+{\sigma }_{\text{s}}+{\sigma }_{\text{ss}}+{\sigma }_d$$

where σ_f is the Peierls–Nabarro stress or lattice friction stress, σ_ss and σ_s are the short-range internal stresses produced, respectively, by interstitial and substitutional elements in solid solution, and σ_d is the contribution of ferrite grain size d.

It is generally assumed that an element going into solid solution produces an increase in strength proportional to its concentration and independent of the presence of other elements [1]. However, the proportionality constants used to express the particular contribution by the unit weight of each element to σ_s and σ_ss are subjected to some degree of variation depending on the source [7], [8], [9], [10], [11]. Substitutional elements produce a moderate strengthening effect, as compared to interstitials like carbon and nitrogen, whose contribution is also enhanced by their strong interaction with dislocations [12], [13].

The grain microstructure contribution for polygonal ferrite follows a Hall–Petch relationship [1], [2], [3], [4], [5]:

$${\sigma }_d={\sigma }_{\text{HP}}+k_{\text{HP}}{d}^{-1/2}$$

in which σ_HP and k_HP are constants and k_HP ≅ 15–18 (MPa mm^0.5) when d is expressed as the mean linear intercept [5]. Similar equations can also be found to express tensile strength. However, the stress–strain (σ–ε) curve is required for a number of applications that can be described by simple empirical equations [14], [15], [16], [17], from which the most broadly applied is the Hollomon equation:

$$\sigma =k_{\text{H}}{\epsilon }^n$$

in which k and n are independent of strain, ε, and can be deduced for ferrite using the following empirical correlations obtained by Morrison at moderate ferrite grain sizes [18]:

$${\text{k}}_{\text{H}}(\text{MPa})=448+436\ast d^{-0.5}\text{and}n=\frac{5}{10+d^{-0.5}}(d\text{in mm})$$

Similar approaches can also be found that extended these equations to include other effects such as composition [19], [20] and precipitates.

More recently, attempts have been made by various authors to develop alternative formulations based on physical principles [11], [21], [22], [23]. This research investigates the work-hardening of ferrite, and develops a microstructure-based predictive model describing the mechanical behaviour under tension of ultra-low and extra-low carbon steels with different ferrite grain sizes.