A Comparative Study of Mathematical Methods for Determining Colding’s Constants for Milling of Steels and Experimental Validation

Abstract

The optimization of cutting parameters is critical for improving machining efficiency and extending tool life. Colding’s equation is one such tool life prediction equation that can be used to optimize the machining parameters. However, the equation is complex and is often challenging to solve to evaluate its mathematical constants. This study investigates three distinct approaches for calculating the five constants (‘K’, ‘H’, ‘M’, ‘N0’, and ‘L’) in Colding’s equation. These techniques include analytical equation calculation and different curve fitting approaches. The primary objective was to assess the accuracy and effectiveness of these methods in predicting cutting parameters. Two workpiece materials, K1045 and Mild Steel, were used with results indicating that the Python 3.13 programming approach outperformed the other methods, including MATLAB R2024b and analytical calculations, achieving error percentages of 9.08% for K1045 and 5.51% for Mild Steel compared to 12.3% and 35.3% for K1045 and 12.4% and 64.3% for Mild Steel, respectively. Furthermore, the constants ‘N0’, ‘M’, and ‘H’ displayed different values for the two materials, indicating their dependence on workpiece material properties. Moreover, it was evident that Mild Steel exhibited better machinability than K1045 at low MRR (40–80 cm³/min), with up to 55.8% longer tool life, but K1045 performed better at medium and high MRRs, suggesting different machinability behaviours under varying cutting conditions. Overall, the findings demonstrate that Colding’s model, when applied with the appropriate computational method, can accurately optimize cutting parameters for different materials, contributing to more efficient and cost-effective machining processes.

1. Introduction

Machining is a critical manufacturing process used to shape and finish parts with high precision and accuracy. The performance and quality of the machining process are influenced by a variety of factors, with cutting parameters playing a pivotal role. Cutting parameters, such as the depth of cut, feed rate, and cutting speed, directly affect the efficiency, cost, and overall quality of the machined part. Furthermore, material properties also have significant influence on the cutting tool performance [1,2,3]. Optimizing these parameters is essential for achieving the desired balance between productivity and workpiece quality. Appropriate cutting parameter optimization not only enhances the tool life but also minimizes energy consumption, reduces material waste, and prevents chatter or excessive wear [4,5,6,7].

Achieving an optimal set of cutting parameters requires a comprehensive understanding of the machining process and its complex interactions. This often involves utilizing theoretical models, empirical data, and advanced computational methods to predict and control the outcomes [8]. Several mathematical models have been developed to characterize the relationship between cutting performance and cutting parameters. Among the earliest is Taylor’s tool life model, which establishes a basic power-law relationship between tool life and cutting speed, providing a simple yet foundational approach to tool wear prediction [9]. Building on this, the extended Taylor’s model incorporates additional variables such as feed rate and depth of cut, improving its accuracy and applicability across a wider range of machining conditions [10]. Usui’s model, grounded in tribological principles, describes tool wear as a function of temperature and stress at the tool–chip interface using a mechanistic wear rate equation, offering insight into the fundamental wear mechanisms during cutting [11]. Colding’s model further enhances predictive capability by integrating thermal effects and material properties, making it particularly effective for capturing the complex interactions in high-temperature machining environments [12].

Among these, Colding’s model is recognized for its high accuracy in predicting tool life, outperforming many traditional models. Its effectiveness is largely due to the use of Woxén’s equivalent chip thickness ($h_e$), which accounts for key cutting parameters such as feed rate, cutting angle, and depth of cut. This leads to more precise predictions compared to models that consider only basic inputs. While it requires more experimental data to establish its constants, Colding’s model offers a reliable and detailed approach to tool wear prediction [13,14,15,16,17].

Colding’s equation consists of five constants (‘K’, ‘H’, ‘M’, ‘N0’, and ‘L’). However, due to the complexity of the equation, it is often times very challenging to determine the values for these constants for a given combination of workpiece material and cutting tool. This paper explores three different methods for calculating the constants in Colding’s equation, a key tool in optimizing cutting parameters. The three approaches used were (1) analytical equations [16], (2) curve fitting approaches via MATLAB-based computational techniques, and (3) Python programming, each offering different advantages and applications in the context of machining parameter optimization. While the model has been previously applied in turning operations [14,15], this study focuses on its implementation in milling, where the intermittent cutting action and fluctuating cutting conditions introduce additional challenges in parameter calibration.

Analytical equations offer a robust analytical method for determining constants, known for its simplicity and effectiveness, making it a preferred choice in industrial applications. Numerous studies have demonstrated the efficacy of analytical models in predicting tool wear and tool life across diverse manufacturing processes. For example, physics-based models have been developed to estimate tool life based on various process parameters and have also been integrated with iterative algorithms for inverse analysis, enabling the identification of optimal conditions [18]. Mathematical models have been proposed to analytically evaluate tool life, aiding in the systematic optimization of machining parameters [19]. Furthermore, analytical approaches have been effectively combined with numerical techniques such as the Finite Difference Method (FDM) and Finite Element Method (FEM), with Usui’s equation widely adopted for practical tool wear estimation [20]. In addition to wear prediction, analytical models have also been utilized to study dynamic phenomena, such as chatter vibration in metal cutting, demonstrating their versatility in machining research [21]. However, while analytical equations are advantageous in terms of ease and speed, they tend to be less flexible and accurate when compared to data-driven methods such as the least squares curve fitting approach.

On the other hand, MATLAB-based computational techniques offer greater flexibility and are capable of handling more complex scenarios, such as nonlinear relationships and multi-variable optimization. MATLAB’s robust toolset, including optimization algorithms and built-in functions, makes it a popular choice for researchers and practitioners alike. In particular, MATLAB provides a powerful environment for implementing least squares methods, supporting both standard and generalized formulations. Its capabilities in uncertainty analysis and optimization have been successfully demonstrated in various applications, including the evaluation of measurements using generalized least squares approaches [22,23].

Python programming, with its open-source libraries like NumPy and SciPy, has emerged as a powerful alternative to MATLAB for performing computational tasks. Python’s versatility, ease of integration with other data analysis tools, and large community support have contributed to its growing adoption in scientific computing and engineering fields. Recent studies have demonstrated that Python provides reliable tools for implementing least squares methods, with evaluations showing its strong performance even when applied to large and complex real-world datasets [24,25]. This continued advancement underscores Python’s suitability for computational modelling, optimization, and data analysis tasks across various domains.

This paper compares the accuracy, computational efficiency, and ease of use of each method to evaluate cutting speed or optimize tool life. By analyzing these approaches, the paper seeks to guide the selection of the most suitable method for calculating constants in Colding’s equation based on the complexity of the problem and available resources. However, the constant identifications for Colding’s equation in this study are specific to the workpiece material and cutting tool combination and therefore cannot be generalized.

2. Theoretical Background

Colding [12] introduced one of the most sophisticated and advanced tool life models in 1980, building on Taylor’s foundational tool life equation [9] by incorporating an empirical curve adjustment to better relate tool life to cutting parameters. Lindstrom’s reformulation of Colding’s equation revealed that with its five constants, it serves as an extended version of Taylor’s model [26]. This relationship is represented mathematically in Equation (1). The effectiveness of Colding’s equation in optimizing cutting processes, particularly for turning operations, has been well-documented [14,15,27].

$$v=e^{\left[K-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-H\right)}^2}{4\ast M}}-\left(N0-L\ast \mathit{ln}\left(h_e\right)\right) \ast \mathit{ln}\left(T\right)\right]}$$

(1)

where ‘K’, ‘H’, ‘M’, ‘N0’, and ‘L’ represent Colding’s constants, while ‘v’, $‘h_e’$, and $‘T’$ denote the cutting speed, equivalent chip thickness, and tool life, respectively. Hägglund [16] introduced an equation specifically tailored for milling operations to accurately compute the equivalent chip thickness, as outlined in Equation (2). This equation incorporates critical cutting parameters such as depth of cut ($a_p)$, nose radius ($r$), and major cutting angle ($k$).

$$h_e={\displaystyle \frac{a_p\ast f_{ze}}{\frac{a_p-r(1-\mathrm{c}\mathrm{o}\mathrm{s}k)}{\mathrm{s}\mathrm{i}\mathrm{n}k}+k\ast r+\frac{f_{ze}}{2}}}$$

(2)

The ‘f_ze’ in the equation represents the function of the crescent shape area (A_cs), and the arc length (l_arc) is given by Equations (3) and (4), respectively.

$$f_{ze}={\displaystyle \frac{A_{cs}}{l_{arc}}}\approx {\displaystyle \frac{a_e\ast f_z}{l_{arc}}}$$

(3)

where ${‘f}_z’$ and a_e in Equation (3) represent feed in mm/tooth and radial depth, respectively.

$$l_{arc}={\mathrm{\varnothing }}_s\ast {\displaystyle \frac{D}{2}}$$

(4)

In Equation (4), ‘${\varnothing }_s$’ and ‘D’ represent the cutting arc angle in radian and tool diameter, respectively. For the non-centred end milling process, the cutting arc angle in degrees can be represented using Equation (5).

$$\mathrm{c}\mathrm{o}\mathrm{s}\left({\mathrm{\varnothing }}_s\right)={\displaystyle \frac{D-2a_e}{D}}$$

(5)

Determining the five constants in Colding’s Equation (1) is inherently challenging, as their values are both interdependent and highly responsive to variations in the machining conditions specific to each experimental setup. Accurate estimation of these constants is crucial for the reliable application of the equation; however, the process is often hindered by inconsistencies in calculation methodologies, which can arise from variations in the selected formula, computational tools, or programming techniques.

2.1. Evaluating Colding’s Constants—Analytical Formula

Hägglund [16] reformulated the analytical Equations (6)–(10) to improve the effectiveness of solving Colding’s equation. These equations offer a more robust framework for determining the constants in Equation (1), enabling a more reliable approach that reduces computational effort without compromising accuracy. While this reformulation provides a systematic calculation platform, the constants must be determined in a specific sequence: ‘L’, ‘N0’, ‘M’, ‘H’, and ‘K’. Additionally, it requires selecting the experimental data points in a planned way, using three different values of equivalent chip thickness (‘$h_e$’). For two of these values, two tests are performed at different cutting speeds, while the third value is tested once, as shown in Figure 1.

The selection of these points follows a structured methodology aimed at reducing the complexity involved in solving Colding’s equation. Although Equation (1) can be used directly to calculate all relevant constants, solving it is analytically intensive and requires considerable time and expertise. To simplify this process, Colding proposed a specific organization of the experimental matrix, which was later refined by Hägglund. In this approach, points 2 and 3 share the same ‘$h_e$’ but differ in cutting speed, as do points 4 and 5, whereas point 1 is positioned between these two sets, as illustrated in Figure 1. This configuration not only facilitates the efficient application of Equations (6)–(10) but also ensures a practical and structured pathway for determining the model constants.

The equivalent chip thickness ‘$h_{\mathrm{e}}$’ is assumed to be identical for data pairs corresponding to points 2 and 3, as well as points 4 and 5, thus $h_{\mathrm{e}23}=h_{\mathrm{e}2}=h_{\mathrm{e}3}$ and $h_{\mathrm{e}45}=h_{\mathrm{e}4}=h_{\mathrm{e}5}$. Then, let the variables be defined as X = ln ($h_{\mathrm{e}})$, Y = ln ($v$), and Z = ln (T). Under these assumptions, the tool life constants can be determined using the following equations [16]:

${\mathrm{X}}_1=\ln \left(h_{\mathrm{e}1}\right)$	${\mathrm{Y}}_1=\ln (v$1)	Z1 = ln (T1)
${\mathrm{X}}_{23}=\ln \left(h_{\mathrm{e}23}\right)$	${\mathrm{Y}}_2=\ln (v$2)	Z2 = ln (T2)
${\mathrm{X}}_{45}=\ln \left(h_{\mathrm{e}45}\right)$	${\mathrm{Y}}_3=\ln (v$3)	Z3 = ln (T3)
	${\mathrm{Y}}_4=\ln (v$4)	Z4 = ln (T4)
	${\mathrm{Y}}_5=\ln (v$5)	Z5 = ln (T5)

where ‘X₁’, ‘X₂₃’, and ‘X₄₅’ represent the natural logarithms of the equivalent chip thicknesses at point 1, points 2 and 3, and points 4 and 5, respectively. ‘Y₁’ to ‘Y₅’ denote the natural logarithms of the cutting speeds at points 1 through 5, and ‘Z₁’ to ‘Z₅’ correspond to the natural logarithms of the experimentally measured tool life values.

$$L={\displaystyle \frac{\frac{Y_4-Y_5}{z_4-Z_5}-\frac{Y_2-Y_3}{z_2-z_3}}{X_{45}-X_{23}}}$$

(6)

$$N0={\displaystyle \frac{Y_2-Y_3-L\ast X_{23}\left(z_2-z_3\right)}{z_3-z_2}}$$

(7)

$$M={\displaystyle \frac{X_{23}-X_{45}}{4\{\frac{\left(Y_1-Y_4-NO\left(z_4-z_1\right)-L\left(X_1\ast Z_1-X_{45}\ast Z_4\right)\right)}{X_1-X_{45}}-\frac{Y_1-Y_2-NO\left(Z_2-Z_1\right)-L\left(X_1\ast Z_1-X_{23}\ast Z_2\right)}{X_1-X_{23}}\}}}$$

(8)

$$H={\displaystyle \frac{4M\left(Y_1-Y_2-N0\left(z_2-z_1\right)-L\left(X_1\ast Z_1-X_{23}\ast z_2\right)\right)}{2\left(X_1-X_{23}\right)}}$$

(9)

$$K=Y_1+{\displaystyle \frac{{\left(X_1-H\right)}^2}{4M}}+\left(N0-L\ast X_1\right)Z_1$$

(10)

2.2. Evaluating Colding’s Constants—MATLAB

MATLAB, as a robust programming and numerical platform for engineering and scientific applications, provides a wide range of built-in functions and optimization techniques for solving complex equations. In this study. it was used to solve Colding’s equation by incorporating optimization methods, such as ‘lsqcurvefit’ [28], which iteratively refines parameters to minimize errors in the model fitting process.

MATLAB’s ‘lsqcurvefit’ function is widely used for nonlinear curve fitting, as it minimizes the sum of squared differences between model predictions and experimental data using a least squares approach [28]. This iterative method is well-suited for identifying parameters in complex models such as Colding’s equation. Additionally, ‘lsqcurvefit’ offers flexibility by allowing users to define constraints on the parameters, such as upper and lower bounds, ensuring that the solutions remain physically meaningful. The iterative nature of ‘lsqcurvefit’ ensures that the parameters converge to values that best fit the data, even when dealing with noisy or limited data [29]. The syntax for implementing ‘lsqcurvefit’ is given by

x = lsqcurvefit(fun, x0, xdata, ydata, lb, ub, options)

where ‘fun’ is the function to be fitted, ‘x0’ is the initial guess for parameters, ‘xdata’ and ‘ydata’ are the data to be fitted, ‘lb’ and ‘ub’ are the lower and upper bounds for the parameters, and ‘options’ is a structure that specifies optimization options.

2.3. Evaluating Colding’s Constants—Python

Python, as a versatile programming language, provides an extensive ecosystem of libraries and functions for solving complex mathematical equations in scientific and engineering applications. With libraries such as NumPy 1.26.4, SciPy 1.13.1, and Matplotlib 3.9.2, Python offers powerful tools for data manipulation, optimization, and visualization. In the context of solving Colding’s equation, Python can be employed by leveraging optimization methods like the ‘least_squares’ function from the SciPy library [30]. This function iteratively adjusts the parameters to minimize errors in the model fitting process, providing accurate solutions for the constants in Colding’s equation.

The ‘least_squares’ function in Python provides a reliable framework for nonlinear optimization by iteratively refining model parameters through the method of least squares. It utilizes an objective function to define variable relationships and minimizes the difference between the model’s predictions and experimental observations. Additionally, ‘least_squares’ allows for parameter constraints, such as upper and lower bounds, ensuring that the solutions remain within physically meaningful limits. This capability ensures that the optimized parameters remain physically meaningful and feasible within the context of the problem. The iterative nature of the algorithm gradually converges towards the most accurate estimates for constants like those in Colding’s equation, even when dealing with noisy or sparse data. This makes Python a valuable tool for solving complex equations in practical applications.

2.4. Assumptions

2.4.1. Tool Failure and Flute Contact Time

This study does not account for the individual Tool Ratio (T_ratio) in the analysis. It is assumed that tool failure occurs when any one of the flutes fails, rather than considering the failure of a single flute as independent from the others. In other words, the failure of the entire tool is presumed once any flute experiences failure. As a result, this research focuses on the overall tool failure and uses the total contact time for the entire tool. Consequently, the time associated with the contact of a single flute is not separately accounted for in the analysis, rather, the total contact time of the tool is considered for calculating the tool’s lifespan.

2.4.2. Cutting Time and Tool Life Calculation

Kantojärvi et al. [31] calculated cutting time only on a reduced section of the workpiece corresponding to constant ‘$h_{\mathrm{e}}$’ values. However, for this study, the total cutting time is defined as the time elapsed from the moment the cutting tool engages with the workpiece until it exits the workpiece. This time span is used to calculate the tool life in order to simulate real-world cutting conditions commonly encountered in industrial settings. In this context, the concept of ‘effective accumulated edge time’, as suggested by Kantojärvi et al. [31] for more academic-oriented analyses, is not explicitly considered. The focus is instead on the total cutting time as a more practical representation of the tool’s operational life, in line with real-world manufacturing processes, where the tool’s effectiveness and longevity are often evaluated based on the total machining time rather than an isolated focus on edge wear.

3. Experimental Details

3.1. Workpiece Materials

Two workpiece materials, K1045 (AS3678) and Mild Steel (AS3678-250) [32], with dimensions of 250 × 250 × 20 mm³ each, were machined under dry face milling conditions. The chemical compositions of these materials are detailed in

Table 1. Chemical composition of K1045 (material data specifications provided by supplier).

C	Si	Mn	P	S	Cr	Ni	B (ppm)	Cu
0.4325	0.242	0.737	0.0018	0.0018	0.014	0.007	3	0.013

and

Table 2. Chemical composition of Mild Steel (material data specifications provided by supplier) (AS3678-250).

C	Si	Mn	P	S	Cr	Ni	Cu	Fe
0.42–0.48	0.15–0.35	0.6–0.9	<0.03	<0.035	<0.2	<0.2	<0.3	Rest

, respectively. The Vickers hardness was measured at 209 ± 10 HV₅ for K1045 and 171 ± 8 HV₅ for Mild Steel. The tensile strength of the workpieces was 647.9 MPa and 454.2 MPa, respectively.

3.2. Tool Geometry

The tool used in the experimental setup was an uncoated carbide end mill, engineered for high-precision milling applications. The detailed geometry of the tool is shown in

Table 3. Basic geometry for the carbide tool (E601-Sutton Tools).

Flute length (mm)	22
Effective teeth	4
Helix angle (°)	30
Material	Tungsten Carbide
Land width (mm)	1.3
Edge radius (μm)	4.26
Length of cut (mm)	22
Rake angle (°)	6
Shank tolerance	h5

.

3.3. Machining Setup

The experiments were conducted using a DMU 50 eVolution Deckel Maho 5-axis milling machine, as shown in Figure 2. A schematic diagram of the cutting process is presented in Figure 3a. The tool life criteria were predefined, with tool failure defined as either when the maximum flank wear (VB_max) reached 0.3 mm or in cases of catastrophic tool failure in accordance with the ISO standard 8688-2 [33]. Tool wear was measured using a Celestron handheld digital microscope as illustrated in Figure 3b.

3.4. Design of Experiments

A total of sixteen experimental trials were carried out for two workpiece materials as presented in

Table 4. Experimental matrix.

Cutting Data	Test No.
	1	2	3	4	5	6	7	8
Cutting speed—$\mathit{v}$ (m/min)	200	300	400	300	300	200/250	300	300
RPM	6366	9549	12,732	9549	9549	6366	9549	9549
Feed—f′ (mm/flute)	0.050	0.050	0.050	0.030	0.040	0.030	0.050	0.050
Radial depth of cut—ae (mm)	4.00	4.00	4.00	4.00	4.00	4.00	4.50	5.00
Axial depth of cut—ap (mm)	15.00
MRR (cm3/min)	76.4	114.6	152.8	68.8	91.7	45.8	128.9	143.2

. The cutting parameters were selected to cover a representative range of milling conditions for materials like K1045 and Mild Steel. By varying cutting speed, feed rate, and radial depth of cut, the experiments produced a wide spread of material removal rate (MRR) values to study their effect on tool wear and dynamic stability. This approach provided a structured basis for analyzing differences in tool wear behaviour, evaluating the consistency of parameter estimation methods, and exploring how process constants respond to changes in cutting conditions and material properties. Additionally, the cutting parameters were selected to replicate the pattern shown in Figure 1, which facilitated easy solving for Colding’s constants from analytical equations.

The study utilized down milling (climb milling) due to its established advantages in minimizing cutting forces, lowering tool wear, and improving surface quality [34,35]. The choice was based on its compatibility with the selected machining conditions and overall process stability. Although up milling (conventional milling) was not explored, its tendency to induce higher mechanical loads and wear [34] presents a potential area for future research. Eight experiments were conducted for K1045, which were repeated twice, and only eight experiments were conducted for Mild Steel. The cutting parameters were the same for both K1045 and Mild Steel, with the exception of test 6, where the cutting speed was set to 200 m/min for K1045 and 250 m/min for Mild Steel. Equivalent chip thickness, ‘$h_e$’, was calculated using the parameters in Table 4 and Equations (2)–(5). After the test data was achieved, the Colding’s constants (‘K’, ‘H’, ‘M’, ‘N0’, and ‘L’) were evaluated using Equations (6)–(10).

As the tests exhibited varying behaviour based on the MRR, the experimental matrix was further categorized into three distinct MRR levels, namely low MRR (40–80 cm³/min), medium MRR (80–120 cm³/min) and high MRR (120–160 cm³/min). This classification is illustrated in Figure 4, providing a clear framework to analyze the impact of different MRRs on the experimental outcomes.

4. Results

4.1. Progressive Tool Wear Graphs

The wear progression for the workpiece materials K1045 and Mild Steel is shown in Figure 5a and Figure 5b, respectively. The red dotted line in Figure 5 signifies the point at which machining was halted upon the tool wear reaching the predefined criterion of a VB_max of 0.3 mm. There was no catastrophic tool failure observed during the experiment.

Regarding the K1045 workpiece material, as illustrated in Figure 5a, the lowest MRR of 45.8 cm³/min (test 6) achieved the longest tool life of 197.7 s, exhibiting a more gradual wear progression till about 150 s of machining followed by rapid wear. In contrast, the high MRR of 143.2 cm³/min (test 8) resulted in the shortest tool life of 47.1 s, with tool wear advancing rapidly and surpassing the 0.3 mm wear threshold.

The tool wear progression in the machining of Mild Steel, as indicated in Figure 5b, was observed to be more gradual throughout the tests, unlike the faster wear rate in K1045. The test exhibiting the shortest tool life was from a higher MRR category, i.e., test 7 (128.9 cm³/min), while the test with the longest tool life came from a lower MRR category, i.e., test 1 (76.4 cm³/min). Machining of Mild Steel showed superior average tool life compared to K1045 when subjected to equivalent MRR values.

Table 5. Tool life obtained for machining K1045 and Mild Steel at different MRRs.

MRR (cm3/min)	Tool Life for Machining K1045 (s)	Tool Life for Machining Mild Steel (s)
45.8	197.7	81.1
68.8	79.2	104.9
76.4	108.9	169.8
91.7	59.1	49.5
114.6	109.4	63.2
128.9	71.6	79.2
143.2	47.1	47.7
152.8	48.3	24.4

presents the average tool life obtained from the experimental trials. The data indicates that, generally, tool life decreases as the MRR increases, particularly when considering the average tool life across the MRR intervals shown in Figure 4.

Overall, Figure 5 illustrates that tools used under low MRR conditions generally demonstrate higher average tool life compared to those operating in the medium or high MRR categories. This observation highlights the conclusion that an increase in MRR accelerates tool wear and reduces tool life. An increase in MRR increases the restraining forces or mechanical stresses exerted on the tool, thereby enhancing its vulnerability to wear [36].

Furthermore, a comparative analysis of the two materials reveals that K1045 consistently exhibits faster tool wear across all MRR levels, indicating it is more abrasive and challenging to machine than Mild Steel, as observed in Table 5. The comparison of cumulative tool life revealed that Mild Steel outperformed K1045, with an overall increase of approximately 18.4%. This finding is supported by the Vickers hardness and tensile strength values presented in Section 3.1, which indicate that the higher hardness of K1045 contributes to more aggressive abrasive interactions with the cutting tool, thereby accelerating tool wear during machining. The increased resistance and friction associated with K1045 enhance tool wear rates, whereas the lower hardness of Mild Steel results in less abrasive contact, placing reduced stress on the cutting tool and leading to comparatively lower wear. Sredanović et al. [37] reported similar findings in their study on micro-milling of cold alloyed tool steel, where softer workpiece materials led to prolonged tool life under identical cutting conditions.

An examination of the individual graphs in Figure 5, along with the corresponding MRR values, reveals that while higher average MRR tends to result in reduced tool life, it is not accurate to generalize that the highest MRR will necessarily lead to the lowest tool life in every case. This suggests that tool wear is not solely dependent on MRR and that factors such as machine tool dynamics, tool setup, and build-up edge formation play a significant role in influencing wear progression.

4.2. Colding’s Constants—Analytical Formula

The constants determined using Equations (6)–(10) are presented in

Table 6. Colding’s constants using analytical formula.

Workpiece Material	L	N0	M	H	K
K1045	−0.47	2.44	−0.11	−3.73	5.63
Mild Steel	0.58	−1.37	−0.04	−3.79	5.54

for both K1045 and Mild Steel workpiece materials. These calculations were performed using data from tests 1, 3, 4, 5, and 6, as the error percentages associated with these test numbers were found to be minimal, ensuring the reliability of the results. Additionally, these tests fully meet the criteria outlined in Figure 1 for applying the specified equations.

The values for each constant obtained using the analytical formula is shown in Table 6 and the significant equations for the workpiece materials K1045 and Mild Steels are illustrated in Equations (11) and (12), respectively. The data in Table 6 indicates that there is a significant difference in the constants ‘L’, ‘N0’, and ‘M’, whereas only minimal difference can be noted on constants ‘H’ and ‘K’. The raw data and calculations for K1045 and Mild Steel are shown in Appendix A and Appendix B, respectively.

$$v=e^{\left[5.63-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-(-3.73)\right)}^2}{4\ast (-0.11)}}-\left(2.44-(-0.47)\ast \mathit{ln}\left(h_e\right)\right)\ast \mathit{ln}\left(T\right)\right]}$$

(11)

$$v=e^{\left[5.54-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-(-3.79)\right)}^2}{4\ast (-0.04)}}-\left((-1.37)-\left(0.58\right)\ast \mathit{ln}\left(h_e\right)\right)\ast \mathit{ln}\left(T\right)\right]}$$

(12)

4.3. Colding’s Constants—MATLAB

In this study, the nonlinear least squares optimization approach, applied through the ‘lsqcurvefit’ function, has been utilized to calculate the five constants in Colding’s equation. During the optimization process, the function iteratively adjusted the parameters ‘K’, ‘H’, ‘M’, ‘N0’, and ‘L’ to minimize the discrepancy between the predicted and observed values of ‘$v$’. Error percentages were also calculated for each data point to quantify the accuracy of the predictions. The relationship between the variables ‘$T$’, ‘$h_{\mathrm{e}}$’, and ‘$v$’ was modelled using an objective function based on Colding’s Equation (1), which was optimized to calculate the constants.

The iterative nature of the process allows for progressively more precise estimates of the constants, ultimately leading to accurate predictions of ‘$v$’ for the given tool life values.

Table 7. Colding’s constants using MATLAB programming.

Workpiece Material	L	N0	M	H	K
K1045	0.15	−0.20	1.10	−3.99	5.77
Mild Steel	0.14	−0.20	9.73	−2.00	5.80

summarizes the constants calculated through MATLAB programming, and the resulting significant equations for K1045 and Mild Steel are provided in Equations (13) and (14), respectively. The program code and implementation of data for K1045 and Mild Steel are shown in Appendix C and Appendix D, respectively.

$$v=e^{\left[5.77-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-(-3.99)\right)}^2}{4\ast \left(1.10\right)}}-\left((-0.20)-\left(0.15\right)\ast \mathit{ln}\left(h_e\right)\right)\ast \mathit{ln}\left(T\right)\right]}$$

(13)

$$v=e^{\left[5.80-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-(-2.00)\right)}^2}{4\ast \left(9.73\right)}}-\left((-0.20)-\left(0.14\right)\ast \mathit{ln}\left(h_e\right)\right)\ast \mathit{ln}\left(T\right)\right]}$$

(14)

4.4. Colding’s Constants—Python

For the calculation of constants using the Python program, test number 1 was excluded for K1045, and test number 3 was omitted for Mild Steel to enhance the model’s accuracy and ensure more reliable parameter estimation. The nonlinear optimization process was executed using NumPy and the ‘least_squares’ functionality provided by SciPy. The approach reduced the residuals between the experimentally observed and predicted cutting speed ‘$v$’ based on Colding’s Equation (1). The constants obtained through Python programming are listed in

Table 8. Colding’s constants using Python programming.

Workpiece Material	L	N0	M	H	K
K1045	0.12	−0.20	0.33	−3.69	5.79
Mild Steel	0.10	−0.11	4.53	−5.00	5.79

. The corresponding major equations for the K1045 and Mild Steel workpiece materials are presented in Equations (15) and (16), respectively. The program code and implementation of data for K1045 and Mild Steel are shown in Appendix E and Appendix F, respectively.

$$v=e^{\left[5.79-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-(-3.69)\right)}^2}{4\ast \left(0.33\right)}}-\left((-0.20)-\left(0.12\right)\ast \mathit{ln}\left(h_e\right)\right)\ast \mathit{ln}\left(T\right)\right]}$$

(15)

$$v=e^{\left[5.79-{\displaystyle \frac{{\left(\mathit{ln}\left(h_e\right)-(-5.00)\right)}^2}{4\ast \left(4.53\right)}}-\left((-0.11)-\left(0.10\right)\ast \mathit{ln}\left(h_e\right)\right)\ast \mathit{ln}\left(T\right)\right]}$$

(16)

5. Discussion

5.1. Material Influence on Tool Wear

The tool wear progression for the workpiece materials K1045 and Mild Steel, as shown in Figure 5, reveals significant differences influenced by material properties and machining conditions. For K1045 (Figure 5a), tool wear advances rapidly, particularly as the MRR increases. Even at lower MRRs, while the initial wear progression is slower, a notable increase in wear occurs over extended machining times, indicating that the tool experiences substantial wear regardless of the machining conditions. In contrast, the tool wear progression for Mild Steel (Figure 5b) is more gradual. Under similar MRR conditions, the wear rate remains consistently lower, and the critical wear threshold is reached later compared to K1045 as illustrated in Table 5.

When categorized based on MRR, the tool life trends for K1045 and Mild Steel, as summarized in Table 5, show distinct behaviours. In the low MRR range (40–80 cm³/min), K1045 exhibited a longer tool life (197.7 s at 45.8 cm³/min and 79.2 s at 68.8 cm³/min) compared to Mild Steel (81.1 s and 104.9 s, respectively), with Mild Steel showing slightly better tool life at 68.8 cm³/min. In the medium MRR range (80–120 cm³/min), Mild Steel generally outperformed K1045, at 76.4 cm³/min and 91.7 cm³/min, while Mild Steel recorded tool lives of 169.8 s and 49.5 s, compared to 108.9 s and 59.1 s for K1045. In the high MRR range (120–160 cm³/min), tool life decreased significantly for both materials; however, K1045 and Mild Steel showed similar trends (e.g., 47.1 s vs. 47.7 s at around 143 cm³/min). Although Mild Steel occasionally achieved longer tool life at specific MRR levels, K1045 consistently exhibited faster wear progression across the tested conditions. This behaviour is attributed to the higher Vickers hardness of K1045 (209 HV₅) compared to Mild Steel (171 HV₅), making K1045 more abrasive and challenging to machine. However, the hardness of the steel may not be the only variable affecting flank wear. Wear rates are consistently higher for certain grades of steel, such as those alloyed with additional elements, even when materials with similar hardness values are machined under similar conditions [2].

The trend in Figure 5 highlights that the material properties of K1045 result in more aggressive tool wear, likely due to its higher hardness. Conversely, the slower wear progression in Mild Steel suggests better machinability and reduced tool stress. The mechanical properties of the workpiece material, particularly hardness and strength, play a critical role in influencing tool performance. Materials with higher hot hardness and strength, such as K1045, can lead to increased tool breakage during machining due to the elevated stresses and temperatures at the cutting edge [1]. In contrast, materials with lower strength require lower cutting forces (F_z—thrust force), thereby reducing the mechanical load on the tool and extending tool life [1]. Further supporting this observation, studies using wear mapping methodologies demonstrated that flank wear becomes significantly more severe when machining harder and tougher steels, such as AISI 4340, compared to softer grades like AISI 1045. It was found that the greater hardness, toughness, and strength of 4340 steel result in higher cutting stresses and tool temperatures, contributing to accelerated tool wear [2]. Moreover, machining alloys with high hardness requires not only greater cutting force but also enhanced tool properties; to resist the severe furrowing and burr formation typically observed with harder workpiece materials, an increase in the hardness of the tool or its coating is necessary. This indicates that the harder the workpiece material, the greater the demand for more wear-resistant and mechanically robust tool materials or coatings to maintain cutting performance [3]. Likewise, in this study it was found that machining K1045 resulted in a higher tool wear rate than when machining a softer Mild Steel workpiece. Overall, these correlations emphasize the importance of carefully matching the mechanical properties of the workpiece with the appropriate tool material and cutting conditions to optimize machining outcomes.

5.2. Effect of Machining Dynamics on Tool Wear

The relationship between machining dynamics and tool wear is complex and fundamentally governed by the interaction of process parameters such as cutting speed, feed rate, and depth of cut. The results from the present milling experiments, summarized in the experimental matrix, Table 5, and Figure 4, illustrate the dynamic influence of varying MRR levels and cutting data on tool wear behaviour across both the work materials K1045 and Mild Steel.

5.2.1. Influence of MRR and Cutting Parameters

MRR, a function of cutting speed (v), feed per tooth (f), and depth of cut (a_e and a_p), serves as a proxy for dynamic loading in machining. Higher MRRs imply greater force input, heat generation, and likelihood of dynamic instability such as chatter [7,38]. Tests 3, 7, and 8 (Table 4), which fall within the high MRR zone (>120 cm³/min) as shown in Figure 4, corresponded with significantly reduced tool life, particularly in K1045 where tool life dropped to below 50 s in test 3 (152.8 cm³/min). These results support the understanding that high MRR conditions increase the likelihood of dynamic instability, such as regenerative chatter, which accelerates tool wear. This observation is consistent with Liu [39], who emphasizes that elevated cutting forces and insufficient damping at high removal rates can lead to chatter-induced degradation of the cutting tool.

Conversely, in test 6, conducted at the lowest MRR (45.8 cm³/min), the tool life was maximized in K1045 (197.7 s), indicating stable machining conditions and low vibrational excitation. Notably, this test also used the lowest feed rate and cutting speed, which collectively would reduce both dynamic and thermal loads.

Surface roughness measurements from test 7, as shown in

Table 9. Surface roughness values (Ra) measured for each pass during test 7.

	Surface Roughness Values for Test 7
Pass Number	Average Surface Roughness (µm)
1	1.11
2	0.74
3	0.89
4	0.43
5	0.42
6	0.23
7	0.2
8	0.24
9	0.22

, provide additional evidence of the influence of machining dynamics, particularly the presence of vibration or chatter. The variation in surface roughness values across the passes suggests intermittent dynamic disturbances during cutting. In high MRR conditions, such as those in test 7 (121.6 cm³/min), the likelihood of regenerative chatter increases due to elevated cutting forces and limited system damping. The observed roughness values are indicative of these dynamic instabilities, which can degrade surface quality and signal unstable cutting conditions.

5.2.2. Effect of Cutting Speed and Feed on Machining Dynamics

Cutting speed has a significant impact on machining dynamics, particularly in relation to vibration sensitivity [40]. This is clearly reflected in test 3, positioned in the high MRR zone (Figure 4), which used the highest cutting speed (400 m/min). This test resulted in the shortest tool life for Mild Steel (24.4 s), suggesting that the combination of high cutting speed and material removal rate likely pushed the process into a dynamically unstable regime, characterized by excessive vibration and accelerated mechanical wear.

In comparison, tests conducted in the low to medium MRR zones, such as tests 1, 2, 4, and 5, which used moderate cutting speeds (200–300 m/min), showed more consistent and prolonged tool life, particularly when paired with lower feed rates. These results indicate that machining under controlled, moderate conditions helps maintain process stability and reduces wear rates.

This trend aligns with established principles in machining dynamics, as discussed by Budak [41], who highlighted the influence of system stiffness, process damping, and chip thickness variation on the occurrence of self-excited vibrations. When feed rate or radial engagement is increased beyond stable limits, especially under high MRR conditions, mechanical oscillations can intensify, leading to irregular cutting forces and premature tool wear.

5.3. Error % Between Various Colding’s Constant Calculation Techniques

To assess the accuracy and reliability of the different methods used for calculating Colding’s constants, error percentages were calculated for each technique based on the comparison between predicted and experimental tool life values. For the analytical method, the constants were determined using five selected experiments out of eight that satisfied the conditions in Figure 1, while the remaining three experiments were reserved for validating the model through error percentage calculations. In contrast, for the MATLAB and Python programming approaches, all available experimental data points were utilized to fit the curve using the least squares method, and the error percentage was calculated individually for each trial. The formula to calculate error percentage is shown in Equation (17).

$$\mathrm{Error} \%=\left({\displaystyle \frac{\left|\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}-\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\right|}{\mathrm{Experimental} \mathrm{c}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}}}\right)\ast 100\%$$

(17)

Figure 6 illustrates the box plot error percentage for each material and calculation method, along with the average error percentages. It can be concluded that the Python-based approach significantly outperformed both the analytical formula and MATLAB for both K1045 and Mild Steel. The average error percentage for this method was 9%, with a maximum error of 13.2% for K1045, whereas for Mild Steel, the average error was 5.5%, with the highest error reaching 12.4%.

The error percentage of Colding’s model, calculated using the constants derived from Equations (6)–(10), resulted in the highest average error, with 35.3% for K1045 and 64.3% for Mild Steel. As expected, this is due to the model not relying on the least squares method. In this approach, out of eight experimental trials for each workpiece material, five trials were used to calculate the constants, while the remaining three were reserved for verification and error percentage calculation. This made the model more reliant on the equations, lacking the ability to account for the inherent variability and complexity of the material. Unlike the least squares method, which adapts the model to best fit the data, equation-based approaches assume fixed relationships that may not fully capture nonlinearities and material-specific factors, resulting in higher error margins.

The average error percentage for MATLAB is relatively close to Python programming as both programs employed the least squares method for curve fitting. However, subtle differences in the optimization algorithms used by each platform can contribute to minor variations in the results. In Python, algorithms such as Trust Region Reflective and Levenberg–Marquardt are commonly used through libraries like SciPy, offering flexible handling of bounds, constraints, and robust convergence criteria. In contrast, MATLAB, although also utilizing methods like Levenberg–Marquardt, applies proprietary versions of these algorithms, which can differ in how constraints are treated, step sizes are adjusted, and convergence is determined. These differences, although seemingly minor, can lead to variations in the final fitted parameters and consequently in the calculated error percentages. In this scenario, Python’s open and flexible optimization framework allowed for more adaptable constraint handling and finer control over the fitting process, contributing to slightly lower and more consistent error percentages compared to MATLAB.

Optimization techniques implemented in Python and MATLAB have proven invaluable not only in machining and tool wear studies but also across a wide spectrum of fields such as data fitting, signal processing, and machine learning. These tools facilitate the resolution of complex, nonlinear problems and the optimization of system performance [42]. Furthermore, Meta-Heuristic (MH) algorithms, widely applied in optimization, have shown significant utility in sectors including engineering, finance, and transportation, where they aid in solving real-world, nonlinear objective functions that traditional optimization methods may fail to address [43]. In geosciences, Python and MATLAB are integral to simulation-based optimization frameworks, particularly in the modelling of groundwater flow and contaminant transport. These approaches, employing both direct-search and heuristic algorithms, have been applied effectively to challenges such as the design of groundwater remediation systems and the calibration of contaminant transport models, offering substantial improvements in optimization efficiency and performance [44].

After comparing the Colding’s constants evaluated using the three different techniques (Table 6, Table 7 and Table 8), it was found that the constants ‘L’ and ‘N0’ were similar for both K1045 and Mild Steel via MATLAB and Python programming (‘L’ is about 0.13 and ‘N0’ is about −0.20 for K1045, whereas ‘L’ is about 0.12 and ‘N0’ is about −0.15 for Mild Steel) and ‘K’ was similar via all techniques irrespective of the workpiece material (‘K’ is about 5.77 for all methods and both workpieces).

The error percentage via Python programming is the lowest, followed by MATLAB, and the analytical approach exhibits a very high error percentage within the defined experimental data, irrespective of the workpiece material. This is because the optimization algorithms in Python and MATLAB uses the least squares regression method to determine the Colding’s constants, whereas the analytical method relies on the constants derived from mathematical equations which may not consider dynamic variability during the machining process. The Colding’s constants ‘L’, N0, and ‘K’ are relatively similar for both workpieces via MATLAB and Python programming, which implies that these constants are not necessarily dependent on the method of evaluation. However, ‘M’ and ‘H’ appear to be sensitive to the method of evaluation. These variations in the constant values can be the attributing factors for prediction error % differences.

5.4. Correlating Colding’s Constants with Changes in Workpiece Material

Since Python programming yields the most accurate results, as shown in Figure 6, this study concentrates solely on the constants derived using this method. Table 8 indicates that the constants ‘L’ and ‘K’ remain relatively consistent between the two workpiece materials, suggesting their robustness to material variation. However, notable deviations in ‘N0’, ‘M’, and ‘H’ can be attributed to the workpiece material properties, such as hardness and microstructure, which directly influence cutting forces, heat generation, and wear mechanisms [45]. Johansson [46] investigated the impact of varying tool coatings on Colding’s constants and proposed that the constant ‘K’ could be utilized to adapt the model for different tool coating variations. Since the tool used in this study was consistent for both workpiece materials, the results indicate that constants like ‘K’ are relatively stable and could be utilized for adjustments in tool-related parameters such as coating, geometry, etc.

As discussed in Section 5.3, Colding’s constants ‘M’ and ‘H’ were dependent on the method of evaluation used. In addition, these two constants appear to be also dependent on workpiece materials. However, due to only two variations in workpiece materials, there is no clear evidence whether the constants ‘M’ and ‘H’ are sensitive to material-specific properties, and more research needs to be carried out to understand the correlation between various cutting and workpiece parameters on Colding’s constants.

6. Conclusions

This research mainly investigated different approaches to solving Colding’s model and understanding the correlations between Colding’s constants and various cutting and workpiece material parameters. The key findings of this study are as follows:

• The study demonstrated that K1045 causes significantly more tool wear than Mild Steel due to its higher Vickers hardness (209 HV₅ vs. 171 HV₅), indicating its more abrasive nature and greater machining difficulty.
• Tool wear increases with higher material removal rate (MRR) for both materials, as elevated MRR levels induce greater mechanical stress and restraining forces.
• Machinability analysis revealed that Mild Steel exhibited better performance at low MRRs (40–80 cm³/min), with an average tool life of 118.6 s compared to 128.6 s for K1045, while K1045 outperformed at medium MRRs (80–120 cm³/min) with an average tool life of 84.2 s versus 56.4 s for Mild Steel, and also showed higher tool life at high MRRs (120–160 cm³/min), averaging 55.6 s compared to 50.4 s for Mild Steel.
• It is essential for machinists to recognize that a tool optimized for one material may not perform efficiently with another; tool selection must consider the specific workpiece material.
• It was evidenced that the use of Colding’s equation presents an error of less than 10% to predict cutting speed.
• Among the approaches used to solve Colding’s model, Python programming achieved the lowest average error percentage (9% for K1045 and 5.5% for Mild Steel), followed by MATLAB (12.3% for K1045 and 12.4% for Mild Steel), while the analytical method showed the highest error (35.3% for K1045 and 64.3% for Mild Steel).
• The lower error in Python and MATLAB is attributed to their use of least squares regression, which better accommodates variability in machining data, unlike the analytical method that uses fixed mathematical equations.
• Colding’s model proved to be effective for cutting parameter optimization, with the highest model error remaining relatively low.
• It was found that Colding’s coefficients were largely consistent across materials when using MATLAB and Python modelling approaches. For K1045, ‘L’ was approximately 0.13 and ‘N0’ was −0.20, while for Mild Steel, ‘L’ was around 0.12 and ‘N0’ was −0.15. The constant ‘K’ remained stable at approximately 5.77 across all techniques and materials, indicating its robustness and lower sensitivity to variations in both modelling method and workpiece material.
• Conversely, the constants ‘M’ and ‘H’ varied significantly with both the modelling approach and material, indicating they are more sensitive and could influence prediction accuracy.
• Further validation of these constants under different cutting conditions is recommended to improve the accuracy and applicability of Colding’s model.