**1. Introduction**

Modern thermomechanically processed steels can be rolled and cooled to a variety of microstructures with properties tailored to many kinds of applications. To minimise scatter in microstructures and properties it is beneficial to have phase transformation models that are as accurate as possible. To be able to predict and understand the onset of phase transformations occuring during the complex cooling path of hot rolled steel for different compositions, a quantitative model is needed. To give a realistic estimate, the model should take into account the severe mechanical deformation occurring in hot rolling prior to cooling. In addition, it should be possible to extend the model and adapt it to specific steels and mechanical treatments prior to cooling. It is also desirable that the input data required for the model are easy to obtain, for example, by constant cooling rate experiments.

Several phase transformation models that can be used for simulations along arbitrary cooling paths have been introduced earlier, for example, references [1–11]. However, there is currently no method that includes the effect of austenite deformation on the transformation start for different steel compositions and which could be used to calculate the transformation start for arbitrary cooling path. Since the nucleation phenomenona during the onset of transformation is different from the successive growth stage, several models separate the growth kinetics of the phase transformation from the onset stage by considering an incubation time or delay for the transformation start [3–5,7,8,10,11]. However, these models do not consider the effect of deformation prior to cooling [6–9], or they are specific to a certain steel composition, or they only take in to account the effect of a few alloying elements [2–5,10,11].

Also, detailed models, such as [12–14], which include the description of the steel microstructure, can be used to calculate phase transformation phenomena. These detailed models include a number

of parameters which can be either obtained experimentally [13,14] or calculated using theory [12]. Such detailed models could benefit from data that can be obtained directly from linear cooling rate experiments in order to find the range of possible values for the parameters.

The method for calculating the start time of transformation from temperature–time transformation (TTT) diagrams has been presented in several articles, e.g., [1,6–8,15], which could be used to calculate the transformation start by applying Scheil's additivity rule for an arbitrary cooling path. However, it has been shown that using such isothermal transformation data to calculate the transformation start for cooling conditions by applying the additivity rule can give incorrect estimates [16,17]. For cooling, the application of the so-called ideal (or true) incubation time [18,19] gives a better estimate for the transformation start for an arbitrary cooling path [20], as it gives exactly the correct continuous cooling transformation (CCT) diagram for constant cooling rates when Scheil's additivity principle is applied.

We present a computational method to calculate an estimate for the ferrite phase transformation start of an arbitrary cooling path and for varying steel compositions containing the following alloying elements: C, Si, Mn, Cr, Cu, Ni, Mo, V, Ti, B, and Nb. The model can be easily adapted for new steels and thermomechanical treatments by using constant cooling rate CCT data as the input for the model. The transformation start time includes the time required for the formation of critical-sized nuclei (incubation time) as well as the time required for the growth of the nuclei to the volume fraction required for the detection of the phase transformation start. The method comprises three parts as follows: (1) Calculation of the CCT start time for different steel compositions and phases using the results presented in [21,22] for steels subjected to strain relevant to hot rolling procedures; (2) construction of the so-called ideal TTT diagram [18–20,23,24] from a constant cooling rate CCT diagram; and (3) application of Scheil's additivity principle to calculate the phase transformation start for an arbitrary cooling path after the ideal TTT diagram has been calculated [18]. We show how it is used to quantitatively compare the manner in which different alloying elements affect the austenite to ferrite phase transformation start time and the corresponding activation energy for a given steel composition. The same calculation is done for steel which has been additionally strained below the no-recrystallization temperature, and the results are compared. We also calculate an estimate for the phase transformation start for different kinds of cooling paths.

Since the model presented in this work is based on experimental CCT diagrams [21], it can be easily adapted and extended for different steels when there is a need to obtain more detailed estimates for given steel compositions or mechanical treatments prior to cooling. If constant cooling rate CCT data is analyzed, it can be used as the basis of this method, and the corresponding results, as presented in this article, can be obtained. We separately analyze the austenite to bainite transformation onset and present the results in [25,26].

## **2. Theory**
