The most commonly known feature of the grain configuration in metallic microstructures is the grain size, which is also the most commonly mentioned factor in relation to mechanical properties among all characteristics of grains. Almost 70 years ago, in the early 1950s, Hall and Petch [1,2] proposed the by now well-known Hall-Petch relation, which gives the relation of yield strength σ_y and average grain size μ_d at ambient temperature as:

where σ_0y is known as the friction stress for the dislocation movement within the grains in a polycrystalline microstructure, while k_y is the Hall-Petch slope describing the local stress needed at the grain boundary for plastic flow [3]. Since the early stage of the development of the Hall-Petch relation, the majority of studies focused only on the mean grain size while the effect of the grain size distribution is not well recognized. Researchers either neglect the additional information that may be present in the grain size distribution, or assume one kind of grain size distribution function like log-normal, which is frequently mentioned [[4], [5], [6], [7]]. The coefficient of variation C_V = s/μ_d of the grain size distribution was proposed by Kurzydłowski and Bucki [8] to represent the grain size distribution, which involves the standard deviation (s) and mean (μ_d) grain size. They state that, beside the mean grain size, the grain size range is important to model mechanical properties. Berbenni et al. [9], Nicaise et al. [10] consider a relative dispersion parameter, defined as p_dis = (d_max - d_min)/μ_d, which includes the influence of the grain size range, i.e. largest grain size d_max and smallest grain size d_min. But just some qualitative trends are proposed between the relative dispersion parameter and mechanical behavior without a clear quantitative relationship. Lehto et al. [11] propose to use the volume-weighted average grain size d_v, which indicates that the influence of each grain on the strength of the material is proportional to the volume of the grain. The volume-weighted average grain size d_v is defined and included in a Hall-Petch-like relation as follows:

where V_T is the total volume of the material, n is the number of grains and V_i is the volume of grains of the grain size d_i. Lehto et al. [11] found that the volumeweighted average grain size has a relation with mean grain size as Eq. (3) shows, where c and f are constants obtained from linear regression of experimental data, which gives c ≈ 1.0. The grain size dispersion parameter here is modified from Berbenni et al. [9] as: ${{p}_{di{{s}_{\text{m}}}}}$=(P_99%-P_1%)/μ_d. P_99% and P_1% are the grain sizes at 99% and 1% probability levels, which is more robust than d_max and d_min. The modified Hall-Petch equation is proposed as Eq. (4) by combining Eqs. (1) and (3). Similar to the volume-weighted average grain size, Raeisinia and Sinclair [12] propose a representative grain size D_R, with relation to mean grain size as follows:

where S is the standard deviation of the assumed log-normal grain size distribution function. Eq. (6) is the modified Hall-Petch equation with the representative grain size incorporated. It helps in explaining why the Hall-Petch slope changes in some cases. Here the representative grain size is calculated based on the assumption that the grain size follows a log-normal distribution function.

General principle for plastic deformation is the movement of dislocations. Materials are strengthened if the dislocation motion is hindered, which can be achieved by different methods, such as grain size reduction, precipitation, solid solution anchoring dislocations and strain hardening by increasing the dislocation density. However, due to the complexity of accurate dislocation density determination, the quantitative relations between dislocation density and mechanical properties are less validated in the literature compared to other microstructure features, such as grain size. Taylor [13] proposed a relation between the 0.2 % proof stress σ_y and the square root of the dislocation density ${\rho}$ as follows:

where α is a constant, which usually ranges from 0.15 to 0.4, M is the Taylor factor, G is the shear modulus and b is the length of the Burgers vector. Cong and Murata studied the dislocation density in low carbon steel in relation to Vickers hardness, which shows that the Vickers hardness increases with the increase of dislocation density and carbon content. This is due to the strengthening of martensite phase with high dislocation density as the carbon content increases and the solid solution strengthening effect of carbon [14]. Kehoe and Kelly [15] determine the dislocation density ρ from TEM micro-graphs of thin foils, for ferritic and martensitic Fe-C alloys at 250 K and 77 K, which shows a clear linear relationship between σ_y and ${\rho}$. Arechabaleta et al. [16] present an accessible and easy way to determine the yield strength and dislocation density, which gives an accurate experimental validation of the Taylor equation and physical interpretation of the parameter α.

It is shown above that many researchers have shown that grain size distribution does have pronounced influence on the strength of materials. But all the mentioned studies concerning grain size distribution have their drawbacks. None of them actually indicates which parameters are the most important to describe the strength effect of the grain size distribution. In the present paper, in order to establish a relationship between grain size distribution, dislocation density and mechanical properties, interstitial free (IF) steel is chosen to be the steel type to manipulate the grain size distribution and dislocation density, minimizing the influence from other aspects, such as carbon content, precipitates and different phases. The ferrite grain size distribution and dislocation density are controlled through different deformation and heat treatment routes. The study results in a combined relation of mean grain size, grain size distribution and dislocation density with hardness.

The chemical composition of the interstitial free steel is shown in Table 1. Different heat treatment routes were applied on the as received hot-rolled IF steel plate with sample dimensions of 10 mm × 4 mm × 2.5 mm. The accurate temperature control during the heat treatment was achieved by a Bähr dilatometer model DIL 805A/D. The detailed heat treatment routes can be seen from Table 2, where the heat treatment temperature, time and cooling rate are indicated. ‘Q’ stands for quenching to room temperature, as measured in the dilatometer, during which the cooling rate above 300 °C is around -200 °C s^-1. Heating and cooling rate during the heat treatment, if not stated in the table, were controlled at 10 °C s^-1 and -30 °C s^-1, respectively.

In order to introduce the influence of dislocations, cold rolling was performed on the 4 mm thick hot-rolled plate. The plate was cold rolled to the final thickness of 3.00 mm, 1.70 mm and 1.13 mm, causing a strain of 25 %, 58 % and 72 %, respectively. The 1.13 mm cold rolled plate was subjected to annealing at 400 °C, 450 °C and 600 °C for time ranging from 1.5 min to 75 min, followed by quenching to room temperature with helium in the dilatometer.

The heat treated samples were mechanically ground with P800, P1200 and P2000 grit abrasive papers, followed by polishing with 3 μm and 1 μm diamond paste. In order to reveal the grain boundaries of IF steel, Marshall’s Reagent was used first, with the etching time around 3-4 s, followed by a 20 s etch in 2 % natal [17]. The nital etching after using Marshall’s Reagent is necessary to reveal the ferrite grain boundaries. The microstructure was captured by the light optical microscope Olympus BX60 M. The grain size analysis was conducted with the ImageJ software, which detects the grain boundaries based on the image contrast of the transformed binary pictures and gives the individual information of each grain, including the visible grain area A_i. The grain size value for each grain i, i.e. equivalent grain diameter d_i, was calculated by assuming that the area equals that of a perfect circle with diameter d_i, where $d_{i}=2\sqrt{A_{i}/\pi}$. The mean grain size μ_d, standard deviation s, skewness $\tilde{\mu}_{3}$ and kurtosis $\tilde{\mu}_{4}$ of the grain size distribution based on n grains are calculated with:

These are the key values, i.e. quantitative measures, which describe the grain size distribution. Skewness represents the asymmetry of the grain size distribution, which equals zero when the distribution is perfectly symmetric. Skewness is negative when the low-value tail of the distribution is longer than the high-value tail and positive when the high-value tail of the distribution is longer than the low-value tail [18]. Kurtosis in this paper actually refers to “excess kurtosis”, which is defined as kurtosis minus 3. Kurtosis measures the “tailedness” of the distribution, which is equal to zero for normal distributions, regardless of the values of its parameters. It is positive for so-called leptokurtic distribution, with fatter tails and negative for so-called platykurtic distribution, with thinner tails, compared to the normal distributions [19].

X-Ray diffraction (XRD) measurements were conducted on a Bruker D8 Advance diffractometer with Bragg-Brentano geometry with Lynxeye position sensitive detector, operating at 45 kV, 40 mA, using CuKα radiation (wavelength = 0.15406 nm) without the scatter screen. The scanning speed was controlled at 0.005°/s between 38° and 90° and 0.013°/s between 90° and 152°. The reflections obtained for single phase body-centered cubic (bcc) ferrite structure are {110}, {200}, {211}, {220}, {310} and {222}. The samples were sticked with small amount of plasticine, due to the small dimension, on a Si{510} wafer holder L40SiB. The obtained data from two different scanning speed stages is merged in Bruker software Diffrac.EVA 4.2.2 and processed with X’Pert Highscore 2.2c. The correction for the instrumental broadening is done by subtraction of the measured peak width of reference sample Lanthanum Hexaboride (LaB₆)-SRM660a. Peak fitting is processed with Topas with the split Pearson VII function, in order to obtain the peak width. The dislocation density calculation is based on the modified Williamson-Hall method, which is explained in detail here [20,21].

In this study, mechanical properties are represented by the hardness value, which was measured with the Struers DualScan 70 auto hardness machine using Vickers hardness standards with 30 N force in order to cover enough grains over the sample surface to reveal the macroscopic hardness of the test sample. There were 8 individual hardness measurements conducted on each sample to minimize the test uncertainty.

Due to the large number of independent variables measured from microstructure features, i.e. mean, standard deviation, skewness and kurtosis of the grain size and dislocation density, and the limited data we can obtain from experiments, we employ a statistical method called Least Absolute Shrinkage and Selection Operator (LASSO), which gives the order of importance for the variables based on the influence of each microstructure variable on the target mechanical property. LASSO was first introduced in the geophysics literature by Santosa and Symes [22] and then independently rediscovered by Tibshirani [23]. LASSO, as a tool widely used in the field of machine learning, is rarely adopted in materials science studies, especially in the experimental field, to perform variable selection. The few studies using LASSO in materials science are computational studies [[24], [25], [26], [27]]. Classically, if there are only a few (or only one) explanatory variables, one can use the method of least squares to estimate the regression parameters. If the number of parameters comes close to or exceeds the number of data points, the least squares estimator becomes unstable or is even not well defined anymore. LASSO is a popular method to circumvent these problems, as it adapts the least squares criterion, leading to a well-defined estimator and at the same time provides a way to select the most relevant explanatory variables from the whole set.

In this study, the glmnet package in R [28] was used. The LASSO prediction of the target mechanical property f at the point x is:

where x_j is the jth variable in the prediction point x, which is a vector containing p variables measured on the microstructure. In order to use this technique and referring to Eqs. (1) and (7), ${{\mu }_{d}}{{^{-1}}^{/}}^{2}~$ and ρ^1/2 were used as variables. The estimate βˆj is the corresponding coefficient in the LASSO which minimizes the objective function:

where n is the number of the experimental data, x_ij is the jth variable of the ith microstructure, y_i is the target mechanical property measured on the sample corresponding to the microstructure with measurement x_i, $\lambda H_{\text{r}}^{2}\underset{j=1}{\overset{p}{\mathop \sum }}\,\left| \frac{{{\beta }_{j}}}{{{\beta }_{j,\lambda =0}}} \right|$ is called the shrinkage penalty which is controlled by the tuning parameter λ [23,25] and in which H_r = 1 HV. This tuning parameter λ determines the decrease of regression coefficients. It can be seen in Eq. (13) that for a positive value of λ, the fit will yield lower values for β_j than for λ = 0. This deteriorates the goodness of the fit. At very large values of λ, all fit parameters β_j will become zero. With gradually increasing λ, the fit parameters that are least important for the trends in the target parameter will decrease to zero more rapidly than the more important parameters. We will present the LASSO analysis as a plot of β_j/(β_j,max - β_j,min) vs λ that shows the change in β values with λ, with β_j,max and β_j,min the maximum and minimum values in the applied λ range. This will be presented and analyzed for the present study in Section 3.4.3.

Based on the microstructure analysis, the 16 samples contain only the ferritic phase without any other phases, as a typical micrograph shows in Fig. 1(a). The XRD measurements also confirm the single phase conclusion by containing just the reflections from bcc structure. The heat treatment routes applied on the hot rolled IF steel plates, as well as the cold rolled plates, result in pronounced differences in hardness values, as shown in Table 2. The hardness varies from (58.6 ± 1.5) HV up to (167.0 ± 1.9) HV. As shown clearly by the data, the hardness is significantly lower for the heat treated samples without cold rolling being applied compared with those after cold rolling, which is due to higher dislocation density induced by cold rolling. The only slowly cooled sample at 1000 °C, 10 min 1 °C s^-1 has the lowest hardness of (58.6 ± 1.5) HV due to the largest mean grain size obtained by slow cooling and low dislocation density. Hardness increases with the increase of rolling thickness reduction and decreases with increasing annealing temperature and time, which is closely related to the dislocation density, as shown in following sections.

Following the microstructure image processing and analysis, the dimensions of around 200 grains are obtained from each sample. An example of grain boundary outlines and corresponding grain size distribution for sample “CR 3 mm” is shown in Fig. 1(b) and (c). The detailed grain size data as well as the histograms of all samples can be seen in Table A1 and Fig. A1 in the Appendix. Both grain size distribution data and histograms show that the distribution of grain size varies from sample to sample, whereas it cannot always be well fitted with a log-normal distribution function, as proposed and applied by many researchers [5,7,9,29,30]. The skewness of all samples is positive, which corresponds to a longer high-value tail of the grain size distributions. Only the sample with slow cooling at 1000 °C, 10 min, 1 °C s^-1 has a negative kurtosis, which corresponds to its thinnest tails as the first histogram shows in Fig. A1. All other samples have positive kurtosis, which means they have fatter tails compared to the normal distribution. The box plot for grain size data of each sample can be seen in Fig. 2, whereas the relation with hardness will be explained in the following section. The boxplot clearly shows that cold rolled samples and those without austenitization stage, i.e. heated up to 800 °C or lower, have lower mean grain size and narrower grain size distribution. Samples that underwent austenitization at 1000 °C have larger mean grain size and broader grain size distribution. This is also shown by the detailed data in Table A1.

The dislocation density, as shown in Table 2, is assumed to be 1 × 10¹² m^-2 [31] for the undeformed plates which is related to the detection limit of the applied XRD method. It increases to 7.5 × 10¹³ m^-2 for cold rolled plates. With the increase of strain, dislocation density increases. By elevating the temperature or extending the time of the annealing treatment after cold rolling, dislocation density generally decreases, except for the condition of 1.13 mm, 400 °C, 9 min.

3.4.1. General trend

The correlation of grain size and hardness can be seen in Fig. 2, Fig. 3. The boxplot in Fig. 2 indicates the grain size range from 25 % to 75 %, while the whisker line shows the 1%-99% range. The box middle line shows the median value of grain size, while the square and the extended middle horizontal line show the mean value. The trend of increasing hardness with decreasing mean grain size is seen in Fig. 2, but can be found in a more quantitative manner in Fig. 3, where the dotted line indicates the Hall-Petch trend for the microstructures that were not plastically deformed. The relation of dislocation density and hardness is shown in Fig. 3, which shows the generally positive effect of dislocation density on hardness, which is linear with $\sqrt{\rho}$, as expressed in the Taylor equation (Eq. (7)). By combining the boxplot in Fig. 2 and dislocation density with hardness plot in Fig. 3, it is clearly shown that cold rolled samples have much higher hardness than those without cold rolling, which is due to higher dislocation density and lower mean grain size.

3.4.2. Overall fitting with mean grain size and dislocation density

For a comprehensive relation between microstructure and mechanical properties, the effect of mean grain size and dislocation density should be considered in combination. Least squares linear regression leads to the following fit with the adjusted R-squared of 0.91 and root mean square error (RMSE) of 11 HV:

where k_d = 187.4 HV μm^1/2, k_ρ = 1.1 × 10^-5 HV m and H₀ = 34.0 HV. The weight for hardness applied in the linear regression is the inverse of standard deviation of hardness. The RMSE is calculated based on the predicted hardness H_p, individual hardness values H_i and number of samples N with the following equation:

3.4.3. Variable selection by LASSO

In order to establish a clear relation among hardness, grain size distribution and dislocation density and also be able to predict hardness under similar conditions, different variables describing both grain size distribution and dislocation density are included in a linear model and the model is fitted using the LASSO method (Section 2.5). These variables are, as defined before, dislocation density, mean grain size, standard deviation, skewness and kurtosis of the grain size distribution, which are introduced in the LASSO Eq. (12). For the LASSO test, μ_d and ρ are brought into the calculation as μ_d^-1/2 and ρ^1/2 in order to apply the linear fitting scheme of the LASSO test equation, equivalent to Eq. (13). The resulting LASSO plot is shown in Fig. 4. In the LASSO plot, from the left to the right direction, the tuning parameter λ becomes larger, hence generates a higher penalty for the fit parameter (β) values, therefore the fit parameter values decrease, eventually to zero. The variable that maintains a non-zero coefficient until the highest value of λ is the dislocation density, which therefore is recognized as the variable that has the most significant impact on hardness. The following variables are mean and kurtosis of the grain size distribution. Skewness does not appear in this LASSO plot, which may be related to the relatively high correlation between skewness and kurtosis.

3.4.4. Final fitting with variables selected by LASSO

Based on the LASSO method, the mean square sum of residuals as a function of λ is shown in Fig. 5 with the number of non-zero fit parameters (β) along the top axis. In general, the aim of model descriptions is to balance accuracy and simplicity. The dashed line on the left in Fig. 5 represents the most accurate model, while the one on the right represents the simplest model with an error within the standard error of the most accurate model, i.e. the model with minimum number of coefficients which gives a good accuracy. In this case, the entire range between the most accurate model and the simplest model implies the use of three variables, which are dislocation density, mean and kurtosis.

Therefore, hardness is fitted with a combination of the Hall-Petch equation and the Taylor equation, while including the influence of the kurtosis $\tilde{\mu}_{4}$. The model can be expressed by:

with adjusted R-Squared of 0.96 and RMSE of 7 HV, where k_d = 127.3 HV μm^1/2, k_ρ =9.9 × 10^-6 HV m, k_k = 7.2 HV and H₀ = 35.7 HV. The positive value of k_k shows the positive effect on hardness from kurtosis of grain size distribution, which means that fatter tails of grain size distribution contribute to higher hardness. A microstructure which has more grains at either low or high grain size values tends to have higher hardness than a microstructure that has more grains having similar grain size values.

Since increasing the number of explanatory variable generally increases R-squared and adjusted R-squared cannot represent the predictive capability of our model, in order to judge the predictive capability of our model, predicted R-squared is calculated, which is driven by the leave-one-out cross-validation [32]. Each data point in turn is removed from the dataset for the fitting and the model is refitted using the remaining data points. Then the hardness value of the removed data point is calculated using the new model, hence leads to the predicted residual error sum of squares, which calculates the predicted R-squared. When comparing the models from Eqs. (14) and (16), the predicted R-squared is increasing from 0.79 to 0.88, which means that including extra explanatory variable, i.e. kurtosis, does improve the predictive capability of our model, to an uncertainty of ±7 HV.

3.4.5. Analysis of the Hall-Petch slope

From Eqs. (14) and (16), two Hall-Petch slopes have been generated: k_d1 = 187.4 HV μm^1/2 and k_d2 = 127.3 HV μm^1/2. The literature [33] indicates that the Hall-Petch slopes for pure iron and low carbon steels range between 150 MPa μm^1/2 and 600 MPa μm^1/2. With 50 ppm solute carbon, the Hall-Petch slope is found to be 560 MPa μm^1/2 [33], which just matches the IF steel with 50 ppm carbon content in this study. Since yield strength can be determined with good precision from Vickers hardness by σ_y = H_V/3 [34,35] and 1 HV = 9.8 MPa, the Hall-Petch slope of 560 MPa μm^1/2 corresponds to 171.4 HV μm^1/2. Therefore, the obtained value k_d1 and k_d2 in this study are both reasonable and comparable with the literature data.

The slope obtained while only using mean grain size and dislocation density (k_d1) is obviously higher than that using mean grain size, dislocation density and kurtosis (k_d2). This is because the Eq. (16) takes the grain size distribution factor kurtosis into the fitting. Since μ_d^-1/2 and $\tilde{\mu}_{4}$ are positively correlated (see Table 3) and kurtosis has a positive influence on hardness, it is a mathematical consequence that the Hall-Petch slope will be lower. This also indicates that the two aspects of the grain size distribution effect on hardness have been separated by adding the grain size distribution term kurtosis.

To understand how the combination of grain size distribution and dislocation density influences the hardness of IF steel, a series of IF steel plates were given different microstructures through different heat treatment routes in combination with cold rolling. Based on the microstructure characterization and hardness measurement, the following conclusions are drawn from this research:

(1) Different heat treatment routes and degrees of cold rolling change the dislocation density and grain size distribution, which contribute to the variation of hardness. Cold rolling plays a more significant role in increasing hardness, due to the decrease of mean grain size and increase of dislocation density.

(2) LASSO, as a relatively new method in experimental materials science, plays an important role as the variable selection tool, which gives further insight into the relative influence of different variables and selects for the simplest model with good accuracy.

(3) The combined contribution of dislocation density and grain size distribution on hardness of IF steel plates can be expressed by the equation ${{H}_{\text{V}}}={{H}_{0}}+{{k}_{\text{d}}}*\mu _{\text{d}}^{-1/2}+{{k}_{\rho }}*{{\rho }^{1/2}}+{{k}_{\text{k}}}*{{\overset{}{\mathop{\mu }}\,}_{4}}$, where k_d = 127.3 HV μm^1/2, k_ρ=9.9 × 10^-6 HV m, k_k=7.2 HV and H₀=35.7 HV.