# Influence of the sheet thickness variability on the deep drawing of a cylindrical cup

PEREIRA André¹,²,a*, PRATES Pedro¹,²,³,⁴, PARREIRA Tomás¹,² and
OLIVEIRA Marta¹,²

¹Centre for Mechanical Engineering, Materials and Processes (CEMMPRE), Department of
Mechanical Engineering, University of Coimbra, Portugal

²Advanced Production and Intelligent Systems Associated Laboratory (ARISE), 4200-465 Porto,
Portugal

³Centre for Mechanical Technology and Automation (TEMA), Mechanical Engineering
Department, University of Aveiro, Portugal

⁴Intelligent Systems Associate Laboratory (LASI), 4800-058 Guimarães, Portugal

aandre.pereira@uc.pt

**Keywords:** Thickness Variability, Cylindrical Cup, Uncertainty

**Abstract.** Sheet metal forming processes are widely used in industry. The quality of formed parts can be significantly affected by various sources of uncertainty inevitably associated with the forming process. The objective of this work is to quantify the influence of thickness variability on the forming process of a cylindrical cup. Using numerical simulation, the influence of the sheet thickness variance on the evolution of the punch force versus displacement, the equivalent plastic strain distribution, the earing profile and the thickness around the cup is studied for a given cup height. Four thickness distributions with different variance values and the same average thickness value were studied. It was concluded that an increase in variance leads to an increase in thickness dispersion (at the base and curvature of the cup) and an increase in equivalent strain dispersion along the cup. The earing profile of the cup is also affected by the thickness variability, but to a lesser extent. On the other hand, the development of the punch force is not affected by the thickness variability.

**Introduction**

Sheet metal forming is a widely used metalworking process in the automotive, aerospace, and metalworking industries [1]. These processes are generally designated and optimized through numerical simulation [2]. The Finite Element Method (FEM) is typically used, although it is based on a deterministic approach [3] that ignores the various sources of uncertainty that are unavoidable in a real industrial environment, such as variations in material properties, sheet thickness or process conditions [4]. These sources of uncertainty can significantly affect the quality of the final product [5-6], which can lead to inefficient manufacturing and costly redesign of the forming process.

Blank thickness is one of the most important factors in the stamping process, as it is dictated by the geometric requirements of the part, but also by its cost. The blanks are usually produced by the cold rolling process, so there is always some variability in the blank thickness along the material coil [7], for example due to mill chatter [8]. Several authors have studied the influence of thickness variability on wrinkling behaviour [9] and buckling [10-11], and conclude that the thickness uncertainty generates uneven stress and strain distributions, which in some cases can affect the forming results. In this sense, the blank thickness uncertainty can be one of the main uncertainty factors affecting the component quality.

This work presents a numerical study to quantify the influence of blank thickness uncertainty on the forming results of a cylindrical cup. The cylindrical cup was chosen because it is a

<figure><img src="image_1738.png" /></figure>
# Influence of the sheet thickness variability on the deep drawing of a cylindrical cup

PEREIRA André¹,²,a*, PRATES Pedro¹,²,³,⁴, PARREIRA Tomás¹,² and
OLIVEIRA Marta¹,²

¹Centre for Mechanical Engineering, Materials and Processes (CEMMPRE), Department of
Mechanical Engineering, University of Coimbra, Portugal

²Advanced Production and Intelligent Systems Associated Laboratory (ARISE), 4200-465 Porto,
Portugal

³Centre for Mechanical Technology and Automation (TEMA), Mechanical Engineering
Department, University of Aveiro, Portugal

⁴Intelligent Systems Associate Laboratory (LASI), 4800-058 Guimarães, Portugal

aandre.pereira@uc.pt

**Keywords:** Thickness Variability, Cylindrical Cup, Uncertainty

**Abstract.** Sheet metal forming processes are widely used in industry. The quality of formed parts can be significantly affected by various sources of uncertainty inevitably associated with the forming process. The objective of this work is to quantify the influence of thickness variability on the forming process of a cylindrical cup. Using numerical simulation, the influence of the sheet thickness variance on the evolution of the punch force versus displacement, the equivalent plastic strain distribution, the earing profile and the thickness around the cup is studied for a given cup height. Four thickness distributions with different variance values and the same average thickness value were studied. It was concluded that an increase in variance leads to an increase in thickness dispersion (at the base and curvature of the cup) and an increase in equivalent strain dispersion along the cup. The earing profile of the cup is also affected by the thickness variability, but to a lesser extent. On the other hand, the development of the punch force is not affected by the thickness variability.

**Introduction**

Sheet metal forming is a widely used metalworking process in the automotive, aerospace, and metalworking industries [1]. These processes are generally designated and optimized through numerical simulation [2]. The Finite Element Method (FEM) is typically used, although it is based on a deterministic approach [3] that ignores the various sources of uncertainty that are unavoidable in a real industrial environment, such as variations in material properties, sheet thickness or process conditions [4]. These sources of uncertainty can significantly affect the quality of the final product [5-6], which can lead to inefficient manufacturing and costly redesign of the forming process.

Blank thickness is one of the most important factors in the stamping process, as it is dictated by the geometric requirements of the part, but also by its cost. The blanks are usually produced by the cold rolling process, so there is always some variability in the blank thickness along the material coil [7], for example due to mill chatter [8]. Several authors have studied the influence of thickness variability on wrinkling behaviour [9] and buckling [10-11], and conclude that the thickness uncertainty generates uneven stress and strain distributions, which in some cases can affect the forming results. In this sense, the blank thickness uncertainty can be one of the main uncertainty factors affecting the component quality.

This work presents a numerical study to quantify the influence of blank thickness uncertainty on the forming results of a cylindrical cup. The cylindrical cup was chosen because it is a

<figure><img src="image_1736.png" /></figure>
commonly used benchmark test to represent sheet metal forming processes [12]. However, to the best of our knowledge, the influence of thickness uncertainty on the forming results of the cylindrical cup is still unexplored. Therefore, the aim of this work is to understand how the thickness variability can affect the forming results, in particular to evaluate the areas of the final part where the initial variations in material thickness have the greatest influence on the forming results. To this end, numerical simulations of the cylindrical cups are used to evaluate the influence of different levels of initial blank thickness variability in order to analyse their impact on the forming process results.

## Numerical Model

In this work, the cylindrical cup drawing benchmark proposed in ESAFORM 2021 [12] is studied. This forming process setup consists of four tools, the punch, the blank holder, the die and the stopper. Fig. 1 illustrates the geometry and dimensions of these tools. The circular blank has an initial average thickness, $t_0$, of 1 mm and a diameter of 107.5 mm. The numerical simulation of the forming process consists of three stages: (i) at the start of the process, the blank holder moves downwards, pressing the blank against the die until a blank holder force of 40 kN is reached; (ii) the punch then moves downwards by 54 mm, drawing the blank completely into the die, while keeping the blank holder force constant at 40 kN; (iii) the last stage consists of the removal of the tools, resulting in the springback of the cup. During the second stage, a stopper of the same thickness as the blank is used to prevent the cup ears from being pinched.

<figure><img src="image_1739.png" /><figcaption>Fig. 1 - Cylindrical cup forming process: (a) numerical model; (b) dimensions of the tools and blank in mm (adapted from [12]).</figcaption></figure>

The numerical simulations were carried out using the software DD3IMP (Deep Drawing 3D Implicit Code) [13]. In order to reduce the computational cost of the numerical simulation, only a quarter of the model was simulated due to the symmetry conditions of the material, geometry and boundary conditions. The friction between the tools and the sheet was assumed to be described by Coulomb's law with a constant friction coefficient of 0.07. The blank is discretised with 67968 (8-node hexahedral solid) elements, with 2 elements in thickness, combined with a selective reduced integration technique [13]. The final mesh is shown in Fig. 2 (a). It can be seen in this figure that the mesh is coarser in the centre because the blank is not significantly deformed in this area (bottom of the cup). In the most discretised area, i.e. the area between the flange and the punch corner, at least one element per 0.3 mm is used to represent the spatial distribution of the thickness variability. Each numerical simulation took an average of 36 hours on a computer equipped with an Intel® Core™ i7-8700K Hexa-Core processor (4.7 GHz).
commonly used benchmark test to represent sheet metal forming processes [12]. However, to the best of our knowledge, the influence of thickness uncertainty on the forming results of the cylindrical cup is still unexplored. Therefore, the aim of this work is to understand how the thickness variability can affect the forming results, in particular to evaluate the areas of the final part where the initial variations in material thickness have the greatest influence on the forming results. To this end, numerical simulations of the cylindrical cups are used to evaluate the influence of different levels of initial blank thickness variability in order to analyse their impact on the forming process results.

## Numerical Model

In this work, the cylindrical cup drawing benchmark proposed in ESAFORM 2021 [12] is studied. This forming process setup consists of four tools, the punch, the blank holder, the die and the stopper. Fig. 1 illustrates the geometry and dimensions of these tools. The circular blank has an initial average thickness, $t_0$, of 1 mm and a diameter of 107.5 mm. The numerical simulation of the forming process consists of three stages: (i) at the start of the process, the blank holder moves downwards, pressing the blank against the die until a blank holder force of 40 kN is reached; (ii) the punch then moves downwards by 54 mm, drawing the blank completely into the die, while keeping the blank holder force constant at 40 kN; (iii) the last stage consists of the removal of the tools, resulting in the springback of the cup. During the second stage, a stopper of the same thickness as the blank is used to prevent the cup ears from being pinched.

<figure><img src="image_1737.png" /><figcaption>Fig. 1 - Cylindrical cup forming process: (a) numerical model; (b) dimensions of the tools and blank in mm (adapted from [12]).</figcaption></figure>

The numerical simulations were carried out using the software DD3IMP (Deep Drawing 3D Implicit Code) [13]. In order to reduce the computational cost of the numerical simulation, only a quarter of the model was simulated due to the symmetry conditions of the material, geometry and boundary conditions. The friction between the tools and the sheet was assumed to be described by Coulomb's law with a constant friction coefficient of 0.07. The blank is discretised with 67968 (8-node hexahedral solid) elements, with 2 elements in thickness, combined with a selective reduced integration technique [13]. The final mesh is shown in Fig. 2 (a). It can be seen in this figure that the mesh is coarser in the centre because the blank is not significantly deformed in this area (bottom of the cup). In the most discretised area, i.e. the area between the flange and the punch corner, at least one element per 0.3 mm is used to represent the spatial distribution of the thickness variability. Each numerical simulation took an average of 36 hours on a computer equipped with an Intel® Core™ i7-8700K Hexa-Core processor (4.7 GHz).
<figure><img src="image_1740.png" /><figcaption>Fig. 2 – Blank model: (a) mesh discretization; (b) thickness distribution (Case 4).</figcaption></figure>

The material under investigation has an isotropic elastic behaviour described by the generalised Hooke's law and an orthotropic plastic behaviour described by the Voce hardening law and the Hill'48 yield criterion. The yield criterion is defined by:

$$F(\sigma_{yy} - \sigma_{zz})^2 + G(\sigma_{zz} - \sigma_{xx})^2 + H(\sigma_{xx} - \sigma_{yy})^2 + 2L\tau_{yz}^2 + 2M\tau_{xz}^2 + 2N\tau_{xy}^2 = Y^2, \quad (1)$$

where $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$, $\tau_{xy}$, $\tau_{yz}$ and $\tau_{xz}$ are the components of the Cauchy stress tensor; Y is the yield stress; F, G, H, L, M and N are anisotropy parameters. The parameters follow the conditions G + H = 1 and L= M = 1.5 (von Mises). The Voce hardening law is defined by:

$$Y = Y_0 + (Y_{sat} - Y_0)(1 - \exp(-c_Y \bar{\epsilon}^p)), \quad (2)$$

where $\bar{\epsilon}^p$ is the equivalent plastic strain; $Y_0$ is the initial yield stress, $Y_{sat}$ is the saturation stress and $c_Y$ is the hardening rate. The material parameters used in the study are given in Table 1. These parameters were identified for an AA 6016-T4 aluminium alloy.

<table><caption>Table 1 – Material Parameters.</caption><thead><tr><th>E [MPa]</th><th>ν</th><th>Y<sub>0</sub> [MPa]</th><th>Y<sub>sat</sub> [MPa]</th><th>C<sub>Y</sub></th><th>F</th><th>G</th><th>H</th><th>L = M</th><th>N</th></tr></thead><tbody><tr><td>65113</td><td>0.33</td><td>157.95</td><td>368.07</td><td>9.311</td><td>0.5980</td><td>0.5981</td><td>0.4019</td><td>1.5</td><td>1.2716</td></tr></tbody></table>

## Thickness Distribution

To assess the influence of thickness variability, four sheet thickness distributions were analysed. These distributions are characterised by different levels of thickness variance but identical average thickness values. For each distribution, the influence of the thickness variability on the numerical results of the cylindrical cup was evaluated, in particular the evolution of the punch force, the earing profile, the cup thickness and the equivalent plastic strain distribution.

The spatial distribution of the blank thickness is generated with a spatial random field following a Gaussian covariance model. This approach ensures a smooth spatial distribution of the random thickness values, which better represents the experimental thickness distribution [14]. In contrast, using a pure random thickness distribution could lead to a spatial distribution where the minimum and maximum thickness values could be in neighbouring points, which is unlikely due to material continuity. The fields are generated numerically using the randomisation method described in [15] and the Gaussian variogram given by:
<figure><img src="image_1738.png" /><figcaption>Fig. 2 – Blank model: (a) mesh discretization; (b) thickness distribution (Case 4).</figcaption></figure>

The material under investigation has an isotropic elastic behaviour described by the generalised Hooke's law and an orthotropic plastic behaviour described by the Voce hardening law and the Hill'48 yield criterion. The yield criterion is defined by:

$$F(\sigma_{yy} - \sigma_{zz})^2 + G(\sigma_{zz} - \sigma_{xx})^2 + H(\sigma_{xx} - \sigma_{yy})^2 + 2L\tau_{yz}^2 + 2M\tau_{xz}^2 + 2N\tau_{xy}^2 = Y^2, \quad (1)$$

where $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$, $\tau_{xy}$, $\tau_{yz}$ and $\tau_{xz}$ are the components of the Cauchy stress tensor; Y is the yield stress; F, G, H, L, M and N are anisotropy parameters. The parameters follow the conditions G + H = 1 and L= M = 1.5 (von Mises). The Voce hardening law is defined by:

$$Y = Y_0 + (Y_{sat} - Y_0)(1 - \exp(-c_Y \bar{\epsilon}^p)), \quad (2)$$

where $\bar{\epsilon}^p$ is the equivalent plastic strain; $Y_0$ is the initial yield stress, $Y_{sat}$ is the saturation stress and $c_Y$ is the hardening rate. The material parameters used in the study are given in Table 1. These parameters were identified for an AA 6016-T4 aluminium alloy.

<table><caption>Table 1 – Material Parameters.</caption><thead><tr><th>E [MPa]</th><th>ν</th><th>Y<sub>0</sub> [MPa]</th><th>Y<sub>sat</sub> [MPa]</th><th>C<sub>Y</sub></th><th>F</th><th>G</th><th>H</th><th>L = M</th><th>N</th></tr></thead><tbody><tr><td>65113</td><td>0.33</td><td>157.95</td><td>368.07</td><td>9.311</td><td>0.5980</td><td>0.5981</td><td>0.4019</td><td>1.5</td><td>1.2716</td></tr></tbody></table>

## Thickness Distribution

To assess the influence of thickness variability, four sheet thickness distributions were analysed. These distributions are characterised by different levels of thickness variance but identical average thickness values. For each distribution, the influence of the thickness variability on the numerical results of the cylindrical cup was evaluated, in particular the evolution of the punch force, the earing profile, the cup thickness and the equivalent plastic strain distribution.

The spatial distribution of the blank thickness is generated with a spatial random field following a Gaussian covariance model. This approach ensures a smooth spatial distribution of the random thickness values, which better represents the experimental thickness distribution [14]. In contrast, using a pure random thickness distribution could lead to a spatial distribution where the minimum and maximum thickness values could be in neighbouring points, which is unlikely due to material continuity. The fields are generated numerically using the randomisation method described in [15] and the Gaussian variogram given by:
$$ \gamma[d] = \xi^2 \left( 1 - \exp \left( -(d/l)^2 \right) \right), \qquad (3) $$

where $\xi^2$ is the random field variance, $d$ is the distance between two points in the field and $l$ is the length scale, which defines the distance at which the correlation between points becomes negligible. It is assumed that $l = 2/\sqrt{\pi}$, meaning that 95% of the random field variance is reached at a distance of 2 mm.

To investigate the influence of the random field variance, four blank thickness distributions were generated. These distributions are characterised by different levels of thickness variance (indicated in Table 2) but identical average thickness values, $t_0 = 1$ mm. For a clear physical meaning, the minimum and maximum thickness values for each case are also given in Table 2. Case 1 represents the ideal situation where the thickness is constant along the blank, while Case 4 represents an extreme situation where the difference between the maximum and minimum thickness values is approximately 0.1 mm. The distribution of Case 4 is shown in Fig. 2 (b).

<table><caption>Table 2 – Characteristics of the blank thickness distributions.</caption><thead><tr><td>Case</td><td>Variance, ξ² [mm²]</td><td>Minimum thickness [mm]</td><td>Maximum thickness [mm]</td></tr></thead><tbody><tr><td>1</td><td>0</td><td>1</td><td>1</td></tr><tr><td>2</td><td>0.00003</td><td>0.987</td><td>1.012</td></tr><tr><td>3</td><td>0.00013</td><td>0.973</td><td>1.025</td></tr><tr><td>4</td><td>0.0004</td><td>0.947</td><td>1.048</td></tr></tbody></table>

## Forming Results

In this section, the influence of the four thickness distributions on the results of the cylindrical cup are analysed, namely the evolution of the punch force versus displacement, the equivalent plastic strain distribution, the earning profile, $\eta$, and the thickness, $t$, around the cup for a given cup height, $h$. Fig. 3 illustrates the measurement locations of the earning profile and thickness.

<figure><img src="image_1741.png" /><figcaption>Fig. 3 – Representation of measurement locations of the earning profile and thickness.</figcaption></figure>

Punch Force Evolution. This subsection analyses the influence of blank thickness variability on the evolution of punch force versus displacement. Fig. 4 shows these results for the 4 cases studied. It can be seen that the force reaches its maximum value of 42.5 kN at a displacement of 19 mm. After this maximum, the punch force decreases in a non-linear behaviour up to a displacement of 27 mm, i.e. the moment when the blank holder loses contact with the blank. After this point, the force continues to decrease, but in a linear trend. When the thicker parts of the blank (caused by
$$ \gamma[d] = \xi^2 \left( 1 - \exp \left( - \left( \frac{d}{l} \right)^2 \right) \right), \qquad (3) $$

where $\xi^2$ is the random field variance, $d$ is the distance between two points in the field and $l$ is the length scale, which defines the distance at which the correlation between points becomes negligible. It is assumed that $l = 2/\sqrt{\pi}$, meaning that 95% of the random field variance is reached at a distance of 2 mm.

To investigate the influence of the random field variance, four blank thickness distributions were generated. These distributions are characterised by different levels of thickness variance (indicated in Table 2) but identical average thickness values, $t_0 = 1$ mm. For a clear physical meaning, the minimum and maximum thickness values for each case are also given in Table 2. Case 1 represents the ideal situation where the thickness is constant along the blank, while Case 4 represents an extreme situation where the difference between the maximum and minimum thickness values is approximately 0.1 mm. The distribution of Case 4 is shown in Fig. 2 (b).

<table><caption>Table 2 – Characteristics of the blank thickness distributions.</caption><thead><tr><td>Case</td><td>Variance, ξ² [mm²]</td><td>Minimum thickness [mm]</td><td>Maximum thickness [mm]</td></tr></thead><tbody><tr><td>1</td><td>0</td><td>1</td><td>1</td></tr><tr><td>2</td><td>0.00003</td><td>0.987</td><td>1.012</td></tr><tr><td>3</td><td>0.00013</td><td>0.973</td><td>1.025</td></tr><tr><td>4</td><td>0.0004</td><td>0.947</td><td>1.048</td></tr></tbody></table>

## Forming Results

In this section, the influence of the four thickness distributions on the results of the cylindrical cup are analysed, namely the evolution of the punch force versus displacement, the equivalent plastic strain distribution, the earning profile, $\eta$, and the thickness, $t$, around the cup for a given cup height, $h$. Fig. 3 illustrates the measurement locations of the earning profile and thickness.

<figure><img src="image_1739.png" /><figcaption>Fig. 3 – Representation of measurement locations of the earning profile and thickness.</figcaption></figure>

Punch Force Evolution. This subsection analyses the influence of blank thickness variability on the evolution of punch force versus displacement. Fig. 4 shows these results for the 4 cases studied. It can be seen that the force reaches its maximum value of 42.5 kN at a displacement of 19 mm. After this maximum, the punch force decreases in a non-linear behaviour up to a displacement of 27 mm, i.e. the moment when the blank holder loses contact with the blank. After this point, the force continues to decrease, but in a linear trend. When the thicker parts of the blank (caused by
the radial deformation of the blank) are embedded in the die, the punching force first increases (between 36 mm and 42 mm) and then decreases (between 42 mm and 44 mm) as the ears are embedded. When the blank is fully embedded (44 mm), the punching force is practically constant to counteract the frictional force. This behaviour is identical in all cases, so it can be concluded that the punching force is not influenced by the thickness variation. This conclusion is to be expected since the force is a global result, i.e. it is influenced by the deformation that occurs along the blank, so it is less sensitive to local thickness variations and more sensitive to the average thickness value.

<figure><img src="image_1742.png" /><figcaption>Fig. 4 – Evolution of the punch force for the 4 thickness distributions.</figcaption></figure>

Equivalent Plastic Strain. Fig. 5 (a) shows the distribution of equivalent plastic strain along the cup for the 4 cases. It can be seen in this figure that as the variance increases, the distribution of equivalent plastic strain becomes less uniform (i.e. relative to Case 1). The maximum value of the equivalent plastic strain also increases with the variance, reaching values of 0.777, 0.807, 0.848 and 0.886 for cases 1, 2, 3 and 4, respectively. For the most extreme variance (Case 4) there is an increase of 14% in the maximum equivalent plastic strain compared to the case with no variability in thickness (Case 1). Comparing Fig. 5 (a) with Fig. 5 (b), it can also be concluded that the regions where the blank thickness is initially smaller (black circles in Fig. 5 (b)) correspond to regions where the equivalent plastic strain reaches higher values (black circles in Fig. 5 (a)). Although this is to be expected in regions subjected to stretching, this observation is also valid in regions that are subjected to ironing (e.g. the upper part of the cup wall).
the radial deformation of the blank) are embedded in the die, the punching force first increases (between 36 mm and 42 mm) and then decreases (between 42 mm and 44 mm) as the ears are embedded. When the blank is fully embedded (44 mm), the punching force is practically constant to counteract the frictional force. This behaviour is identical in all cases, so it can be concluded that the punching force is not influenced by the thickness variation. This conclusion is to be expected since the force is a global result, i.e. it is influenced by the deformation that occurs along the blank, so it is less sensitive to local thickness variations and more sensitive to the average thickness value.

<figure><img src="image_1740.png" /><figcaption>Fig. 4 – Evolution of the punch force for the 4 thickness distributions.</figcaption></figure>

Equivalent Plastic Strain. Fig. 5 (a) shows the distribution of equivalent plastic strain along the cup for the 4 cases. It can be seen in this figure that as the variance increases, the distribution of equivalent plastic strain becomes less uniform (i.e. relative to Case 1). The maximum value of the equivalent plastic strain also increases with the variance, reaching values of 0.777, 0.807, 0.848 and 0.886 for cases 1, 2, 3 and 4, respectively. For the most extreme variance (Case 4) there is an increase of 14% in the maximum equivalent plastic strain compared to the case with no variability in thickness (Case 1). Comparing Fig. 5 (a) with Fig. 5 (b), it can also be concluded that the regions where the blank thickness is initially smaller (black circles in Fig. 5 (b)) correspond to regions where the equivalent plastic strain reaches higher values (black circles in Fig. 5 (a)). Although this is to be expected in regions subjected to stretching, this observation is also valid in regions that are subjected to ironing (e.g. the upper part of the cup wall).
<figure><img src="image_1743.png" /><figcaption>Fig. 5 – Distribution of the: (a) equivalent plastic strain for the 4 cases; (b) thickness distribution for the Case 4.</figcaption></figure>

Thickness distribution. This subsection analyses the influence of the initial thickness variance on the thickness results around the cup for a given cup height, $h$. The thickness evolution in the circumferential direction was evaluated for 5 different cup heights, 10, 15, 20, 25 and 30 mm (as shown in Fig. 3). Fig. 6 shows the evolution of the cup thickness as a function of the angle with the rolling direction for the 4 cases. It can be seen that the increasing variance of the initial blank thickness leads to a greater dispersion of the thickness distribution at the bottom and in the curvature zone of the cup. However, for a cup height of 30 mm, there is almost no influence of the initial thickness variance on the thickness distribution. In fact, as the height of the cup over which the thickness distribution is analysed increases, there is less variation in the thickness results, because the ironing of the blank tends to uniformise the blank thickness in the upper part of the cup wall.
<figure><img src="image_1741.png" /><figcaption>Fig. 5 – Distribution of the: (a) equivalent plastic strain for the 4 cases; (b) thickness distribution for the Case 4.</figcaption></figure>

Thickness distribution. This subsection analyses the influence of the initial thickness variance on the thickness results around the cup for a given cup height, $h$. The thickness evolution in the circumferential direction was evaluated for 5 different cup heights, 10, 15, 20, 25 and 30 mm (as shown in Fig. 3). Fig. 6 shows the evolution of the cup thickness as a function of the angle with the rolling direction for the 4 cases. It can be seen that the increasing variance of the initial blank thickness leads to a greater dispersion of the thickness distribution at the bottom and in the curvature zone of the cup. However, for a cup height of 30 mm, there is almost no influence of the initial thickness variance on the thickness distribution. In fact, as the height of the cup over which the thickness distribution is analysed increases, there is less variation in the thickness results, because the ironing of the blank tends to uniformise the blank thickness in the upper part of the cup wall.
<figure><img src="image_1744.png" /><figcaption>Fig. 6 – Thickness distribution evaluated for cup heights, *h*, of 10, 15, 20, 25 and 30 mm, for the 4 cases.</figcaption></figure>

Earing profile. This subsection analyses the influence of the variability of the initial blank thickness on the earing profile. Fig. 7 shows the evolution of the earing profile as a function of the angle with the rolling direction for the 4 cases. It can be seen in this figure that the earing profile has a similar evolution for the 4 cases, with four ears in total, due to the fact that it is an anisotropic material described by the Hill'48 criterion with a minimum anisotropy coefficient at 45º with the rolling direction. It can be seen in Fig. 7 that increasing the variance of the initial thickness leads to an increase in the dispersion of the earing profile. In particular, it can be seen that the cup height at 0º decreases as the variance increases.

<figure><img src="image_1745.png" /><figcaption>Fig. 7 – Earing profile of the cup for the 4 cases.</figcaption></figure>

**Conclusion**

This work numerically analyses the influence of blank thickness variability on the forming results of a cylindrical cup. For this purpose, four blank thickness distributions characterised by different
<figure><img src="image_1742.png" /><figcaption>Fig. 6 – Thickness distribution evaluated for cup heights, *h*, of 10, 15, 20, 25 and 30 mm, for the 4 cases.</figcaption></figure>

Earing profile. This subsection analyses the influence of the variability of the initial blank thickness on the earing profile. Fig. 7 shows the evolution of the earing profile as a function of the angle with the rolling direction for the 4 cases. It can be seen in this figure that the earing profile has a similar evolution for the 4 cases, with four ears in total, due to the fact that it is an anisotropic material described by the Hill'48 criterion with a minimum anisotropy coefficient at 45º with the rolling direction. It can be seen in Fig. 7 that increasing the variance of the initial thickness leads to an increase in the dispersion of the earing profile. In particular, it can be seen that the cup height at 0º decreases as the variance increases.

<figure><img src="image_1743.png" /><figcaption>Fig. 7 – Earing profile of the cup for the 4 cases.</figcaption></figure>

**Conclusion**

This work numerically analyses the influence of blank thickness variability on the forming results of a cylindrical cup. For this purpose, four blank thickness distributions characterised by different
levels of thickness variance but identical average thickness values were studied. The influence of the thickness variability was evaluated in the evolution of the punch force versus displacement, the equivalent plastic strain distribution, the earing profile and the thickness distribution around the cup for a given cup height.

From the numerical results of the cylindrical cup, it could be concluded that the results of thickness distribution, equivalent plastic strain and earing profile are influenced by the thickness variability, while the influence in the evolution of punch force versus displacement is negligible. As the thickness variance increases, there is an increase in the scatter of the thickness distribution, equivalent plastic strain and earing profile. However, some regions of the cylindrical cup are more sensitive to this variance than others. In particular, the thickness distribution is more affected at the bottom and in the curvature zone of the cup. On the upper part of the cup wall, the influence of the variance on the variability of the thickness distribution is insignificant because it is subjected to ironing.

In future work, the influence of the thickness variability on the results of the cylindrical cup will be analysed for different thickness distributions. For example, using distributions generated for different variogram models or characterised by anisotropy, which can be caused by the rolling process. It would also be interesting to carry out a study similar to the one presented in this work, but considering the numerical simulation of the entire cup (i.e. without assuming geometrical symmetries in the numerical model), with the aim of understanding, for example, whether the thickness variability can lead to a decentralisation of the cup during stamping.

## Acknowledgements

This work was sponsored by FEDER funds through the program COMPETE (Programa Operacional Factores de Competitividade), by national funds through FCT (Fundação para a Ciência e a Tecnologia) under the projects UIDB/00285/2020, UIDB/00481/2020, UIDP/00481/2020, CENTRO-01-0145-FEDER-022083, LA/P/0104/2020 and LA/P/0112/2020. It was also supported by the project RealForm (reference 2022.02370.PTDC), funded by Portuguese Foundation for Science and Technology. All supports are gratefully acknowledged.

## References

[1] A. E. Marques, P. A. Prates, A. F. G. Pereira, M. C. Oliveira, J. V. Fernandes, and B. M. Ribeiro, "Performance comparison of parametric and non-parametric regression models for uncertainty analysis of sheet metal forming processes," *Metals*, vol. 10, no. 4, p. 457, 2020. https://doi.org/10.3390/met10040457

[2] A. F. G. Pereira, M. F. Ruivo, M. C. Oliveira, J. V. Fernandes, and P. A. Prates, "Numerical study of the square cup stamping process: a stochastic analysis," *ESAFORM 2021, 2021*. https://doi.org/10.25518/esaform21.2158

[3] P. A. Prates, A. S. Adaixo, M. C. Oliveira, and J. V. Fernandes, "Numerical study on the effect of mechanical properties variability in sheet metal forming processes," *The International Journal of Advanced Manufacturing Technology*, vol. 96, no. 1-4, pp. 561-580, 2018. https://doi.org/10.1007/s00170-018-1604-y

[4] A. Col, "Investigation on press forming scatter origin," in *Proceedings of the 6th international conference on material forming*, 2003, pp. 183-6.

[5] W. Hancock, and M. Zayko, M. Autio, and D. Ponagajba, "Analysis of components of variation in automotive stamping processes," *Qual Eng*, vol. 10, no. 1, pp. 115-124, 1997. https://doi.org/10.1080/08982119708919114
levels of thickness variance but identical average thickness values were studied. The influence of the thickness variability was evaluated in the evolution of the punch force versus displacement, the equivalent plastic strain distribution, the earing profile and the thickness distribution around the cup for a given cup height.

From the numerical results of the cylindrical cup, it could be concluded that the results of thickness distribution, equivalent plastic strain and earing profile are influenced by the thickness variability, while the influence in the evolution of punch force versus displacement is negligible. As the thickness variance increases, there is an increase in the scatter of the thickness distribution, equivalent plastic strain and earing profile. However, some regions of the cylindrical cup are more sensitive to this variance than others. In particular, the thickness distribution is more affected at the bottom and in the curvature zone of the cup. On the upper part of the cup wall, the influence of the variance on the variability of the thickness distribution is insignificant because it is subjected to ironing.

In future work, the influence of the thickness variability on the results of the cylindrical cup will be analysed for different thickness distributions. For example, using distributions generated for different variogram models or characterised by anisotropy, which can be caused by the rolling process. It would also be interesting to carry out a study similar to the one presented in this work, but considering the numerical simulation of the entire cup (i.e. without assuming geometrical symmetries in the numerical model), with the aim of understanding, for example, whether the thickness variability can lead to a decentralisation of the cup during stamping.

## Acknowledgements

This work was sponsored by FEDER funds through the program COMPETE (Programa Operacional Factores de Competitividade), by national funds through FCT (Fundação para a Ciência e a Tecnologia) under the projects UIDB/00285/2020, UIDB/00481/2020, UIDP/00481/2020, CENTRO-01-0145-FEDER-022083, LA/P/0104/2020 and LA/P/0112/2020. It was also supported by the project RealForm (reference 2022.02370.PTDC), funded by Portuguese Foundation for Science and Technology. All supports are gratefully acknowledged.

## References

[1] A. E. Marques, P. A. Prates, A. F. G. Pereira, M. C. Oliveira, J. V. Fernandes, and B. M. Ribeiro, "Performance comparison of parametric and non-parametric regression models for uncertainty analysis of sheet metal forming processes," *Metals*, vol. 10, no. 4, p. 457, 2020. https://doi.org/10.3390/met10040457

[2] A. F. G. Pereira, M. F. Ruivo, M. C. Oliveira, J. V. Fernandes, and P. A. Prates, "Numerical study of the square cup stamping process: a stochastic analysis," *ESAFORM 2021, 2021*. https://doi.org/10.25518/esaform21.2158

[3] P. A. Prates, A. S. Adaixo, M. C. Oliveira, and J. V. Fernandes, "Numerical study on the effect of mechanical properties variability in sheet metal forming processes," *The International Journal of Advanced Manufacturing Technology*, vol. 96, no. 1-4, pp. 561-580, 2018. https://doi.org/10.1007/s00170-018-1604-y

[4] A. Col, "Investigation on press forming scatter origin," in *Proceedings of the 6th international conference on material forming*, 2003, pp. 183-6.

[5] W. Hancock, and M. Zayko, M. Autio, and D. Ponagajba, "Analysis of components of variation in automotive stamping processes," *Qual Eng*, vol. 10, no. 1, pp. 115-124, 1997. https://doi.org/10.1080/08982119708919114
[6] K. D. Majeske and P. C. Hammett, “Identifying sources of variation in sheet metal stamping,” International Journal of Flexible Manufacturing Systems, vol. 15, no. 1, pp. 5–18, 2003. https://doi.org/10.1023/A:1023993806025

[7] H. Müllerschön, W. Roux, D. Lorenz, and K. Roll, “Stochastic analysis of uncertainties for metal forming processes with LS-OPT,” Proceedings NUMISHEET, Interlaken, Switzerland, 2008.

[8] V. Panjkovic, “Steel Rolling: Chatter,” in Encyclopedia of Iron, Steel, and Their Alloys, CRC Press, 2016, pp. 3333–3345. doi: 10.1081/E-EISA-120050422

[9] H. Li, H. Sun, H. Liu, and N. Liu, “Loading conditions constrained wrinkling behaviors of thin-walled sheet/tube parts during metal forming,” J Mater Process Technol, vol. 296, p. 117199, 2021. https://doi.org/10.1016/j.jmatprotec.2021.117199

[10] V. Papadopoulos and M. Papadrakakis, “The effect of material and thickness variability on the buckling load of shells with random initial imperfections,” Comput Methods Appl Mech Eng, vol. 194, no. 12–16, pp. 1405–1426, 2005. https://doi.org/10.1016/j.cma.2004.01.043

[11] Y. Luo, J. Zhan, and P. Liu, “Buckling assessment of thin-walled plates with uncertain geometrical imperfections based on non-probabilistic field model,” Thin-Walled Structures, vol. 145, p. 106435, 2019. https://doi.org/10.1016/j.tws.2019.106435

[12] A. M. Habraken et al., “Analysis of ESAFORM 2021 cup drawing benchmark of an Al alloy, critical factors for accuracy and efficiency of FE simulations,” International Journal of Material Forming, vol. 15, no. 5, p. 61, 2022. https://doi.org/10.1007/s12289-022-01672-w

[13] L. F. Menezes and C. Teodosiu, “Three-dimensional numerical simulation of the deep-drawing process using solid finite elements,” J Mater Process Technol, vol. 97, no. 1–3, pp. 100–106, 2000. https://doi.org/10.1016/S0924-0136(99)00345-3

[14] B. Engel and R. Steinheimer, “Obtaining more precise flow curves from uniaxial tensile tests for FE-Simulations,” 2008.

[15] F. Heße, V. Prykhodko, S. Schlüter, and S. Attinger, “Generating random fields with a truncated power-law variogram: A comparison of several numerical methods,” Environmental Modelling & Software, vol. 55, pp. 32–48, 2014. https://doi.org/10.1016/j.envsoft.2014.01.013
[6] K. D. Majeske and P. C. Hammett, “Identifying sources of variation in sheet metal stamping,” International Journal of Flexible Manufacturing Systems, vol. 15, no. 1, pp. 5–18, 2003. https://doi.org/10.1023/A:1023993806025

[7] H. Müllerschön, W. Roux, D. Lorenz, and K. Roll, “Stochastic analysis of uncertainties for metal forming processes with LS-OPT,” Proceedings NUMISHEET, Interlaken, Switzerland, 2008.

[8] V. Panjkovic, “Steel Rolling: Chatter,” in Encyclopedia of Iron, Steel, and Their Alloys, CRC Press, 2016, pp. 3333–3345. doi: 10.1081/E-EISA-120050422

[9] H. Li, H. Sun, H. Liu, and N. Liu, “Loading conditions constrained wrinkling behaviors of thin-walled sheet/tube parts during metal forming,” J Mater Process Technol, vol. 296, p. 117199, 2021. https://doi.org/10.1016/j.jmatprotec.2021.117199

[10] V. Papadopoulos and M. Papadrakakis, “The effect of material and thickness variability on the buckling load of shells with random initial imperfections,” Comput Methods Appl Mech Eng, vol. 194, no. 12–16, pp. 1405–1426, 2005. https://doi.org/10.1016/j.cma.2004.01.043

[11] Y. Luo, J. Zhan, and P. Liu, “Buckling assessment of thin-walled plates with uncertain geometrical imperfections based on non-probabilistic field model,” Thin-Walled Structures, vol. 145, p. 106435, 2019. https://doi.org/10.1016/j.tws.2019.106435

[12] A. M. Habraken et al., “Analysis of ESAFORM 2021 cup drawing benchmark of an Al alloy, critical factors for accuracy and efficiency of FE simulations,” International Journal of Material Forming, vol. 15, no. 5, p. 61, 2022. https://doi.org/10.1007/s12289-022-01672-w

[13] L. F. Menezes and C. Teodosiu, “Three-dimensional numerical simulation of the deep-drawing process using solid finite elements,” J Mater Process Technol, vol. 97, no. 1–3, pp. 100–106, 2000. https://doi.org/10.1016/S0924-0136(99)00345-3

[14] B. Engel and R. Steinheimer, “Obtaining more precise flow curves from uniaxial tensile tests for FE-Simulations,” 2008.

[15] F. Heße, V. Prykhodko, S. Schlüter, and S. Attinger, “Generating random fields with a truncated power-law variogram: A comparison of several numerical methods,” Environmental Modelling & Software, vol. 55, pp. 32–48, 2014. https://doi.org/10.1016/j.envsoft.2014.01.013
# Constitutive model validity evaluation for MT 2.0 applications

PESHAVE Amar¹,ᵃ*, LAVA Pascal¹,ᵇ, COPPIETERS Sam²,ᶜ and
PIERRON Fabrice¹,ᵈ

¹MatchID NV, Leiekaai 25a, 9000 Ghent, Belgium

²KU Leuven, Mechatronic System Dynamics (LMSD), Celestijnenlaan 300, 3001 Leuven,
Belgium

ᵃamar.peshave@matchid.eu, ᵇpascal.lava@matchid.eu, ᶜsam.coppieters@kuleuven.be,

ᵈfabrice.pierron@matchid.eu

**Keywords:** Digital Image Correlation, Virtual Fields Method, Material Testing 2.0, Model Validity Evaluation

**Abstract.** This paper demonstrates a methodology to discriminate between the performances of different material models within the framework of Material Testing 2.0, which consists in coupling heterogeneous test configurations, full-field measurements using for instance Digital Image Correlation (DIC) and inverse identification like the Virtual Fields Method (VFM). The methodology relies on using a set of different virtual fields for parameter identification with a selected model, and to evaluate the performance of the model. The paper illustrates this methodology on anisotropic metal plasticity.

## Introduction

The necessity to identify ever more complex plasticity model in sheet metal forming leads to extensive and costly experimental campaigns, sometimes requiring up to more than ten different test configurations [1,2]. There is therefore a need to speed up the process by revisiting classical testing. This is now facilitated by the wide availability of camera-based deformation measurements like Digital Image Correlation (DIC), providing a map of deformation with spatial density comparable to that of simulation. This allows for the use of more complex test geometries and loadings that lead to heterogeneous stress and strain states. The downside of this new test paradigm is that an inverse identification technique has to be used, like the Virtual Fields Method (VFM, [3]) for instance, which in turn requires an *a priori* choice of a constitutive model. There have been recent efforts in the literature to circumvent this limitation. For instance, Flaschel *et al.* [4] consider a library of terms in a generic constitutive model. For plasticity, they use a Fourier expansion of the yield surface and they simultaneously identify the parameters and the number of terms. To do so, their cost function uses an equivalent of the VFM with local FE-based virtual fields, akin to the Equilibrium Gap Indicator (EGI) described below. Another approach, Data-Driven Identification [6], consists in using advanced statistical tools to build up a manifold relating stress to strain to identify stress fields directly. As opposed to the above, this technique is purely non-parametric. The downside is that the results can be difficult to handle as they are provided as point clouds in the stress/strain space. Finally, under the condition of isotropy, thus using the fact that stress and strain have the same principal directions, Cameron and Tasan [6] directly integrate stress equilibrium to obtain stress fields. None of the above however have tackled anisotropic plasticity so far.

The present approach is akin to that of Flaschel *et al.* [4] except that the two steps, model formulation and identification, are conducted in series rather than in parallel. Different models are first identified with a set of virtual fields, and then a second set of virtual fields ranks them lead to a final choice of models. This procedure extends from initial work aimed at hyperelastic laws [7].

<figure><img src="image_1746.png" /></figure>
# Constitutive model validity evaluation for MT 2.0 applications

PESHAVE Amar¹,ᵃ*, LAVA Pascal¹,ᵇ, COPPIETERS Sam²,ᶜ and
PIERRON Fabrice¹,ᵈ

¹MatchID NV, Leiekaai 25a, 9000 Ghent, Belgium

²KU Leuven, Mechatronic System Dynamics (LMSD), Celestijnenlaan 300, 3001 Leuven,
Belgium

ᵃamar.peshave@matchid.eu, ᵇpascal.lava@matchid.eu, ᶜsam.coppieters@kuleuven.be,

ᵈfabrice.pierron@matchid.eu

**Keywords:** Digital Image Correlation, Virtual Fields Method, Material Testing 2.0, Model Validity Evaluation

**Abstract.** This paper demonstrates a methodology to discriminate between the performances of different material models within the framework of Material Testing 2.0, which consists in coupling heterogeneous test configurations, full-field measurements using for instance Digital Image Correlation (DIC) and inverse identification like the Virtual Fields Method (VFM). The methodology relies on using a set of different virtual fields for parameter identification with a selected model, and to evaluate the performance of the model. The paper illustrates this methodology on anisotropic metal plasticity.

**Introduction**

The necessity to identify ever more complex plasticity model in sheet metal forming leads to extensive and costly experimental campaigns, sometimes requiring up to more than ten different test configurations [1,2]. There is therefore a need to speed up the process by revisiting classical testing. This is now facilitated by the wide availability of camera-based deformation measurements like Digital Image Correlation (DIC), providing a map of deformation with spatial density comparable to that of simulation. This allows for the use of more complex test geometries and loadings that lead to heterogeneous stress and strain states. The downside of this new test paradigm is that an inverse identification technique has to be used, like the Virtual Fields Method (VFM, [3]) for instance, which in turn requires an *a priori* choice of a constitutive model. There have been recent efforts in the literature to circumvent this limitation. For instance, Flaschel *et al*. [4] consider a library of terms in a generic constitutive model. For plasticity, they use a Fourier expansion of the yield surface and they simultaneously identify the parameters and the number of terms. To do so, their cost function uses an equivalent of the VFM with local FE-based virtual fields, akin to the Equilibrium Gap Indicator (EGI) described below. Another approach, Data-Driven Identification [6], consists in using advanced statistical tools to build up a manifold relating stress to strain to identify stress fields directly. As opposed to the above, this technique is purely non-parametric. The downside is that the results can be difficult to handle as they are provided as point clouds in the stress/strain space. Finally, under the condition of isotropy, thus using the fact that stress and strain have the same principal directions, Cameron and Tasan [6] directly integrate stress equilibrium to obtain stress fields. None of the above however have tackled anisotropic plasticity so far.

The present approach is akin to that of Flaschel *et al*. [4] except that the two steps, model formulation and identification, are conducted in series rather than in parallel. Different models are first identified with a set of virtual fields, and then a second set of virtual fields ranks them lead to a final choice of models. This procedure extends from initial work aimed at hyperelastic laws [7].

<figure><img src="image_1744.png" /></figure>
Methodology

Test configuration. The selected test is similar to that in [7] and is shown on Fig. 1. It is an asymmetric double notch tensile specimen first considered by Meuwissen et al. [8] to create heterogeneous stress and strain fields for elasto-plastic identification. The specimen dimensions are provided directly in Fig. 1.

Synthetic data. The methodology is illustrated here using simulated data. This is a necessary first step for which the ground truth is known (model and parameters) and the procedure can be verified. A finite element model of the test was performed on Abaqus using a Hill48 yield surface and Swift hardening. The parameters are listed in Tab. 1. To simulate the experimental process robustly, synthetic image deformation has been performed using a numerically generated speckle pattern that is numerically deformed to encode the finite element displacement data. This was first imagined by Lava et al. [9] and refined later for small strain by Rossi et al. [10]. Grey level camera noise can then be added to the images directly for realistic simulations of actual image recordings. The complete procedure was performed using the MatchID FEDEF module (www.matchid.eu).

<figure><img src="image_1747.png" /><figcaption>Fig. 1. Test configuration.</figcaption></figure>

<table><caption>Table 1. Test parameters.</caption><tbody><tr><td rowspan="2">Elastic behaviour</td><td>E (GPa)</td><td>219</td></tr><tr><td>v (-)</td><td>0.3</td></tr><tr><td rowspan="4">Hill48 yield surface</td><td>F (-)</td><td>0.23</td></tr><tr><td>G (-)</td><td>0.35</td></tr><tr><td>H (-)</td><td>0.65</td></tr><tr><td>N (-)</td><td>1.41</td></tr><tr><td rowspan="3">Swift hardening</td><td>ε<sub>0</sub> (MPa)</td><td>0.0059</td></tr><tr><td>K (MPa)</td><td>563</td></tr><tr><td>n (-)</td><td>0.276</td></tr></tbody></table>
## Methodology

Test configuration. The selected test is similar to that in [7] and is shown on Fig. 1. It is an asymmetric double notch tensile specimen first considered by Meuwissen et al. [8] to create heterogeneous stress and strain fields for elasto-plastic identification. The specimen dimensions are provided directly in Fig. 1.

Synthetic data. The methodology is illustrated here using simulated data. This is a necessary first step for which the ground truth is known (model and parameters) and the procedure can be verified. A finite element model of the test was performed on Abaqus using a Hill48 yield surface and Swift hardening. The parameters are listed in Tab. 1. To simulate the experimental process robustly, synthetic image deformation has been performed using a numerically generated speckle pattern that is numerically deformed to encode the finite element displacement data. This was first imagined by Lava et al. [9] and refined later for small strain by Rossi et al. [10]. Grey level camera noise can then be added to the images directly for realistic simulations of actual image recordings. The complete procedure was performed using the MatchID FEDEF module (www.matchid.eu).

<figure><img src="image_1745.png" /><figcaption>Fig. 1. Test configuration.</figcaption></figure>

<table><caption>Table 1. Test parameters.</caption><tr><td>Elastic behaviour</td><td></td><td></td></tr><tr><td></td><td>E (GPa)</td><td>219</td></tr><tr><td></td><td>v (-)</td><td>0.3</td></tr><tr><td>Hill48 yield surface</td><td></td><td></td></tr><tr><td></td><td>F (-)</td><td>0.23</td></tr><tr><td></td><td>G (-)</td><td>0.35</td></tr><tr><td></td><td>H (-)</td><td>0.65</td></tr><tr><td></td><td>N (-)</td><td>1.41</td></tr><tr><td>Swift hardening</td><td></td><td></td></tr><tr><td></td><td>ε<sub>0</sub> (MPa)</td><td>0.0059</td></tr><tr><td></td><td>K (MPa)</td><td>563</td></tr><tr><td></td><td>n (-)</td><td>0.276</td></tr></table>
Data processing. The numerically deformed images were processed with DIC to produce strain maps. The DIC parameters used are reported in Tab. 2. 50 load steps have been simulated. From the strain and load data, the VFM with Sensitivity-Based Virtual Fields (SBVFs, [11]) was used to identify the parameters of both Von Mises and Hill48 criteria, with the correct formulation of hardening (Ludwick).

<table><caption>Table 2. DIC parameters.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Virtual camera resolution</td><td>2448×2048</td></tr><tr><td>Software</td><td>MatchID 2024.0</td></tr><tr><td>Pixel to mm conversion</td><td>0.016 mm. pixel<sup>-1</sup></td></tr><tr><td>Subset size</td><td>15 pixel</td></tr><tr><td>Step Size</td><td>5 Pixel</td></tr><tr><td>Interpolation</td><td>Cubic spline</td></tr><tr><td>Shape function</td><td>Quadratic</td></tr><tr><td>Correlation criterion</td><td>ZNSSD</td></tr><tr><td>Pre smoothening</td><td>Gaussian 5×5 pixel</td></tr><tr><td>Strain window</td><td>21 data points</td></tr></tbody></table>

## Model ranking

Equilibrium Gap Indicator (EGI). The EGI corresponds to a particular case of the VFM with local piecewise virtual fields. It was first used in [12] to identify damage maps in composites tested in-plane, then adapted to plate bending to identify barely visible damage in impacted composite plates. It was also used to detect heterogeneities in thin paper webs loaded in-plane. More recently, it was used to discriminate between hyperelastic constitutive models [7]. The EGI can detect inconsistencies in the local distribution of stresses leading to a violation of local stress equilibrium, suggesting that the constitutive model is inadequate. This paper is the first to apply this concept to anisotropic plasticity.

Force Reconstruction Error (FRE). This is obtained using piecewise virtual fields deforming one row of data across the specimen. In [14], this was used to identify elasto-plastic parameters. Here, it is used to rank the performance of material models, as in [7] for hyperelastic materials. It can also be seen as comparing the globally applied load measured by the load cell to individual cross-sectional loads reconstructed from the measured strains and the identified model. This indicator provides a 1D plot along the specimen loading axis.

Compared to the procedure in [4], where EGI virtual fields are used to select and identify the models, here, the models are identified with a set of virtual fields (SBVFs) and ranked with another set of virtual fields. It would be interesting to benchmark the two techniques to get an insight into respective efficiency.

## Results

Figs. 2 and 3 show stress maps obtained from the FEDEF module. The conference presentation will provide EGI and FRE results as these are currently being processed.
Data processing. The numerically deformed images were processed with DIC to produce strain maps. The DIC parameters used are reported in Tab. 2. 50 load steps have been simulated. From the strain and load data, the VFM with Sensitivity-Based Virtual Fields (SBVFs, [11]) was used to identify the parameters of both Von Mises and Hill48 criteria, with the correct formulation of hardening (Ludwick).

<table><caption>Table 2. DIC parameters.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Virtual camera resolution</td><td>2448×2048</td></tr><tr><td>Software</td><td>MatchID 2024.0</td></tr><tr><td>Pixel to mm conversion</td><td>0.016 mm. pixel<sup>-1</sup></td></tr><tr><td>Subset size</td><td>15 pixel</td></tr><tr><td>Step Size</td><td>5 Pixel</td></tr><tr><td>Interpolation</td><td>Cubic spline</td></tr><tr><td>Shape function</td><td>Quadratic</td></tr><tr><td>Correlation criterion</td><td>ZNSSD</td></tr><tr><td>Pre smoothening</td><td>Gaussian 5×5 pixel</td></tr><tr><td>Strain window</td><td>21 data points</td></tr></tbody></table>

## Model ranking

Equilibrium Gap Indicator (EGI). The EGI corresponds to a particular case of the VFM with local piecewise virtual fields. It was first used in [12] to identify damage maps in composites tested in-plane, then adapted to plate bending to identify barely visible damage in impacted composite plates. It was also used to detect heterogeneities in thin paper webs loaded in-plane. More recently, it was used to discriminate between hyperelastic constitutive models [7]. The EGI can detect inconsistencies in the local distribution of stresses leading to a violation of local stress equilibrium, suggesting that the constitutive model is inadequate. This paper is the first to apply this concept to anisotropic plasticity.

Force Reconstruction Error (FRE). This is obtained using piecewise virtual fields deforming one row of data across the specimen. In [14], this was used to identify elasto-plastic parameters. Here, it is used to rank the performance of material models, as in [7] for hyperelastic materials. It can also be seen as comparing the globally applied load measured by the load cell to individual cross-sectional loads reconstructed from the measured strains and the identified model. This indicator provides a 1D plot along the specimen loading axis.

Compared to the procedure in [4], where EGI virtual fields are used to select and identify the models, here, the models are identified with a set of virtual fields (SBVFs) and ranked with another set of virtual fields. It would be interesting to benchmark the two techniques to get an insight into respective efficiency.

## Results

Figs. 2 and 3 show stress maps obtained from the FEDEF module. The conference presentation will provide EGI and FRE results as these are currently being processed.
<figure><img src="image_1748.png" /><figcaption>Fig. 2. Strain.</figcaption></figure>

<figure><img src="image_1749.png" /><figcaption>Fig. 3. FRE for Von Mises (left) and Hill48 (right) models.</figcaption></figure>

**Conclusion**

This study is part of a larger corpus of work aiming at not only using MT2.0 for faster identification of anisotropic plasticity models, but also at evaluating the performance of each model to select the most appropriate. To this purpose, specific metrics have been developed (EGI and FRE). The presentation will propose different results combining increasing levels of complexity (Von Mises, Hill48, Yld2000-2D) and show how these metrics can discriminate between these models.

**References**

[1] Y. Zhang, A. Yamanaka, S. Cooreman, T. Kuwabara and S. Coppieters, S., Inverse identification of plastic anisotropy through multiple non-conventional mechanical experiments, Int. J. of Sol. and Struct., 285 (2023) 112534.

https://doi.org/10.1016/j.ijsolstr.2023.112534
<figure><img src="image_1746.png" /><figcaption>Fig. 2. Strain.</figcaption></figure>

<figure><img src="image_1747.png" /><figcaption>Fig. 3. FRE for Von Mises (left) and Hill48 (right) models.</figcaption></figure>

**Conclusion**

This study is part of a larger corpus of work aiming at not only using MT2.0 for faster identification of anisotropic plasticity models, but also at evaluating the performance of each model to select the most appropriate. To this purpose, specific metrics have been developed (EGI and FRE). The presentation will propose different results combining increasing levels of complexity (Von Mises, Hill48, Yld2000-2D) and show how these metrics can discriminate between these models.

**References**

[1] Y. Zhang, A. Yamanaka, S. Cooreman, T. Kuwabara and S. Coppieters, S., Inverse identification of plastic anisotropy through multiple non-conventional mechanical experiments, Int. J. of Sol. and Struct., 285 (2023) 112534.
https://doi.org/10.1016/j.ijsolstr.2023.112534
[2] M. Rossi, A. Lattanzi, L. Morichelli, J.M.P Martins, S. Thuillier, A. Andrade-Campos and S. Coppieters, Testing methodologies for the calibration of advanced plasticity models for sheet metals: A review. Strain, 58(6) (2022) e12426. https://doi.org/10.1111/str.12426

[3] F. Pierron and M. Grédiac, The Virtual Fields Method, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1824-5

[4] M. Flaschel, S. Kumar and L. De Lorenzis, Discovering plasticity models without stress data, Comp. Mat., 8(1) (2022) 91. https://doi.org/10.1038/s41524-022-00752-4

[5] R. Langlois, M. Coret and J. Réthoré, Non-parametric stress field estimation for history-dependent materials: Application to ductile material exhibiting Piobert-Lüders localization bands, Strain, 58(4) (2022) e12410. https://doi.org/10.1111/str.12410

[6] B.C. Cameron and C.C Tasan, Full-field stress computation from measured deformation fields: A hyperbolic formulation. J. Mech. Phys. Sol., 147 (2021) 104186. https://doi.org/10.1016/j.jmps.2020.104186

[7] A. Peshave, F. Pierron, P. Lava, D. Moens, D. Vandepitte, Metrics to evaluate constitutive model fitness based on DIC, in revision, Strain (2023).

[8] M.H.H Meuwissen, C.W.J. Oomens, F.P.T. Baaijens, R. Petterson, and J.D. Janssen, Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method, J. Mat. Proc. Tech., 75(1-3) (1998) 204-211. https://doi.org/10.1016/S0924-0136(97)00366-X

[9] P. Lava, S. Cooreman, S. Coppieters, M. De Strycker and D. Debruyne, Assessment of measuring errors in DIC using deformation fields generated by plastic FEA. Opt. Las. Eng., 47(7-8) (2009) 747-753. https://dx.doi.org/10.1016/j.optlaseng.2009.03.007

[10] M. Rossi, P. Lava, F. Pierron, D. Debruyne and M. Sasso, Effect of DIC spatial resolution, noise and interpolation error on identification results with the VFM, Strain, 51(3) (2015) 206-222. https://doi.org/10.1111/str.12134

[11] D. Claire, F. Hild, F. and S. Roux, A finite element formulation to identify damage fields: The equilibrium gap method. Int. J. for Num. Meth. in Eng., 61(2) (2004) 189-208. https://doi.org/10.1002/nme.1057

[12] C. Devivier, F. Pierron, and M. Wisnom, Impact damage detection in composite plates using deflectometry and the Virtual Fields Method, Comp. A, 48 (2013) 201-218. http://dx.doi.org/10.1016/j.compositesa.2013.01.011

[13] J. Considine, F. Pierron, K. Turner, P. Lava, and X. Tang, Smoothly varying inplane stiffness heterogeneity evaluated under uniaxial tensile stress, Strain, 53(5) (2017) e12237. http://dx.doi.org/10.1111/str.12237

[14] Y. Pannier, S. Avril, R. Rotinat and F. Pierron, Identification of elasto-plastic constitutive parameters from statically undetermined tests using the virtual fields method, Exp. Mech., 46(6) (2006) 735-755. http://dx.doi.org/10.1007/s11340-006-9822-x
[2] M. Rossi, A. Lattanzi, L. Morichelli, J.M.P Martins, S. Thuillier, A. Andrade-Campos and S. Coppieters, Testing methodologies for the calibration of advanced plasticity models for sheet metals: A review. Strain, 58(6) (2022) e12426. https://doi.org/10.1111/str.12426

[3] F. Pierron and M. Grédiac, The Virtual Fields Method, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1824-5

[4] M. Flaschel, S. Kumar and L. De Lorenzis, Discovering plasticity models without stress data, Comp. Mat., 8(1) (2022) 91. https://doi.org/10.1038/s41524-022-00752-4

[5] R. Langlois, M. Coret and J. Réthoré, Non-parametric stress field estimation for history-dependent materials: Application to ductile material exhibiting Piobert-Lüders localization bands, Strain, 58(4) (2022) e12410. https://doi.org/10.1111/str.12410

[6] B.C. Cameron and C.C Tasan, Full-field stress computation from measured deformation fields: A hyperbolic formulation. J. Mech. Phys. Sol., 147 (2021) 104186. https://doi.org/10.1016/j.jmps.2020.104186

[7] A. Peshave, F. Pierron, P. Lava, D. Moens, D. Vandepitte, Metrics to evaluate constitutive model fitness based on DIC, in revision, Strain (2023).

[8] M.H.H Meuwissen, C.W.J. Oomens, F.P.T. Baaijens, R. Petterson, and J.D. Janssen, Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method, J. Mat. Proc. Tech., 75(1-3) (1998) 204-211. https://doi.org/10.1016/S0924-0136(97)00366-X

[9] P. Lava, S. Cooreman, S. Coppieters, M. De Strycker and D. Debruyne, Assessment of measuring errors in DIC using deformation fields generated by plastic FEA. Opt. Las. Eng., 47(7-8) (2009) 747-753. https://dx.doi.org/10.1016/j.optlaseng.2009.03.007

[10] M. Rossi, P. Lava, F. Pierron, D. Debruyne and M. Sasso, Effect of DIC spatial resolution, noise and interpolation error on identification results with the VFM, Strain, 51(3) (2015) 206-222. https://doi.org/10.1111/str.12134

[11] D. Claire, F. Hild, F. and S. Roux, A finite element formulation to identify damage fields: The equilibrium gap method. Int. J. for Num. Meth. in Eng., 61(2) (2004) 189-208. https://doi.org/10.1002/nme.1057

[12] C. Devivier, F. Pierron, and M. Wisnom, Impact damage detection in composite plates using deflectometry and the Virtual Fields Method, Comp. A, 48 (2013) 201-218. http://dx.doi.org/10.1016/j.compositesa.2013.01.011

[13] J. Considine, F. Pierron, K. Turner, P. Lava, and X. Tang, Smoothly varying inplane stiffness heterogeneity evaluated under uniaxial tensile stress, Strain, 53(5) (2017) e12237. http://dx.doi.org/10.1111/str.12237

[14] Y. Pannier, S. Avril, R. Rotinat and F. Pierron, Identification of elasto-plastic constitutive parameters from statically undetermined tests using the virtual fields method, Exp. Mech., 46(6) (2006) 735-755. http://dx.doi.org/10.1007/s11340-006-9822-x
Machining, cutting and severe plastic
deformation processes

<figure><img src="image_1750.png" /></figure>
Machining, cutting and severe plastic
deformation processes

<figure><img src="image_1748.png" /></figure>
# Numerical modelling of the micro-cutting in the abrasion process with pyramidal indenter

WERCHFENI Achref¹,ª*, MOUFKI Abdelhadi¹,ᵇ, LEFEBVRE André¹,ᶜ and SINOT Olivier¹,ᵈ

¹University of Lorraine, LEM3 laboratory, France

ªachref.werchfeni@univ-lorraine.fr, ᵇabdelhadi.moufki@univ-lorraine.fr,

ᶜandre.lefebvre@univ-lorraine.fr, ᵈolivier.sinot@univ-lorraine.fr

**Keywords:** Abrasion Process, Finite Element Method, CEL (Coupled Eulerian-Lagrangian) Approach, Single Grit Grinding, Grain/Material, Interaction

**Abstract.** A 3-D finite element model of abrasion process based on a CEL (Coupled Eulerian-Lagrangian) approach was developed. A scratch test on elastic-perfectly plastic materials with a pyramidal indenter was simulated. The influence of the interfacial friction coefficient *f*, the geometric parameters of the indenter and the cutting conditions on the overall friction coefficient *μ* were studied. Subsequently, the finite element simulation results were compared with an analytical model. It was found that the *μ* increases linearly with the friction coefficient *f* and the attack angle of grit *β*. The FE model results present a good agreement with the analytical model results.

## Introduction

Grinding is a material removal process using a grinding wheel consisting of agglomerate grains. It enables surface quality and tight dimensional tolerances to be achieved on very hard materials. Almost all of the energy required is converted into heat, which can lead to high temperatures at the wheel/part interface, geometric defects and thermal damage. Thermal instrumentation with ground thermocouples allows the inverse determination of heat flow densities, partition coefficients, and convection exchange coefficients at the wheel/workpiece/lubricant interface [1]. Grinding forces can be used as an indicator to monitor the process. Indeed, these forces are related to the rise in temperature and the appearance of vibrations. Analytical modeling has great importance in order to analyze the work material and the grinding wheel. Therefore, modelling can be used to study the effects of cutting conditions in order to improve the quality of the grinding.

The grinding wheel is a complex tool consisting of abrasive grains with undefined cutting edges and geometry, which makes the modelling process much more complex than in conventional machining. The process of abrasion by a single grit grinding is broken down in the literature into several phases: elastic and then plastic deformations with the formation of microchips and side beads [2,3]. The production of microchips, with a thickness in the order of one micrometer, results from micro-cutting process due to the abrasive action of the active grains at the wheel/part interface for which the attack angle is larger enough.

Xie and Williams [4] modeled the local abrasion process with a pyramidal indenter, taking into account the angle of attack and the coefficient of interfacial friction. This model has been developed to predict the overall coefficient of friction (i.e. the force ratio: ratio of the tangential grinding force to the normal grinding force) and the wear rate when a surface is sliding against a randomly rough harder surface. They showed that the overall coefficient of friction of an asperity contact depend on the deformation mode, the operating condition and the mechanical properties of the soft surface. This model provides a qualitative prediction on the wear process and wear rate.

On a global scale of abrasion process, current approaches, such as those proposed by [5,6], takes into account the actual topography of the grinding wheel with statistical approaches to

<figure><img src="image_1751.png" /></figure>
# Numerical modelling of the micro-cutting in the abrasion process with pyramidal indenter

WERCHFENI Achref¹,ª*, MOUFKI Abdelhadi¹,ᵇ, LEFEBVRE André¹,ᶜ and SINOT Olivier¹,ᵈ

¹University of Lorraine, LEM3 laboratory, France

ªachref.werchfeni@univ-lorraine.fr, ᵇabdelhadi.moufki@univ-lorraine.fr,

ᶜandre.lefebvre@univ-lorraine.fr, ᵈolivier.sinot@univ-lorraine.fr

**Keywords:** Abrasion Process, Finite Element Method, CEL (Coupled Eulerian-Lagrangian) Approach, Single Grit Grinding, Grain/Material, Interaction

**Abstract.** A 3-D finite element model of abrasion process based on a CEL (Coupled Eulerian-Lagrangian) approach was developed. A scratch test on elastic-perfectly plastic materials with a pyramidal indenter was simulated. The influence of the interfacial friction coefficient *f*, the geometric parameters of the indenter and the cutting conditions on the overall friction coefficient *μ* were studied. Subsequently, the finite element simulation results were compared with an analytical model. It was found that the *μ* increases linearly with the friction coefficient *f* and the attack angle of grit *β*. The FE model results present a good agreement with the analytical model results.

## Introduction

Grinding is a material removal process using a grinding wheel consisting of agglomerate grains. It enables surface quality and tight dimensional tolerances to be achieved on very hard materials. Almost all of the energy required is converted into heat, which can lead to high temperatures at the wheel/part interface, geometric defects and thermal damage. Thermal instrumentation with ground thermocouples allows the inverse determination of heat flow densities, partition coefficients, and convection exchange coefficients at the wheel/workpiece/lubricant interface [1]. Grinding forces can be used as an indicator to monitor the process. Indeed, these forces are related to the rise in temperature and the appearance of vibrations. Analytical modeling has great importance in order to analyze the work material and the grinding wheel. Therefore, modelling can be used to study the effects of cutting conditions in order to improve the quality of the grinding.

The grinding wheel is a complex tool consisting of abrasive grains with undefined cutting edges and geometry, which makes the modelling process much more complex than in conventional machining. The process of abrasion by a single grit grinding is broken down in the literature into several phases: elastic and then plastic deformations with the formation of microchips and side beads [2,3]. The production of microchips, with a thickness in the order of one micrometer, results from micro-cutting process due to the abrasive action of the active grains at the wheel/part interface for which the attack angle is larger enough.

Xie and Williams [4] modeled the local abrasion process with a pyramidal indenter, taking into account the angle of attack and the coefficient of interfacial friction. This model has been developed to predict the overall coefficient of friction (i.e. the force ratio: ratio of the tangential grinding force to the normal grinding force) and the wear rate when a surface is sliding against a randomly rough harder surface. They showed that the overall coefficient of friction of an asperity contact depend on the deformation mode, the operating condition and the mechanical properties of the soft surface. This model provides a qualitative prediction on the wear process and wear rate.

On a global scale of abrasion process, current approaches, such as those proposed by [5,6], takes into account the actual topography of the grinding wheel with statistical approaches to

<figure><img src="image_1749.png" /></figure>
determine the height distribution of grain heights and slopes. In this context, Torrance and Badger [5] proposed a grinding forces model based on the 3D model of Xie and Williams [7,4], in combination with a statistical approach. The model was used to evaluate the likely effects of wheel grade, coolant, etc. on the grinding forces and roughness of the workpiece.

From the literature review, it can be noted that the model of Xie and Williams, based on various assumptions, is used by several authors [3,8,9,10]. Consequently, the validity analysis of the model is important, by comparing its results with those obtained from a more precise model such as a finite element model. In the present work, the model of Xie and Williams is compared with a numerical modelling of the abrasion process at grain scale based on a CEL (Coupled Eulerian-Lagrangian) approach. The aim of this study is to determine the effects of overlapping and indenter geometry on the formation of lateral ridges, taking into account the work material behavior, with respect to the cutting direction (feed direction), the tribological conditions and the operating conditions. Additionally, the study seeks to evaluate the validity of Xie and Williams' model [7,4] by comparing the obtained results with a finite element model of abrasion machining using a pyramidal indenter.

## Finite element simulation

To analyze the grain/material interaction, a 3D finite element simulation of abrasion machining on a local scale is carried out by using ABAQUS/CAE software. The abrasive grain is modelled by using a rigid pyramid-shaped indenter with a square base, a pyramid angle of $2\alpha$, an attack angle $\beta$ and a radius of 10 µm at the tip as shown in Fig. 1. The workpiece is modelled with a deformable part and dimensions of length 0.5 mm, width 0.5 mm and height 0.16 mm. The workpiece material is 42CD4 steel, whose elastic material properties are Young's modulus $E=200$ GPa, Poisson's ratio $\nu=0.3$, Density $\rho=7800$ Kg.m$^{-3}$ and yield stresses are 612 MPa. A first scratch is pre-machined in the geometry of the workpiece. As for the FEM boundary condition, the workpiece bottom was set to encastre type. The displacement load of the indenter was loaded on reference point RP, while the indenter could only move in X and Z directions. The cutting path of single grain for FEM simulation is defined as in Fig. 2 to simulate different parameters of grain/material interaction.

<figure><img src="image_1752.png" /><figcaption>Figure 1. The workpiece and the pyramidal indenter used for the Finite Element Model (FEM)</figcaption></figure>

In developed 3D FEM, a CEL (Coupled Eulerian-Lagrangian) approach and a progressive fine mesh for the interaction zone are used. Finer meshes over the interaction area provide better conformity of contact grain/material and maximum accuracy of FE results. The grain and Eulerian part (work material) are modelled by using quadratic tetrahedral elements of type C3D10M and linear hexahedral elements of type EC3D8R, respectively. For the numerical stability of the CEL approach, the rules for the orientation and the dimensions of the mesh, proposed by Ducobu et al [11, 12], have been adopted in the present work. The simulation conditions correspond to those
determine the height distribution of grain heights and slopes. In this context, Torrance and Badger [5] proposed a grinding forces model based on the 3D model of Xie and Williams [7,4], in combination with a statistical approach. The model was used to evaluate the likely effects of wheel grade, coolant, etc. on the grinding forces and roughness of the workpiece.

From the literature review, it can be noted that the model of Xie and Williams, based on various assumptions, is used by several authors [3,8,9,10]. Consequently, the validity analysis of the model is important, by comparing its results with those obtained from a more precise model such as a finite element model. In the present work, the model of Xie and Williams is compared with a numerical modelling of the abrasion process at grain scale based on a CEL (Coupled Eulerian-Lagrangian) approach. The aim of this study is to determine the effects of overlapping and indenter geometry on the formation of lateral ridges, taking into account the work material behavior, with respect to the cutting direction (feed direction), the tribological conditions and the operating conditions. Additionally, the study seeks to evaluate the validity of Xie and Williams' model [7,4] by comparing the obtained results with a finite element model of abrasion machining using a pyramidal indenter.

## Finite element simulation

To analyze the grain/material interaction, a 3D finite element simulation of abrasion machining on a local scale is carried out by using ABAQUS/CAE software. The abrasive grain is modelled by using a rigid pyramid-shaped indenter with a square base, a pyramid angle of $2\alpha$, an attack angle $\beta$ and a radius of 10 µm at the tip as shown in Fig. 1. The workpiece is modelled with a deformable part and dimensions of length 0.5 mm, width 0.5 mm and height 0.16 mm. The workpiece material is 42CD4 steel, whose elastic material properties are Young's modulus $E=200$ GPa, Poisson's ratio $\nu=0.3$, Density $\rho=7800$ Kg.m$^{-3}$ and yield stresses are 612 MPa. A first scratch is pre-machined in the geometry of the workpiece. As for the FEM boundary condition, the workpiece bottom was set to encastre type. The displacement load of the indenter was loaded on reference point RP, while the indenter could only move in X and Z directions. The cutting path of single grain for FEM simulation is defined as in Fig. 2 to simulate different parameters of grain/material interaction.

<figure><img src="image_1750.png" /><figcaption>Figure 1. The workpiece and the pyramidal indenter used for the Finite Element Model (FEM)</figcaption></figure>

In developed 3D FEM, a CEL (Coupled Eulerian-Lagrangian) approach and a progressive fine mesh for the interaction zone are used. Finer meshes over the interaction area provide better conformity of contact grain/material and maximum accuracy of FE results. The grain and Eulerian part (work material) are modelled by using quadratic tetrahedral elements of type C3D10M and linear hexahedral elements of type EC3D8R, respectively. For the numerical stability of the CEL approach, the rules for the orientation and the dimensions of the mesh, proposed by Ducobu et al [11, 12], have been adopted in the present work. The simulation conditions correspond to those
used by Xie and Williams [7] in their 3D model for the interaction of single hard pyramid-shaped asperity in sliding contact with a softer surface [4]. In order to respect the assumptions of Xie's model and to decouple the grinding parameter effects during the grain/material interaction in the present work, the workpiece material behavior is assumed elastic-perfectly plastic.

<figure><img src="image_1753.png" /><figcaption>Figure 2. A single grain simulation path</figcaption></figure>

Several numerical simulations were carried out according to the geometric parameters and the friction coefficient used in Xie and Williams's model. In this context, three modes of material deformation and three contact regimes were defined by Xie and Williams [7]: (i) elastic shakedown involves non-permanent elastic deformation of the material, (ii) ploughing involves permanent plastic deformation without material removal, and (iii) micro-cutting involves material being simultaneously sheared beneath the grit, being pushed sideways, and being removed as chips. For each mode, an abrasive wear coefficient K and a force ratio or an overall friction coefficient μ are developed. These models are defined as a function of the attack angle of grit β, the relative overlap of successive grinding scratches l, the shear yield strength of the work surface k, the ratio of the surface hardness Hs to the bulk hardness Hb and the Tresca's friction coefficient f (ratio between the shear strength of the rake interface to the shear strength of the workpiece).

$$ \mu = \sqrt{\frac{2}{\pi} \frac{\tan \beta}{l^{0.25}}} \left(1 - \frac{f}{3\sqrt{3}} \left[1 + \frac{\pi}{4(\tan \beta)^2}\right]\right)^{0.5} \quad (1) $$

$$ K = 0.003 \frac{3\sqrt{3}(\tan \beta)^3}{fk\sqrt{l}} \sqrt{\frac{H_b}{H_s}} \quad (2) $$

$$ l = \frac{L}{(h+h') \tan \alpha} \quad (3) $$

$$ F_n = \frac{H_s h^2}{(\tan \beta)^2} \quad (4) $$

Torrance and Badger [5] have confirmed that the attack angles in a grinding wheel vary in the range of 5° to 30° (angles greater than 30° are quite rare). Thus, in this FE study, we have ignored
used by Xie and Williams [7] in their 3D model for the interaction of single hard pyramid-shaped asperity in sliding contact with a softer surface [4]. In order to respect the assumptions of Xie's model and to decouple the grinding parameter effects during the grain/material interaction in the present work, the workpiece material behavior is assumed elastic-perfectly plastic.

<figure><img src="image_1751.png" /><figcaption>Figure 2. A single grain simulation path</figcaption></figure>

Several numerical simulations were carried out according to the geometric parameters and the friction coefficient used in Xie and Williams's model. In this context, three modes of material deformation and three contact regimes were defined by Xie and Williams [7]: (i) elastic shakedown involves non-permanent elastic deformation of the material, (ii) ploughing involves permanent plastic deformation without material removal, and (iii) micro-cutting involves material being simultaneously sheared beneath the grit, being pushed sideways, and being removed as chips. For each mode, an abrasive wear coefficient K and a force ratio or an overall friction coefficient μ are developed. These models are defined as a function of the attack angle of grit β, the relative overlap of successive grinding scratches l, the shear yield strength of the work surface k, the ratio of the surface hardness Hs to the bulk hardness Hb and the Tresca's friction coefficient f (ratio between the shear strength of the rake interface to the shear strength of the workpiece).

$$ \mu = \sqrt{\frac{2}{\pi} \frac{\tan \beta}{l^{0.25}}} \left(1 - \frac{f}{3\sqrt{3}} \left[1 + \frac{\pi}{4(\tan \beta)^2}\right]\right)^{0.5} \quad (1) $$

$$ K = 0.003 \frac{3\sqrt{3}(\tan \beta)^3}{fk\sqrt{l}} \sqrt{\frac{H_b}{H_s}} \quad (2) $$

$$ l = \frac{L}{(h+h') \tan \alpha} \quad (3) $$

$$ F_n = \frac{H_s h^2}{(\tan \beta)^2} \quad (4) $$

Torrance and Badger [5] have confirmed that the attack angles in a grinding wheel vary in the range of 5° to 30° (angles greater than 30° are quite rare). Thus, in this FE study, we have ignored
some physical mechanisms considered by Xie and Williams (ploughing and rubbing; and rubbing alone). In the present work, the study is focused on the micro-cutting mode for a multiple-pass test when the model of *K* and *μ* are defined by equations (Eq.1) and (Eq.2). In this type of test, the relative overlap of successive grinding scratches *l* has an important role and a direct effect on the defined models. This parameter is defined as the distance between adjacent scratches *L* divided by the half-width of the plowed grooves, so it can be written as Eq.3, where *h* is the scratch depth and *h'* is the pre-existing ridge height, as shown below in Fig. 3.

<figure><img src="image_1754.png" /><figcaption>Figure 3. Geometry of a wear track produced in material already deformed by the production of an earlier track</figcaption></figure>

## Results and discussion

A set of 25 simulations were carried out according to different geometric parameters of Xie and Williams's model. For each FE simulation, a single parameter is varied in order to determine the evolution of the normal $F_n$, tangential $F_t$ and lateral $F_l$ forces generated by the indenter penetration in the workpiece. The values of the parameters were chosen to be consistent with Xie's experimental conditions. Table 1 presents the characteristics of the FE simulations. Subsequently, the numerical results of the overall friction coefficient are compared to the model (Eq.1 and Eq.2) for each parameter (*L*, *h*, *h'*, *β* and *f*). The Fig. 4 (a) and (b) shows an example of the results of an FE simulation and the corresponding cutting forces.

<table><thead><tr><th>Parameter</th><th>Values</th><th>Number of simulations</th><th>Simulation duration</th></tr></thead><tbody><tr><td><i>L</i> [µm]</td><td>10, 15, 20, 25, 30</td><td>5</td><td>69 hours</td></tr><tr><td><i>h</i> [µm]</td><td>12, 16, 20, 24, 28</td><td>5</td><td>70,5 hours</td></tr><tr><td><i>h'</i> [µm]</td><td>0, 2, 4, 6, 8</td><td>5</td><td>70,5 hours</td></tr><tr><td>β [°]</td><td>15°, 20°, 25.5°, 30°, 35°</td><td>5</td><td>69 hours</td></tr><tr><td><i>f</i></td><td>0, 0.2, 0.34, 0.48, 0.57</td><td>5</td><td>70,5 hours</td></tr><tr><td></td><td></td><td>Total</td><td>349,5 hours</td></tr></tbody></table>
some physical mechanisms considered by Xie and Williams (ploughing and rubbing; and rubbing alone). In the present work, the study is focused on the micro-cutting mode for a multiple-pass test when the model of *K* and *μ* are defined by equations (Eq.1) and (Eq.2). In this type of test, the relative overlap of successive grinding scratches *l* has an important role and a direct effect on the defined models. This parameter is defined as the distance between adjacent scratches *L* divided by the half-width of the plowed grooves, so it can be written as Eq.3, where *h* is the scratch depth and *h'* is the pre-existing ridge height, as shown below in Fig. 3.

<figure><img src="image_1752.png" /><figcaption>Figure 3. Geometry of a wear track produced in material already deformed by the production of an earlier track</figcaption></figure>

## Results and discussion

A set of 25 simulations were carried out according to different geometric parameters of Xie and Williams's model. For each FE simulation, a single parameter is varied in order to determine the evolution of the normal $F_n$, tangential $F_t$ and lateral $F_l$ forces generated by the indenter penetration in the workpiece. The values of the parameters were chosen to be consistent with Xie's experimental conditions. Table 1 presents the characteristics of the FE simulations. Subsequently, the numerical results of the overall friction coefficient are compared to the model (Eq.1 and Eq.2) for each parameter (*L*, *h*, *h'*, *β* and *f*). The Fig. 4 (a) and (b) shows an example of the results of an FE simulation and the corresponding cutting forces.

<table><thead><tr><th>Parameter</th><th>Values</th><th>Number of simulations</th><th>Simulation duration</th></tr></thead><tbody><tr><td><i>L</i> [µm]</td><td>10, 15, 20, 25, 30</td><td>5</td><td>69 hours</td></tr><tr><td><i>h</i> [µm]</td><td>12, 16, 20, 24, 28</td><td>5</td><td>70,5 hours</td></tr><tr><td><i>h'</i> [µm]</td><td>0, 2, 4, 6, 8</td><td>5</td><td>70,5 hours</td></tr><tr><td>β [°]</td><td>15°, 20°, 25.5°, 30°, 35°</td><td>5</td><td>69 hours</td></tr><tr><td><i>f</i></td><td>0, 0.2, 0.34, 0.48, 0.57</td><td>5</td><td>70,5 hours</td></tr><tr><td></td><td></td><td>Total</td><td>349,5 hours</td></tr></tbody></table>
<figure><img src="image_1755.png" /><figcaption>Figure 4. An example of an FE simulation where L=20µm, h=12µm, h'=6µm, β=20° and f=0.2:
(a) The equivalent plastic strain in the workpiece (PEEQ). (b) The normal, tangential and lateral force generated by the indenter</figcaption></figure>

**Effect of the interfacial friction coefficient f:** To analyze the influence of the local friction coefficient *f*, on the magnitude of the overall friction coefficient *μ*, *f* was varied between 0 and 0.57, while keeping the scratch depth, the attack angle, the pre-existing ridge height and the distance between adjacent scratches at 12µm, 25.5°, 6µm and 20µm, respectively, during the scratch test. The Fig. 5 shows the numerical and model results of the force ratio (μ = Ft/Fn) as a function of the friction. It can be observed that the numerical force ratio increases as the interfacial friction coefficient increases. Clearly, the normal forces were not significantly affected by *f*. The interfacial friction had a more effect on the tangential forces. This was also confirmed by the work of Anderson et al [13]. Which also shows that the cutting tool is doing more ploughing or cutting as the interfacial friction increases. Except this is not the case for Xie and Williams's model, the empirical force ratio gradually decreases with the increase in friction.

<figure><img src="image_1756.png" /><figcaption>Figure 5. Variation of the force ratios with the interfacial friction coefficient</figcaption></figure>

**Effect of the attack angle of grit β:** The results of the simulations are summarized in Fig. 6 (a). It is seen that the force ratios increase rapidly with *β* when the other parameters are fixed (*f*=0.2, *h*=12µm, *L*=20µm and *h*'=6µm). This can be explained by the increase in the value of forces generated by the indenter with a decrease in the attack angle, meaning that the normal force is much greater for *β* = 15° than those for larger *β* values, while the tangential force is not
<figure><img src="image_1753.png" /><figcaption>Figure 4. An example of an FE simulation where L=20µm, h=12µm, h'=6µm, β=20° and f=0.2:
(a) The equivalent plastic strain in the workpiece (PEEQ). (b) The normal, tangential and lateral force generated by the indenter</figcaption></figure>

**Effect of the interfacial friction coefficient f:** To analyze the influence of the local friction coefficient *f*, on the magnitude of the overall friction coefficient *μ*, *f* was varied between 0 and 0.57, while keeping the scratch depth, the attack angle, the pre-existing ridge height and the distance between adjacent scratches at 12µm, 25.5°, 6µm and 20µm, respectively, during the scratch test. The Fig. 5 shows the numerical and model results of the force ratio (μ = Ft/Fn) as a function of the friction. It can be observed that the numerical force ratio increases as the interfacial friction coefficient increases. Clearly, the normal forces were not significantly affected by *f*. The interfacial friction had a more effect on the tangential forces. This was also confirmed by the work of Anderson et al [13]. Which also shows that the cutting tool is doing more ploughing or cutting as the interfacial friction increases. Except this is not the case for Xie and Williams's model, the empirical force ratio gradually decreases with the increase in friction.

<figure><img src="image_1754.png" /><figcaption>Figure 5. Variation of the force ratios with the interfacial friction coefficient</figcaption></figure>

**Effect of the attack angle of grit β:** The results of the simulations are summarized in Fig. 6 (a). It is seen that the force ratios increase rapidly with *β* when the other parameters are fixed (f=0.2, h=12µm, L=20µm and h'=6µm). This can be explained by the increase in the value of forces generated by the indenter with a decrease in the attack angle, meaning that the normal force is much greater for *β* = 15° than those for larger *β* values, while the tangential force is not
significantly affected by $\beta$. This is because for a given scratch depth, the contact area between the indenter and the workpiece increases rapidly with decrease in the attack angle, and hence, much larger normal forces are required to remove the material for a low angle $\beta$, as shown in Fig. 6 (b). This was confirmed by the Torrance and Badger's model [5] of the normal force on the grain (Eq.4). These numerical results also consistent with the experimental results of Subhash and Zhang [14].

<figure><img src="image_1757.png" /><figcaption>Figure 6. Variation of the force ratios (a), the normal and tangential forces generated by the indenter (b) with the attack angle of grit</figcaption></figure>

**Effect of the scratch depth h:** As could be seen in Fig. 7, as the depth of cut increases, practically, there is no important effect on the evolution of the force ratios. All other parameters are considered as constant ($\beta$=25.5°, $f$=0.2, $L$=20µm and $h'$=6µm), and only the effect of scratch depth on the force ratios are investigated. These results are not surprising, it was expected that the increase in the scratch depth leads to a more severe material removal condition and consequently a proportional increase on the normal and tangential forces generated by the indenter, which gives a numerical force ratio does not depend on the variation of the scratch depth.

<figure><img src="image_1758.png" /><figcaption>Figure 7. Variation of the force ratios with the scratch depth</figcaption></figure>

**Effect of L and h':** Fig. 8 (a) and (b) shows the variation of the force ratios with the distance between adjacent scratches (distance between the pre-existing scratch and the machined scratch) and the pre-existing ridge height, respectively, at the depth of 12µm, the Tresca's friction
significantly affected by $\beta$. This is because for a given scratch depth, the contact area between the indenter and the workpiece increases rapidly with decrease in the attack angle, and hence, much larger normal forces are required to remove the material for a low angle $\beta$, as shown in Fig. 6 (b). This was confirmed by the Torrance and Badger's model [5] of the normal force on the grain (Eq.4). These numerical results also consistent with the experimental results of Subhash and Zhang [14].

<figure><img src="image_1755.png" /><figcaption>Figure 6. Variation of the force ratios (a), the normal and tangential forces generated by the indenter (b) with the attack angle of grit</figcaption></figure>

**Effect of the scratch depth h:** As could be seen in Fig. 7, as the depth of cut increases, practically, there is no important effect on the evolution of the force ratios. All other parameters are considered as constant ($\beta$=25.5°, $f$=0.2, $L$=20µm and $h'$=6µm), and only the effect of scratch depth on the force ratios are investigated. These results are not surprising, it was expected that the increase in the scratch depth leads to a more severe material removal condition and consequently a proportional increase on the normal and tangential forces generated by the indenter, which gives a numerical force ratio does not depend on the variation of the scratch depth.

<figure><img src="image_1756.png" /><figcaption>Figure 7. Variation of the force ratios with the scratch depth</figcaption></figure>

**Effect of L and h':** Fig. 8 (a) and (b) shows the variation of the force ratios with the distance between adjacent scratches (distance between the pre-existing scratch and the machined scratch) and the pre-existing ridge height, respectively, at the depth of 12µm, the Tresca's friction
coefficient of 0.2 and the attack angle of 25.5°. It is obvious that the force ratios slightly change with *L* and *h'*. An increase in geometric parameters systematically increases the grain/material contact area, this can make the normal and tangential force increasing with the same rate but the force ratios has still the same values. In this case, the force ratio depends only on the modes of material deformation.

<figure><img src="image_1759.png" /><figcaption>Figure 8. Variation of the force ratios with: (a) the distance between adjacent scratches, (b) the pre-existing ridge height ($L=20\mu m$)</figcaption></figure>

## Conclusion

A 3-D finite element analysis of abrasion process at grain scale, based on the CEL approach, was presented. The numerical results were compared with the model of Xie and Williams [7], which has been utilized in several works in the literature. The study was limited to the case of micro-cutting which corresponds to the configuration where the attack angle is larger enough. From the FE simulation of scratching with a pyramidal grain, it was found that the overall friction coefficient $\mu$ increases linearly with the interfacial friction coefficient $f$ and the attack angle of grit $\beta$. It can be noted that the calculated values of $\mu$ as a function of the different parameters from the Xie and Williams's model agree well with FE model, except in the case of the variation of $\mu$ with $f$. In this case, the model of $\mu$, given by Eq.1 and Eq.2, is half that obtained by FE simulation for high Tresca's friction coefficient and decreases rapidly with the increase of $f$, which is the opposite of the numerical results. This point will be clarified in future work by integrating the experimental results into our study.

The numerical and analytical results with a pyramidal indenter revealed that the overall friction coefficient $\mu$ cannot be completely decoupled into geometric variables. It was found to be independent from distance between adjacent scratches, pre-existing ridge height and the depth of cut. Therefore, the overall friction coefficient $\mu$ can be determined purely based on the geometry of the grain (i.e. the attack angle and the shape) and the interfacial friction of the contact.

## Acknowledgments

The authors would like to thank the University of Lorraine and the LEM3 laboratory for their financial and material support of this research.

## References

[1] A. Lefebvre, P. Lipinski, P. Vieville and C. Lescalier, "Experimental analysis of temperature in grinding at the global and local scales," *Machining Science and Technology*, pp. 12:1, 1-14, 2008. https://doi.org/10.1080/10910340701873489

[2] A. Khellouki, J. Rech and H. Zahouani, "Micro-scale investigation on belt finishing cutting mechanisms," *Wear 308*, pp. 17-28, 2013. https://doi.org/10.1016/j.wear.2013.09.016
coefficient of 0.2 and the attack angle of 25.5°. It is obvious that the force ratios slightly change with *L* and *h'*. An increase in geometric parameters systematically increases the grain/material contact area, this can make the normal and tangential force increasing with the same rate but the force ratios has still the same values. In this case, the force ratio depends only on the modes of material deformation.

<figure><img src="image_1757.png" /><figcaption>Figure 8. Variation of the force ratios with: (a) the distance between adjacent scratches, (b) the pre-existing ridge height ($L=20\mu m$)</figcaption></figure>

## Conclusion

A 3-D finite element analysis of abrasion process at grain scale, based on the CEL approach, was presented. The numerical results were compared with the model of Xie and Williams [7], which has been utilized in several works in the literature. The study was limited to the case of micro-cutting which corresponds to the configuration where the attack angle is larger enough. From the FE simulation of scratching with a pyramidal grain, it was found that the overall friction coefficient $\mu$ increases linearly with the interfacial friction coefficient $f$ and the attack angle of grit $\beta$. It can be noted that the calculated values of $\mu$ as a function of the different parameters from the Xie and Williams's model agree well with FE model, except in the case of the variation of $\mu$ with $f$. In this case, the model of $\mu$, given by Eq.1 and Eq.2, is half that obtained by FE simulation for high Tresca's friction coefficient and decreases rapidly with the increase of $f$, which is the opposite of the numerical results. This point will be clarified in future work by integrating the experimental results into our study.

The numerical and analytical results with a pyramidal indenter revealed that the overall friction coefficient $\mu$ cannot be completely decoupled into geometric variables. It was found to be independent from distance between adjacent scratches, pre-existing ridge height and the depth of cut. Therefore, the overall friction coefficient $\mu$ can be determined purely based on the geometry of the grain (i.e. the attack angle and the shape) and the interfacial friction of the contact.

## Acknowledgments

The authors would like to thank the University of Lorraine and the LEM3 laboratory for their financial and material support of this research.

## References

[1] A. Lefebvre, P. Lipinski, P. Vieville and C. Lescalier, "Experimental analysis of temperature in grinding at the global and local scales," *Machining Science and Technology*, pp. 12:1, 1-14, 2008. https://doi.org/10.1080/10910340701873489

[2] A. Khellouki, J. Rech and H. Zahouani, "Micro-scale investigation on belt finishing cutting mechanisms," *Wear 308*, pp. 17-28, 2013. https://doi.org/10.1016/j.wear.2013.09.016
[3] J. A. Badger and A. A. Torrance, "A comparison of two models to predict grinding forces from wheel surface topography," *International Journal of Machine Tools & Manufacture* 40, p. 1099-1120, 2000. https://doi.org/10.1016/S0890-6955(99)00116-9

[4] Y. Xie and J. A. Williams, "The generation of worn surfaces by the repeated interaction of parallel grooves," *Wear* 162-164, pp. 864-872, 1993. https://doi.org/10.1016/0043-1648(93)90088-4

[5] A. A. Torrance and J. A. Badger, "The relation between the traverse dressing of vitrified grinding wheels and their performance," *International Journal of Machine Tools & Manufacture* 40, p. 1787-1811, 2000. https://doi.org/10.1016/S0890-6955(00)00015-8

[6] A. Abdolhamid and M. Mehrdad, "Modeling and analysis of grinding forces based on the single grit scratch," *Int J Adv Manuf Technol*, p. 78:1223-1231, 2015. https://doi.org/10.1007/s00170-014-6729-z

[7] Y. Xie and J. Williams, "The prediction of friction and wear when a soft surface slides against a harder rough surface," *wear* 196, pp. 21-34, 1996. https://doi.org/10.1016/0043-1648(95)06830-9

[8] A. G. Mamalis, A. K. Vortselas and C. N. Panagopoulos, "Analytical and Numerical Wear Modeling of Metallic Interfaces: A Statistical Asperity Approach," *Tribology Transactions*, pp. 56: 121-129, 2013. https://doi.org/10.1080/10402004.2012.732198

[9] D. V. De Pellegrin and A. A. Torrance, "Characterisation of abrasive particles and surfaces in grinding," *In Jennings, M (Ed.) Proceedings of the 1st International Industrial Diamond at Work Conference.*, pp. 1-15., 2005.

[10] L. Zhang, H. Y. Tann, C. M. Yuan, Y. P. Chen and Z. D. Zhou, "An investigation of material removal in polishing with fixed abrasives," *An investigation of material removal in polishing with fixed abrasives. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture*, pp. 216(1), 103-112, 2002. https://doi.org/10.1243/0954405021519591

[11] F. Ducobu, E. Rivière-Lorphèvre and E. Filippi, "Mesh influence in orthogonal cutting modelling with the Coupled," *European Journal of Mechanics A/Solids* 65, pp. 324-335, 2017. https://doi.org/10.1016/j.euromechsol.2017.05.007

[12] F. Ducobu, E. Rivière-Lorphèvre, M. Galindo-Fernandez, S. Ayvar-Soberanis, P. J. Arrazola and H. Ghadbeigi, "Coupled Eulerian-Lagrangian (CEL) simulation for modelling of chip formation in AA2024-T3," *Procedia CIRP* 82, pp. 142-147, 2019. https://doi.org/10.1016/j.procir.2019.04.071

[13] D. Anderson, A. Warkentin and R. Bauer, "Experimental and numerical investigations of single abrasive-grain cutting," *International Journal of Machine Tools & Manufacture* 51, pp. 898-910, 2011. https://doi.org/10.1016/j.ijmachtools.2011.08.006

[14] G. Subhash and W. Zhang, "Investigation of the overall friction coefficient in single-pass scratch test," *Wear* 252, pp. 123-134, 2002. https://doi.org/10.1016/S0043-1648(01)00852-3
[3] J. A. Badger and A. A. Torrance, "A comparison of two models to predict grinding forces from wheel surface topography," *International Journal of Machine Tools & Manufacture* 40, p. 1099-1120, 2000. https://doi.org/10.1016/S0890-6955(99)00116-9

[4] Y. Xie and J. A. Williams, "The generation of worn surfaces by the repeated interaction of parallel grooves," *Wear* 162-164, pp. 864-872, 1993. https://doi.org/10.1016/0043-1648(93)90088-4

[5] A. A. Torrance and J. A. Badger, "The relation between the traverse dressing of vitrified grinding wheels and their performance," *International Journal of Machine Tools & Manufacture* 40, p. 1787-1811, 2000. https://doi.org/10.1016/S0890-6955(00)00015-8

[6] A. Abdolhamid and M. Mehrdad, "Modeling and analysis of grinding forces based on the single grit scratch," *Int J Adv Manuf Technol*, p. 78:1223-1231, 2015. https://doi.org/10.1007/s00170-014-6729-z

[7] Y. Xie and J. Williams, "The prediction of friction and wear when a soft surface slides against a harder rough surface," *wear* 196, pp. 21-34, 1996. https://doi.org/10.1016/0043-1648(95)06830-9

[8] A. G. Mamalis, A. K. Vortselas and C. N. Panagopoulos, "Analytical and Numerical Wear Modeling of Metallic Interfaces: A Statistical Asperity Approach," *Tribology Transactions*, pp. 56: 121-129, 2013. https://doi.org/10.1080/10402004.2012.732198

[9] D. V. De Pellegrin and A. A. Torrance, "Characterisation of abrasive particles and surfaces in grinding," *In Jennings, M (Ed.) Proceedings of the 1st International Industrial Diamond at Work Conference.*, pp. 1-15., 2005.

[10] L. Zhang, H. Y. Tann, C. M. Yuan, Y. P. Chen and Z. D. Zhou, "An investigation of material removal in polishing with fixed abrasives," *An investigation of material removal in polishing with fixed abrasives. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture*, pp. 216(1), 103-112, 2002. https://doi.org/10.1243/0954405021519591

[11] F. Ducobu, E. Rivière-Lorphèvre and E. Filippi, "Mesh influence in orthogonal cutting modelling with the Coupled," *European Journal of Mechanics A/Solids* 65, pp. 324-335, 2017. https://doi.org/10.1016/j.euromechsol.2017.05.007

[12] F. Ducobu, E. Rivière-Lorphèvre, M. Galindo-Fernandez, S. Ayvar-Soberanis, P. J. Arrazola and H. Ghadbeigi, "Coupled Eulerian-Lagrangian (CEL) simulation for modelling of chip formation in AA2024-T3," *Procedia CIRP* 82, pp. 142-147, 2019. https://doi.org/10.1016/j.procir.2019.04.071

[13] D. Anderson, A. Warkentin and R. Bauer, "Experimental and numerical investigations of single abrasive-grain cutting," *International Journal of Machine Tools & Manufacture* 51, pp. 898-910, 2011. https://doi.org/10.1016/j.ijmachtools.2011.08.006

[14] G. Subhash and W. Zhang, "Investigation of the overall friction coefficient in single-pass scratch test," *Wear* 252, pp. 123-134, 2002. https://doi.org/10.1016/S0043-1648(01)00852-3
# Axial drilling investigations and the potential of orbital techniques for enhanced hole quality in orthopedics

HUSSEIN Raafat<fnref n="1" />,ª*, ARAUJO Anna Carla<fnref n="1" />,ᵇ, LANDON Yann<fnref n="1" />,ᶜ and
RIBEIRO DA SILVA Cristiane Evelise<fnref n="2" />,ᵈ

**Keywords:** Biomanufacturing, Machining Bone, Drilling, Hole Accuracy, Delamination

**Abstract.** Precision in surgical bone drilling is essential for restoring bones mobility and function. However, the intricate nature and fiber-reinforced composite structure of bones inherently pose drilling-induced mechanical damage to the bone surface, affecting the primary stability necessary for implant anchorage and therefore leading to implant failure. The critical need for enhanced hole quality and damage reduction has spurred investigations into the optimal drilling parameters, novel drilling tools and alternative machining techniques. This study rigorously investigates the effect of the cutting speed and feed rate during axial drilling employing a center drill. It extends toward a comprehensive analysis of forces, temperature and mechanical damage, with a particular emphasis on delamination assessment. Then, the optimal parameters are established using the Tool-Material Couple (COM) optimization strategy. Subsequently, a novel approach of orbital drilling in bones is introduced for hole quality enhancement when compared to the conventional technique. This investigation serves as a foundational step for a more comprehensive study that ventures into the innovative application of orbital drilling in orthopedics.

## Introduction

In orthopedic biomanufacturing, the optimization of drilling techniques holds paramount significance, particularly in the context of implant integration. Axial drilling has long been the conventional approach, necessitating a meticulous examination of tool specifications and drilling parameters to enhance hole accuracy and mitigate the adverse effects of delamination. The majority of drill bits and drilling techniques commonly employed in surgical procedures share design principles and geometric characteristics akin to those originally devised for metal cutting applications [1]. Consequently, addressing the inherent issues in conventional bone drilling remains challenging due to the intricate material properties of bones and the operational conditions involved. Therefore, for optimal hole quality, achieving a delicate balance among drilling techniques, parameters, and tool specifications is essential.

Over the past decade, researchers have extensively delved into the fundamental principles of bone cutting to enhance tool specifications and geometry. Liao et al. [2] introduced a milling cutter with a main cutting edge and micro-cutting edges arranged on an Archimedes spiral, combining fracture cutting with shear cutting and significantly reducing surface damage and cutting temperature. Shu et al. [3] proposed a three-step drill structure to modify cutting conditions at the entrance and exit of drilling aiming at reducing mechanical and thermal damages and improve position accuracy in bone drilling.

Drilling parameters play a critical role in bone drilling processes, affecting both temperature generation and drilling forces. Udiljak et al. [4] investigated high-speed drilling techniques to mitigate thermal damage, revealing that cutting speed had no discernible impact on axial drilling

<figure><img src="image_1760.png" /></figure>

<footnote n="1">¹Institut Clément Ader (ICA), CNRS UMR 5312, Toulouse, France</footnote>

<footnote n="2">²Materials Division, Instituto Nacional de Tecnologia, Rio de Janeiro, Brazil</footnote>

<footnote>ªraafat.hussein@univ-tlse3.fr, ᵇaraujo@insa-toulouse.fr, ᶜyann.landon@univ-tlse3.fr,</footnote>

<footnote>ᵈcristiane.evelise@int.gov.br</footnote>
# Axial drilling investigations and the potential of orbital techniques for enhanced hole quality in orthopedics

HUSSEIN Raafat<fnref n="1" />,ª*, ARAUJO Anna Carla<fnref n="1" />,ᵇ, LANDON Yann<fnref n="1" />,ᶜ and
RIBEIRO DA SILVA Cristiane Evelise<fnref n="2" />,ᵈ

**Keywords:** Biomanufacturing, Machining Bone, Drilling, Hole Accuracy, Delamination

**Abstract.** Precision in surgical bone drilling is essential for restoring bones mobility and function. However, the intricate nature and fiber-reinforced composite structure of bones inherently pose drilling-induced mechanical damage to the bone surface, affecting the primary stability necessary for implant anchorage and therefore leading to implant failure. The critical need for enhanced hole quality and damage reduction has spurred investigations into the optimal drilling parameters, novel drilling tools and alternative machining techniques. This study rigorously investigates the effect of the cutting speed and feed rate during axial drilling employing a center drill. It extends toward a comprehensive analysis of forces, temperature and mechanical damage, with a particular emphasis on delamination assessment. Then, the optimal parameters are established using the Tool-Material Couple (COM) optimization strategy. Subsequently, a novel approach of orbital drilling in bones is introduced for hole quality enhancement when compared to the conventional technique. This investigation serves as a foundational step for a more comprehensive study that ventures into the innovative application of orbital drilling in orthopedics.

## Introduction

In orthopedic biomanufacturing, the optimization of drilling techniques holds paramount significance, particularly in the context of implant integration. Axial drilling has long been the conventional approach, necessitating a meticulous examination of tool specifications and drilling parameters to enhance hole accuracy and mitigate the adverse effects of delamination. The majority of drill bits and drilling techniques commonly employed in surgical procedures share design principles and geometric characteristics akin to those originally devised for metal cutting applications [1]. Consequently, addressing the inherent issues in conventional bone drilling remains challenging due to the intricate material properties of bones and the operational conditions involved. Therefore, for optimal hole quality, achieving a delicate balance among drilling techniques, parameters, and tool specifications is essential.

Over the past decade, researchers have extensively delved into the fundamental principles of bone cutting to enhance tool specifications and geometry. Liao et al. [2] introduced a milling cutter with a main cutting edge and micro-cutting edges arranged on an Archimedes spiral, combining fracture cutting with shear cutting and significantly reducing surface damage and cutting temperature. Shu et al. [3] proposed a three-step drill structure to modify cutting conditions at the entrance and exit of drilling aiming at reducing mechanical and thermal damages and improve position accuracy in bone drilling.

Drilling parameters play a critical role in bone drilling processes, affecting both temperature generation and drilling forces. Udiljak et al. [4] investigated high-speed drilling techniques to mitigate thermal damage, revealing that cutting speed had no discernible impact on axial drilling

<figure><img src="image_1758.png" /></figure>

<footnote n="1">¹Institut Clément Ader (ICA), CNRS UMR 5312, Toulouse, France</footnote>

<footnote n="2">²Materials Division, Instituto Nacional de Tecnologia, Rio de Janeiro, Brazil</footnote>

<footnote>ªraafat.hussein@univ-tlse3.fr, ᵇaraujo@insa-toulouse.fr, ᶜyann.landon@univ-tlse3.fr,</footnote>

<footnote>ᵈcristiane.evelise@int.gov.br</footnote>
force but significantly increased bone drilling temperature. Conversely, higher feed rate per tooth led to increased axial drilling force while reducing machining time and friction duration, resulting in lower temperatures. They recommended a minimum cutting speed of 6 m/min and a high drilling feed rate of 0.1 mm/tooth for effective temperature reduction in conventional drilling. Li et al. [5] further explored temperature trends, demonstrating that maximum temperature decreased with feed rate but increased with spindle speed. Simultaneously, exposure duration decreased with both feed rate and spindle speed. Samarasinghe et al. [6] observed contrasting effects of spindle speed and feed rate on temperature and force: higher spindle speeds increased temperature but minimized forces, whereas higher feed rates decreased temperature but maximized forces.

Singh et al. [7] conducted an analysis on the influence of different drilling methods, including conventional drilling (CD), ultrasonic-assisted drilling (UAD), and Robotic-assisted drilling (RAD) on bones. Their findings revealed significant reduction in thrust force when applying UAD compared to CD, as it increased hole dimension accuracy and reduced surface roughness. However, the ongoing quest for optimized hole quality and reduced heat generation in bone drilling techniques, has spurred researchers to explore alternative approaches [8]. In titanium alloys, CFRP, and aerospace materials, for example, orbital drilling has been widely used and demonstrated superior benefits over conventional and non-conventional drilling techniques due to its ability to reduce cutting forces and temperatures while improving hole quality [9]. However, to the author's knowledge, no studies have explored the application of orbital drilling in bones and its potential to enhance surgical procedures.

This paper presents a systematic investigation, substantiated by experimental data, to delineate the best conditions for axial drilling in orthopedics. The ultimate goal is to facilitate a future comparative analysis with orbital drilling and assess the potential need for procedural adjustments when employing robotic assistance in surgery. Axial drilling experiments were conducted on bone specimens using a center drill. The experimental procedure COM (Tool-Material Couple methodology) is used to define cutting speed and feed parameters. The results of hole quality of these experiments are analyzed to define the optimal range with higher precision. Additionally, some holes were drilled using orbital drilling technique to assess its efficiency compared to the conventional technique. This paper is an innovative work toward the unexplored domain of orbital drilling techniques in orthopedic applications.

## Axial and Orbital Drilling Kinematics

Mechanistic force models offer a quantitative approach to predicting cutting forces based on the uncut chip thickness. Following experimental tests, these models facilitate the estimation of specific force coefficients for various sets of variables, as detailed in existing literature [10]. The forces in orthogonal cutting are delineated by a mechanistic force model that exclusively considers the cutting action of the cutting edge while neglecting ploughing and chisel edge effects. The machining force vector ($\vec{F}$) is computed by decomposing it into tangential ($dF_t$), radial ($dF_r$), and axial ($dF_f$) components for small elements of the cutting edge along the flute. The local force at a specific point on the cutting edge is a function of the uncut chip thickness ($h$) and the specific force coefficients ($K_c, K_r$, and $K_f$), given by Eq. 1.

$$dF_j = K_j \cdot h \cdot db \qquad (1)$$

Here, $j$ represents the tangential, radial and axial components (c, r and f) and the product $h \cdot db$ represents the elementary uncut chip load calculated at the specific point over the elemental length ($db$) of the cutting edge. In drilling operations, the specific force coefficients are commonly identified in a global frame, generalized from the local reference in each tool rotation.

$$K_c = \frac{F_c}{A_c} = \frac{120 \cdot P_c}{V_c \cdot f_z \cdot D} \qquad (2)$$
force but significantly increased bone drilling temperature. Conversely, higher feed rate per tooth led to increased axial drilling force while reducing machining time and friction duration, resulting in lower temperatures. They recommended a minimum cutting speed of 6 m/min and a high drilling feed rate of 0.1 mm/tooth for effective temperature reduction in conventional drilling. Li et al. [5] further explored temperature trends, demonstrating that maximum temperature decreased with feed rate but increased with spindle speed. Simultaneously, exposure duration decreased with both feed rate and spindle speed. Samarasinghe et al. [6] observed contrasting effects of spindle speed and feed rate on temperature and force: higher spindle speeds increased temperature but minimized forces, whereas higher feed rates decreased temperature but maximized forces.

Singh et al. [7] conducted an analysis on the influence of different drilling methods, including conventional drilling (CD), ultrasonic-assisted drilling (UAD), and Robotic-assisted drilling (RAD) on bones. Their findings revealed significant reduction in thrust force when applying UAD compared to CD, as it increased hole dimension accuracy and reduced surface roughness. However, the ongoing quest for optimized hole quality and reduced heat generation in bone drilling techniques, has spurred researchers to explore alternative approaches [8]. In titanium alloys, CFRP, and aerospace materials, for example, orbital drilling has been widely used and demonstrated superior benefits over conventional and non-conventional drilling techniques due to its ability to reduce cutting forces and temperatures while improving hole quality [9]. However, to the author's knowledge, no studies have explored the application of orbital drilling in bones and its potential to enhance surgical procedures.

This paper presents a systematic investigation, substantiated by experimental data, to delineate the best conditions for axial drilling in orthopedics. The ultimate goal is to facilitate a future comparative analysis with orbital drilling and assess the potential need for procedural adjustments when employing robotic assistance in surgery. Axial drilling experiments were conducted on bone specimens using a center drill. The experimental procedure COM (Tool-Material Couple methodology) is used to define cutting speed and feed parameters. The results of hole quality of these experiments are analyzed to define the optimal range with higher precision. Additionally, some holes were drilled using orbital drilling technique to assess its efficiency compared to the conventional technique. This paper is an innovative work toward the unexplored domain of orbital drilling techniques in orthopedic applications.

## Axial and Orbital Drilling Kinematics

Mechanistic force models offer a quantitative approach to predicting cutting forces based on the uncut chip thickness. Following experimental tests, these models facilitate the estimation of specific force coefficients for various sets of variables, as detailed in existing literature [10]. The forces in orthogonal cutting are delineated by a mechanistic force model that exclusively considers the cutting action of the cutting edge while neglecting ploughing and chisel edge effects. The machining force vector ($\vec{F}$) is computed by decomposing it into tangential ($dF_t$), radial ($dF_r$), and axial ($dF_f$) components for small elements of the cutting edge along the flute. The local force at a specific point on the cutting edge is a function of the uncut chip thickness ($h$) and the specific force coefficients ($K_c$, $K_r$, and $K_f$), given by Eq. 1.

$$dF_j=K_j \cdot h \cdot db \quad (1)$$

Here, $j$ represents the tangential, radial and axial components (c, r and f) and the product $h \cdot db$ represents the elementary uncut chip load calculated at the specific point over the elemental length ($db$) of the cutting edge. In drilling operations, the specific force coefficients are commonly identified in a global frame, generalized from the local reference in each tool rotation.

$$K_c=\frac{F_c}{A_c}=\frac{120 \cdot P_c}{V_c \cdot f_z \cdot D} \quad (2)$$
$$K_f = \frac{F_f}{A_c} = \frac{F_f}{f_z \cdot \frac{D}{2}} = \frac{F_z}{f_z \cdot D} \qquad (3)$$

The local specific force coefficients exhibit variability along the cutting edge due to changes in cutting speed and varying cutting angles. For a fixed tool geometry of two flutes (Z=2), an average specific force coefficient is calculated using the total chip load per cutting edge ($A_c$), where $A_c = \int h \cdot db = f_z \cdot D/2$, $f_z$ is the axial feed per tooth or per cutting edge and D is the nominal tool diameter. Notably, experimental measurement of the cutting force for a single cutting edge is impractical in drilling, necessitating the consideration of both edges and their opposite directions. The average specific cutting force ($K_c$ in N/mm²) for a tool-workpiece pair is then calculated using the experimental cutting power ($P_c$), incorporating contributions from both cutting edges as in Eq. 2. Conversely, measuring the total force in z direction ($F_z$) allows for the direct calculation of the feed force for each cutting edge ($F_f$), where $F_z = Z \cdot F_f$, (Z=2). The specific feed force ($K_f$ in N/mm²) is determined accordingly as in Eq. 3.

Orbital drilling, also called helical milling, introduces an innovative hole making technique in orthopedics where the end mill cutter traces a helical path around the hole axis within the workpiece. This method involves two primary movements: spindle rotation ($N_{rev}$) in round per minute (rpm) and the feed movement along the helical trajectory ($Vf_{traj}$) in mm per minute (mm/min), which can be decomposed into tangential feed ($Vf_{xy}$) and axial feed ($Vf_z$). The diameter of the helical path ($D_e$ in (mm)) is determined by the hole diameter ($D_H$) and tool diameter ($D_t$) according to Eq. 4. Understanding cutting forces in orbital drilling necessitates defining four crucial parameters: the pitch ($P$ in mm/rot), representing the axial depth per orbital rotation, the ramp angle ($\alpha$) of the path, axial feed per tooth ($f_z$), and tangential feed per tooth ($f_{xy}$). The relationship between $\alpha$, $P$, and $e$ is described by Eq. 5, where $P$ is a function of the feed as in Eq. 6. $f_z$ and $f_{xy}$ are local parameters used to determine the feed rate along the helical trajectory as in Eq. 7, where Z is the tooth number of the end mill.

$$D_e = 2e = D_H - D_t \qquad (4)$$

$$\alpha = \tan^{-1} \frac{P}{2 \cdot \pi \cdot e} \qquad (5)$$

$$P = \frac{f_z}{f_{xy}} \cdot 2 \cdot \pi \cdot e \qquad (6)$$

$$Vf_{traj} = N_{rev} \cdot Z \cdot \sqrt{f_z^2 + f_{xy}^2} \qquad (7)$$
$$K_f = \frac{F_f}{A_c} = \frac{F_f}{f_z \cdot \frac{D}{2}} = \frac{F_z}{f_z \cdot D} \qquad (3)$$

The local specific force coefficients exhibit variability along the cutting edge due to changes in cutting speed and varying cutting angles. For a fixed tool geometry of two flutes (Z=2), an average specific force coefficient is calculated using the total chip load per cutting edge ($A_c$), where $A_c = \int h \cdot db = f_z \cdot D/2$, $f_z$ is the axial feed per tooth or per cutting edge and D is the nominal tool diameter. Notably, experimental measurement of the cutting force for a single cutting edge is impractical in drilling, necessitating the consideration of both edges and their opposite directions. The average specific cutting force ($K_c$ in N/mm²) for a tool-workpiece pair is then calculated using the experimental cutting power ($P_c$), incorporating contributions from both cutting edges as in Eq. 2. Conversely, measuring the total force in z direction ($F_z$) allows for the direct calculation of the feed force for each cutting edge ($F_f$), where $F_z = Z \cdot F_f$, (Z=2). The specific feed force ($K_f$ in N/mm²) is determined accordingly as in Eq. 3.

Orbital drilling, also called helical milling, introduces an innovative hole making technique in orthopedics where the end mill cutter traces a helical path around the hole axis within the workpiece. This method involves two primary movements: spindle rotation ($N_{rev}$) in round per minute (rpm) and the feed movement along the helical trajectory ($Vf_{traj}$) in mm per minute (mm/min), which can be decomposed into tangential feed ($Vf_{xy}$) and axial feed ($Vf_z$). The diameter of the helical path ($D_e$ in (mm)) is determined by the hole diameter ($D_H$) and tool diameter ($D_t$) according to Eq. 4. Understanding cutting forces in orbital drilling necessitates defining four crucial parameters: the pitch ($P$ in mm/rot), representing the axial depth per orbital rotation, the ramp angle ($\alpha$) of the path, axial feed per tooth ($f_z$), and tangential feed per tooth ($f_{xy}$). The relationship between $\alpha$, $P$, and $e$ is described by Eq. 5, where $P$ is a function of the feed as in Eq. 6. $f_z$ and $f_{xy}$ are local parameters used to determine the feed rate along the helical trajectory as in Eq. 7, where Z is the tooth number of the end mill.

$$D_e = 2e = D_H - D_t \qquad (4)$$

$$\alpha = \tan^{-1} \frac{P}{2 \cdot \pi \cdot e} \qquad (5)$$

$$P = \frac{f_z}{f_{xy}} \cdot 2 \cdot \pi \cdot e \qquad (6)$$

$$Vf_{traj} = N_{rev} \cdot Z \cdot \sqrt{f_z^2 + f_{xy}^2} \qquad (7)$$
<figure><img src="image_1761.png" /><figcaption>Figure 1: Bovine bone samples preparation.</figcaption></figure>

## Materials and Methods

The selection of bone specimens involved a meticulous process to ensure relevance to real-world clinical scenarios. This specific phase of the project was conducted in collaboration with the National Institute of Technology (INT) of Rio de Janeiro (Brazil), ensuring the highest standards of precision and adherence to scientific protocols throughout the sample preparation process. Bovine femurs (Fig. 1-1), which exhibit structural similarities to human bones, were obtained from a local slaughterhouse, specifically sourced from Nelore breed cattle aged 2 years. The bovine bones were derived ethically where no animals were harmed for the purpose of this in-vitro study.

The initial step of the preparatory procedure involved the precise removal of bone epiphyses, focusing exclusively on the bone diaphysis segment for subsequent sample preparation as shown in Fig. 1-2. Following this, a meticulous dissection procedure was implemented, encompassing comprehensive internal and external cleansing to eliminate bone marrow and soft tissue components. The thorough cleaning protocol employed a solution of hydrogen peroxide (35% p.a), as illustrated in Fig. 1-3, appropriately diluted with distilled water in a 50% ratio. This formulation was instrumental in achieving the efficient removal of soft tissue while preserving the integrity of the bone structure. After the cleansing process, each diaphysis femur underwent a precise division into four longitudinal sections, resulting in a total of 100 samples. Following the dissection and cleansing procedures, the prepared femur specimens underwent a milling procedure (Fig. 1-4) to refine their shape to have a tensile test geometry according to ASTM D638 standard, type IV (Fig. 1-5). Each sample, approximately measuring 25 mm (length) × 6.22 mm (width) with a thickness of 3.2 mm, emphasizing the need for meticulous control over the thickness (error = ±0.05 mm) to eliminate the thickness-induced error in the temperature elevation. This critical step aimed to ensure uniformity and compliance with standardized testing protocols for subsequent traction tests before drilling experiments to determine the bones specimens' elastic modulus. This step was crucial in characterizing the material mechanical properties of the bones, providing valuable data on their resistance to deformation under axial loading and their mechanical behavior.

Complementary to the traction tests, the density and specific heat of each bone specimen were measured. These parameters are essential for assessing thermal responses during the drilling process. The density data contributes to understanding the structural composition of the bone, while specific heat measurements provide insights into the material's thermal characteristics. The density was measured by Helium psychometry (Quantachrome Ultrapycnmeter 1000) with five measurements recorded for each sample. Heat capacity of the bovine bone sample, which represents the amount of heat required to raise the temperature of one gram of the bovine bone sample by one Kelvin, was determined through a simultaneous thermal analyzer, utilizing a sapphire standard for calibration. The properties listed in Table 1, including elastic modulus,
<figure><img src="image_1759.png" /><figcaption>Figure 1: Bovine bone samples preparation.</figcaption></figure>

## Materials and Methods

The selection of bone specimens involved a meticulous process to ensure relevance to real-world clinical scenarios. This specific phase of the project was conducted in collaboration with the National Institute of Technology (INT) of Rio de Janeiro (Brazil), ensuring the highest standards of precision and adherence to scientific protocols throughout the sample preparation process. Bovine femurs (Fig. 1-1), which exhibit structural similarities to human bones, were obtained from a local slaughterhouse, specifically sourced from Nelore breed cattle aged 2 years. The bovine bones were derived ethically where no animals were harmed for the purpose of this in-vitro study.

The initial step of the preparatory procedure involved the precise removal of bone epiphyses, focusing exclusively on the bone diaphysis segment for subsequent sample preparation as shown in Fig. 1-2. Following this, a meticulous dissection procedure was implemented, encompassing comprehensive internal and external cleansing to eliminate bone marrow and soft tissue components. The thorough cleaning protocol employed a solution of hydrogen peroxide (35% p.a), as illustrated in Fig. 1-3, appropriately diluted with distilled water in a 50% ratio. This formulation was instrumental in achieving the efficient removal of soft tissue while preserving the integrity of the bone structure. After the cleansing process, each diaphysis femur underwent a precise division into four longitudinal sections, resulting in a total of 100 samples. Following the dissection and cleansing procedures, the prepared femur specimens underwent a milling procedure (Fig. 1-4) to refine their shape to have a tensile test geometry according to ASTM D638 standard, type IV (Fig. 1-5). Each sample, approximately measuring 25 mm (length) × 6.22 mm (width) with a thickness of 3.2 mm, emphasizing the need for meticulous control over the thickness (error = ±0.05 mm) to eliminate the thickness-induced error in the temperature elevation. This critical step aimed to ensure uniformity and compliance with standardized testing protocols for subsequent traction tests before drilling experiments to determine the bones specimens' elastic modulus. This step was crucial in characterizing the material mechanical properties of the bones, providing valuable data on their resistance to deformation under axial loading and their mechanical behavior.

Complementary to the traction tests, the density and specific heat of each bone specimen were measured. These parameters are essential for assessing thermal responses during the drilling process. The density data contributes to understanding the structural composition of the bone, while specific heat measurements provide insights into the material's thermal characteristics. The density was measured by Helium psychometry (Quantachrome Ultrapycnmeter 1000) with five measurements recorded for each sample. Heat capacity of the bovine bone sample, which represents the amount of heat required to raise the temperature of one gram of the bovine bone sample by one Kelvin, was determined through a simultaneous thermal analyzer, utilizing a sapphire standard for calibration. The properties listed in Table 1, including elastic modulus,
density, and specific heat, are crucial factors that vary among different types of bones and influence their drilling behavior. As bones exhibit a wide range of properties depending on factors such as bone density, mineral content, and microstructure, it is essential to account for these variations in the design and optimization of drilling processes for bone tissues [11]. The elastic modulus dictates bone stiffness and resistance to deformation during drilling, density influences material removal rates, chip formation, and tool wear and specific heat reflects the low thermal conductivity of bones which affects heat dissipation.

<table><caption>Table 1: Material characteristics of the bone samples.</caption><thead><tr><td>Material properties</td><td>Bovine bones specimens</td></tr></thead><tbody><tr><td>Elastic modulus</td><td>24 ± 1.15 [GPa]</td></tr><tr><td>Density</td><td>2.2799 ± 0.0151 [g/cc]</td></tr><tr><td>Specific heat</td><td>3.7796 ± 0.1038 [J/Kg°C]</td></tr></tbody></table>

The drilling experiments were executed on a CNC DMG-DMU85eVo, employing a rigorous methodology to ensure precision and accurate data acquisition. Workpieces were securely mounted onto a 9257B Kistler dynamometer, a 6-component force measurement device. This dynamometer, seamlessly connected to a 5070 Kistler amplifier, facilitated the meticulous capture and recording of forces generated during drilling experiments. For the acquisition of force data with utmost precision, a 9201 National Instruments acquisition module operating at a 10 kHz acquisition rate and employing analog-to-digital conversion was deployed. The entire data acquisition and analysis process were conducted using the LabView platform, ensuring a comprehensive understanding of drilling dynamics. All drilling tests were conducted at room temperature (approximately 25 °C). Type K thermocouples, with a temperature range of -40 °C to 500 °C, were inserted to measure temperature during the drilling operation. Thermocouple holes (diameter of 1.4 mm) were drilled on one side of the cross-section of specimens to position the thermocouple bead at a depth of 1.5 mm from the top bone surface and 1 mm from the surface of the hole wall (Fig. 2-a). Temperature data were recorded using the TC-08 data logger, connected to a USB port on the computer, and the Pico log software was utilized for data visualization. As the drilling depth did not extend to levels typically associated with significant thermal effects, especially considering the short duration of the longest experiment, which was less than 30 seconds, the study prioritized the comparison of the maximum temperature value achieved under each set of drilling conditions. Notably, manual irrigation methods, due to their limited efficiency in cooling during cortical bone drilling, and considering the small thickness of the specimens, were intentionally excluded from the current experimental setup.

A series of axial drilling experiments was executed using a 2-fluted, 4 mm-diameter solid carbide center drill from Fraisa (Fig. 2-b) to investigate the effect of such drill specifications on the mechanical damage. The conical shape and wider margin of center drills play a role in minimizing the risk of fractures or damage to the bone, ensuring a more controlled and accurate drilling procedure. The orbital drilling experiment was conducted as an insightful random test, allowing us to assess its efficiency and draw meaningful comparisons with axial drilling, despite the inherently distinct parameters associated with each technique. A 4-fluted, 3-mm-diameter end mill of 30° helix angle from Fraisa (Fig. 2-c) was used to conduct orbital drilling experiment at a cutting speed of 20 m/min, a local axial feed ($f_z$) of 0.008 mm/th and a local tangential feed ($f_{xy}$) of 0.09 mm/th.
density, and specific heat, are crucial factors that vary among different types of bones and influence their drilling behavior. As bones exhibit a wide range of properties depending on factors such as bone density, mineral content, and microstructure, it is essential to account for these variations in the design and optimization of drilling processes for bone tissues [11]. The elastic modulus dictates bone stiffness and resistance to deformation during drilling, density influences material removal rates, chip formation, and tool wear and specific heat reflects the low thermal conductivity of bones which affects heat dissipation.

<table><caption>Table 1: Material characteristics of the bone samples.</caption><thead><tr><td>Material properties</td><td>Bovine bones specimens</td></tr></thead><tbody><tr><td>Elastic modulus</td><td>24 ± 1.15 [GPa]</td></tr><tr><td>Density</td><td>2.2799 ± 0.0151 [g/cc]</td></tr><tr><td>Specific heat</td><td>3.7796 ± 0.1038 [J/Kg°C]</td></tr></tbody></table>

The drilling experiments were executed on a CNC DMG-DMU85eVo, employing a rigorous methodology to ensure precision and accurate data acquisition. Workpieces were securely mounted onto a 9257B Kistler dynamometer, a 6-component force measurement device. This dynamometer, seamlessly connected to a 5070 Kistler amplifier, facilitated the meticulous capture and recording of forces generated during drilling experiments. For the acquisition of force data with utmost precision, a 9201 National Instruments acquisition module operating at a 10 kHz acquisition rate and employing analog-to-digital conversion was deployed. The entire data acquisition and analysis process were conducted using the LabView platform, ensuring a comprehensive understanding of drilling dynamics. All drilling tests were conducted at room temperature (approximately 25 °C). Type K thermocouples, with a temperature range of -40 °C to 500 °C, were inserted to measure temperature during the drilling operation. Thermocouple holes (diameter of 1.4 mm) were drilled on one side of the cross-section of specimens to position the thermocouple bead at a depth of 1.5 mm from the top bone surface and 1 mm from the surface of the hole wall (Fig. 2-a). Temperature data were recorded using the TC-08 data logger, connected to a USB port on the computer, and the Pico log software was utilized for data visualization. As the drilling depth did not extend to levels typically associated with significant thermal effects, especially considering the short duration of the longest experiment, which was less than 30 seconds, the study prioritized the comparison of the maximum temperature value achieved under each set of drilling conditions. Notably, manual irrigation methods, due to their limited efficiency in cooling during cortical bone drilling, and considering the small thickness of the specimens, were intentionally excluded from the current experimental setup.

A series of axial drilling experiments was executed using a 2-fluted, 4 mm-diameter solid carbide center drill from Fraisa (Fig. 2-b) to investigate the effect of such drill specifications on the mechanical damage. The conical shape and wider margin of center drills play a role in minimizing the risk of fractures or damage to the bone, ensuring a more controlled and accurate drilling procedure. The orbital drilling experiment was conducted as an insightful random test, allowing us to assess its efficiency and draw meaningful comparisons with axial drilling, despite the inherently distinct parameters associated with each technique. A 4-fluted, 3-mm-diameter end mill of 30° helix angle from Fraisa (Fig. 2-c) was used to conduct orbital drilling experiment at a cutting speed of 20 m/min, a local axial feed ($f_z$) of 0.008 mm/th and a local tangential feed ($f_{xy}$) of 0.09 mm/th.
<figure><img src="image_1762.png" /><figcaption>Figure 2: a) Experimental setup, b) center drill for axial drilling experiment, c) end mill for orbital drilling experiment and d) design of experiment according to COM</figcaption></figure>

The Tool-Material Couple (called as COM in French standard NF E66-520) is a methodical approach designed to establish the appropriate machining conditions for a given material. This standardized procedure consists in minimizing the specific cutting energy, and in the case of axial drilling, it relies on minimizing the specific axial coefficient, denoted as $K_f$. It is used in this case to analyze the quality requirements, such as, the delamination, where the operating conditions considered are the cutting speed ($V_c$ in m/min) and axial feed ($f_a$ in mm/rev). The maximum temperature achieved was also recorded in all experiments because the limiting temperature and heating time are important to avoid bone damage. Establishing the optimal tool operating domain involves a two-step process:

* First step involves adjusting the cutting speed while maintaining a fixed axial feed. The axial feed was maintained at a constant rate of 0.02 mm/rev and the cutting speed was changed in each experiment across a spectrum, including values of 6.5, 20, 40, 60, 80, 100, 140, and 200 m/min, each was repeated twice. The optimum cutting speed value is defined, during the experiment, to the one corresponding to the minimal delamination.

* Subsequently, the cutting speed was fixed at the optimum value determined in the previous step and the axial feed was increased for each experiment across a spectrum, including values of 0.04, 0.06, 0.1, 0.14, 0.2 and 0.3 mm/rev to determine the feed corresponding to the minimal delamination. Notably, each experiment was repeated twice.

During the drilling operations, forces and temperature were recorded and analyzed. Each drilled hole was then examined by a Dino-Lite Digital Microscope (AM8117MZT) to check the delamination around the hole edge. High-resolution images of the bone specimens, encompassing both pre- and post-drilling states on both sides, were acquired using an Olympus digital microscope. The analytical emphasis then shifted towards the precise identification and quantification of delamination areas through a combination of ImageJ and Python scripts, purposefully designed for image segmentation. The utilization of these scripts enabled thorough and detailed processing of the digital images, extracting crucial information for subsequent evaluation. The procedure for image segmentation can be described as follows (Fig. 3):

1. High-resolution microscope images of the specimen were captured before and after drilling.

2. Using Python, image segmentation was performed on both pre- and post-drilling images.

3. Both images were compared, and regions or pixels unrelated to the delaminated area were identified and removed.

4. The images were converted to grayscale to facilitate the marking of the delaminated area.

5. Image parameters, such as circle center and radius, were determined through methods like image enhancement and circle fitting. The least squares method was used to ensure the accurate fitting of the circle based on the detected edges.
<figure><img src="image_1760.png" /><figcaption>Figure 2: a) Experimental setup, b) center drill for axial drilling experiment, c) end mill for orbital drilling experiment and d) design of experiment according to COM</figcaption></figure>

The Tool-Material Couple (called as COM in French standard NF E66-520) is a methodical approach designed to establish the appropriate machining conditions for a given material. This standardized procedure consists in minimizing the specific cutting energy, and in the case of axial drilling, it relies on minimizing the specific axial coefficient, denoted as $K_f$. It is used in this case to analyze the quality requirements, such as, the delamination, where the operating conditions considered are the cutting speed ($V_c$ in m/min) and axial feed ($f_a$ in mm/rev). The maximum temperature achieved was also recorded in all experiments because the limiting temperature and heating time are important to avoid bone damage. Establishing the optimal tool operating domain involves a two-step process:

* First step involves adjusting the cutting speed while maintaining a fixed axial feed. The axial feed was maintained at a constant rate of 0.02 mm/rev and the cutting speed was changed in each experiment across a spectrum, including values of 6.5, 20, 40, 60, 80, 100, 140, and 200 m/min, each was repeated twice. The optimum cutting speed value is defined, during the experiment, to the one corresponding to the minimal delamination.

* Subsequently, the cutting speed was fixed at the optimum value determined in the previous step and the axial feed was increased for each experiment across a spectrum, including values of 0.04, 0.06, 0.1, 0.14, 0.2 and 0.3 mm/rev to determine the feed corresponding to the minimal delamination. Notably, each experiment was repeated twice.

During the drilling operations, forces and temperature were recorded and analyzed. Each drilled hole was then examined by a Dino-Lite Digital Microscope (AM8117MZT) to check the delamination around the hole edge. High-resolution images of the bone specimens, encompassing both pre- and post-drilling states on both sides, were acquired using an Olympus digital microscope. The analytical emphasis then shifted towards the precise identification and quantification of delamination areas through a combination of ImageJ and Python scripts, purposefully designed for image segmentation. The utilization of these scripts enabled thorough and detailed processing of the digital images, extracting crucial information for subsequent evaluation. The procedure for image segmentation can be described as follows (Fig. 3):

1. High-resolution microscope images of the specimen were captured before and after drilling.

2. Using Python, image segmentation was performed on both pre- and post-drilling images.

3. Both images were compared, and regions or pixels unrelated to the delaminated area were identified and removed.

4. The images were converted to grayscale to facilitate the marking of the delaminated area.

5. Image parameters, such as circle center and radius, were determined through methods like image enhancement and circle fitting. The least squares method was used to ensure the accurate fitting of the circle based on the detected edges.
6. Based on the determined circle parameters, the delamination was accurately extracted through threshold segmentation.

7. The images were then processed in ImageJ: the contrast was adjusted, the hole's area was removed, and further threshold segmentation was applied as shown in Fig. 3-2,3,4 respectively.

8. Noise filtering techniques were employed to refine the segmentation results Fig. 3-5.

9. As a result of these steps, only the delaminated area remained in the images.

<figure><img src="image_1763.png" /><figcaption>Figure 3: Image segmentation process to calculate the delaminated area around the hole.</figcaption></figure>

Specific parameters crucial to the assessment of delamination were considered. The delamination factor (Fd) was proposed by [12] as the ratio of the maximum diameter of delamination ($D_{max}$) and the nominal diameter ($D_0$) as shown in Eq. 8. However, this conventional delamination factor fails to provide an accurate representation of delamination magnitude as it neglects the consideration of the area and severity of delamination. In response, [13] introduced an improved approach, termed the adjusted delamination factor (Fda), expressed in Eq. 9. Fda incorporates the conventional delamination factor (Fd) and a term reflecting the damaged area extent Ad as depicted in Fig. 3. Notably, it considers both the maximum crack length ($A_{max}$) and the initial area ($A_0$) corresponding to $D_{max}$ and $D_0$, respectively.

$$Fd = \frac{D_{max}}{D_0} \qquad (6)$$

$$Fda = Fd + \frac{Ad(Fd^2 - Fd)}{A_{max} - A_0} \qquad (7)$$

## Results and Discussion

Axial drilling experiments were conducted to assess the influence of cutting conditions, specifically cutting speed and axial feed, on the mechanical integrity of bone tissue, as denoted by hole quality. The examination of delamination, as depicted in Fig. 4, reveals a relationship between delamination and the cutting parameters. Both Fd and Fda exhibit an initial peak at lower cutting speeds, subsequently diminishing with the increase in cutting speed up to 60 m/min. However, an upswing is observed at 80 m/min, followed by a declining trend at higher cutting speeds, notably
6. Based on the determined circle parameters, the delamination was accurately extracted through threshold segmentation.

7. The images were then processed in ImageJ: the contrast was adjusted, the hole's area was removed, and further threshold segmentation was applied as shown in Fig. 3-2,3,4 respectively.

8. Noise filtering techniques were employed to refine the segmentation results Fig. 3-5.

9. As a result of these steps, only the delaminated area remained in the images.

<figure><img src="image_1761.png" /><figcaption>Figure 3: Image segmentation process to calculate the delaminated area around the hole.</figcaption></figure>

Specific parameters crucial to the assessment of delamination were considered. The delamination factor ($Fd$) was proposed by [12] as the ratio of the maximum diameter of delamination ($D_{max}$) and the nominal diameter ($D_0$) as shown in Eq. 8. However, this conventional delamination factor fails to provide an accurate representation of delamination magnitude as it neglects the consideration of the area and severity of delamination. In response, [13] introduced an improved approach, termed the adjusted delamination factor ($Fda$), expressed in Eq. 9. $Fda$ incorporates the conventional delamination factor ($Fd$) and a term reflecting the damaged area extent $Ad$ as depicted in Fig. 3. Notably, it considers both the maximum crack length ($A_{max}$) and the initial area ($A_0$) corresponding to $D_{max}$ and $D_0$, respectively.

$$Fd = \frac{D_{max}}{D_0} \qquad (6)$$

$$Fda = Fd + \frac{Ad(Fd^2 - Fd)}{A_{max} - A_0} \qquad (7)$$

## Results and Discussion

Axial drilling experiments were conducted to assess the influence of cutting conditions, specifically cutting speed and axial feed, on the mechanical integrity of bone tissue, as denoted by hole quality. The examination of delamination, as depicted in Fig. 4, reveals a relationship between delamination and the cutting parameters. Both $Fd$ and $Fda$ exhibit an initial peak at lower cutting speeds, subsequently diminishing with the increase in cutting speed up to 60 m/min. However, an upswing is observed at 80 m/min, followed by a declining trend at higher cutting speeds, notably
<figure><img src="image_1764.png" /><figcaption>Figure 4. The effect of cutting speed and axial feed on</figcaption></figure>

140 m/min. This evolution is difficult to explain. It can be assumed that at low cutting speeds, the cutting mechanism is degraded and ruled by ploughing effect, generating delamination more easily. At 80 m/min, the evolution of the delamination results indicates a change in the cutting mechanism, with probably higher forces and higher temperature generated in the drilling zone, leading to fiber and matrix softening. In contrast, delamination concerning the feed rate demonstrates a decreasing trend until reaching 0.14 mm/rev, after which a notable increase is observed. This can be explained by the axial force. At low feed rates, the ploughing effect due to the low chip thickness is preponderant and generates excessive axial forces. This phenomenon stabilizes around 0.08 mm/rev. Over 0.14 mm/rev, the increase of the feed leads to the increase of the axial force, as the mechanistic force models well represent it. Finally, the minimal delamination was observed at a cutting speed (Vc) of 60 m/min and an axial feed (fa) of 0.14 mm/rev. Moreover, the observation from Fig. 4 reveals that the adjusted delamination Fda consistently surpasses the

<figure><img src="image_1765.png" /><figcaption>Figure 5. The effect of cutting conditions on the specific axial force coefficient and the maximum temperature.</figcaption></figure>

conventional delamination factor Fd for all cutting speeds and feed rates. This superiority stems from the consideration of both the maximum crack length and the delaminated damaged area.
<figure><img src="image_1762.png" /><figcaption>Figure 4. The effect of cutting speed and axial feed on</figcaption></figure>

140 m/min. This evolution is difficult to explain. It can be assumed that at low cutting speeds, the cutting mechanism is degraded and ruled by ploughing effect, generating delamination more easily. At 80 m/min, the evolution of the delamination results indicates a change in the cutting mechanism, with probably higher forces and higher temperature generated in the drilling zone, leading to fiber and matrix softening. In contrast, delamination concerning the feed rate demonstrates a decreasing trend until reaching 0.14 mm/rev, after which a notable increase is observed. This can be explained by the axial force. At low feed rates, the ploughing effect due to the low chip thickness is preponderant and generates excessive axial forces. This phenomenon stabilizes around 0.08 mm/rev. Over 0.14 mm/rev, the increase of the feed leads to the increase of the axial force, as the mechanistic force models well represent it. Finally, the minimal delamination was observed at a cutting speed (Vc) of 60 m/min and an axial feed (fa) of 0.14 mm/rev. Moreover, the observation from Fig. 4 reveals that the adjusted delamination Fda consistently surpasses the

<figure><img src="image_1763.png" /><figcaption>Figure 5. The effect of cutting conditions on the specific axial force coefficient and the maximum temperature.</figcaption></figure>

conventional delamination factor Fd for all cutting speeds and feed rates. This superiority stems from the consideration of both the maximum crack length and the delaminated damaged area.
<figure><img src="image_1766.png" /><figcaption>Figure 6. A Comparison between axial drilling technique under the optimized determined conditions of $V_c=60$ m/min and $f_a=0.14$ mm/rev and orbital drilling technique under $V_c=20$ m/min, $f_z = 0.008$ and $f_{xy} = 0.09$ mm/th according to the average forces and delamination.</figcaption></figure>

The analysis of the specific axial force coefficient in axial drilling Fig. 5 reveals the lowest value reached at 60 m/min suggesting an optimal tool-bone interaction that balances tool penetration and chip formation. This confirms the previous analysis of the impact of the cutting speed on delamination. However, the non-uniform trend beyond 60 m/min may be attributed to the complex interaction of factors, including heat generation influencing bone properties and cutting dynamics. Conversely, the decreasing trend in the specific axial force coefficient with increasing feed rate indicates more efficient chip evacuation and reduced cutting resistance. The plateau observed at 0.14 mm/rev signifies an optimal feed rate where further increases have minimal impact. It can be related directly to the previous observations on delamination. This behavior also aligns with the parallel plateau response observed in maximal temperature, indicating a harmonized influence of feed rate on both cutting forces and heat generation.

The comparative analysis between orbital and axial drilling techniques is elucidated in Fig. 6. Orbital drilling reduces axial forces to approximately one-third of the corresponding axial forces in optimal axial drilling conditions. This reduction confirms the enhanced efficiency of orbital drilling and explains the results obtained for delamination indicating a reduction in the drilling-induced mechanical damage. The observed elevation in the tangential forces in orbital drilling is anticipated, given its inherent distribution of cutting forces between axial and tangential feeds. Moreover, the investigation into delamination reveals a noteworthy decrease of up to 9% when compared to delamination occurring at the optimal parameters of axial drilling. This reduction in delamination further emphasizes the advantages of orbital drilling, particularly in minimizing detrimental effects associated with mechanical damage during the drilling process, which in turn affects the primary implant stability. These findings show the potential of orbital drilling as an advanced technique with improved force characteristics and diminished delamination tendencies.

## Conclusion

In this paper, the Tool-Material Couple (COM) optimization strategy was employed to establish the appropriate cutting speed and feed during axial drilling in cortical bones. This study aims to minimize mechanical damage and enhance the overall hole quality of bone. Results show that at a cutting speed ($V_c$) of 60 m/min and an axial feed ($f_a$) of 0.14 mm/rev a minimal delamination and cutting force coefficient with higher accuracy was detected. These findings provide insights into optimizing cutting conditions for minimal mechanical damage and efficient drilling performance in axial drilling applications for bones femur cortical bones as characterized in this study. Then, an orbital drilling experiment was conducted on bones to check the efficiency of this technique as an alternative to the conventional drilling techniques in orthopedics and it was clear that this novel
<figure><img src="image_1764.png" /><figcaption>Figure 6. A Comparison between axial drilling technique under the optimized determined conditions of $V_c=60$ m/min and $f_a=0.14$ mm/rev and orbital drilling technique under $V_c=20$ m/min, $f_z = 0.008$ and $f_{xy} = 0.09$ mm/th according to the average forces and delamination.</figcaption></figure>

The analysis of the specific axial force coefficient in axial drilling Fig. 5 reveals the lowest value reached at 60 m/min suggesting an optimal tool-bone interaction that balances tool penetration and chip formation. This confirms the previous analysis of the impact of the cutting speed on delamination. However, the non-uniform trend beyond 60 m/min may be attributed to the complex interaction of factors, including heat generation influencing bone properties and cutting dynamics. Conversely, the decreasing trend in the specific axial force coefficient with increasing feed rate indicates more efficient chip evacuation and reduced cutting resistance. The plateau observed at 0.14 mm/rev signifies an optimal feed rate where further increases have minimal impact. It can be related directly to the previous observations on delamination. This behavior also aligns with the parallel plateau response observed in maximal temperature, indicating a harmonized influence of feed rate on both cutting forces and heat generation.

The comparative analysis between orbital and axial drilling techniques is elucidated in Fig. 6. Orbital drilling reduces axial forces to approximately one-third of the corresponding axial forces in optimal axial drilling conditions. This reduction confirms the enhanced efficiency of orbital drilling and explains the results obtained for delamination indicating a reduction in the drilling-induced mechanical damage. The observed elevation in the tangential forces in orbital drilling is anticipated, given its inherent distribution of cutting forces between axial and tangential feeds. Moreover, the investigation into delamination reveals a noteworthy decrease of up to 9% when compared to delamination occurring at the optimal parameters of axial drilling. This reduction in delamination further emphasizes the advantages of orbital drilling, particularly in minimizing detrimental effects associated with mechanical damage during the drilling process, which in turn affects the primary implant stability. These findings show the potential of orbital drilling as an advanced technique with improved force characteristics and diminished delamination tendencies.

## Conclusion

In this paper, the Tool-Material Couple (COM) optimization strategy was employed to establish the appropriate cutting speed and feed during axial drilling in cortical bones. This study aims to minimize mechanical damage and enhance the overall hole quality of bone. Results show that at a cutting speed ($V_c$) of 60 m/min and an axial feed ($f_a$) of 0.14 mm/rev a minimal delamination and cutting force coefficient with higher accuracy was detected. These findings provide insights into optimizing cutting conditions for minimal mechanical damage and efficient drilling performance in axial drilling applications for bones femur cortical bones as characterized in this study. Then, an orbital drilling experiment was conducted on bones to check the efficiency of this technique as an alternative to the conventional drilling techniques in orthopedics and it was clear that this novel
technique results in reduction of forces and elimination of delamination. Following the tendency of orbital drilling technique for mechanical damage reduction, an optimization strategy for the cutting parameters of orbital drilling is among the work perspectives of this study.

## References

[1] Z. Jia, C. Zhang, F. Wang, R. Fu, and C. Chen, "An investigation of the effects of step drill geometry on drilling induced delamination and burr of Ti/CFRP stacks," *Composite Structures*, vol. 235, p. 111786, Mar. 2020. https://doi.org/10.1016/j.compstruct.2019.111786.

[2] Z. Liao, D. A. Axinte, and D. Gao, "A novel cutting tool design to avoid surface damage in bone machining," *International Journal of Machine Tools and Manufacture*, vol. 116, pp. 52–59, May 2017. https://doi.org/10.1016/j.ijmachtools.2017.01.003.

[3] L. Shu et al., "A novel self-centring drill bit design for low-trauma bone drilling," *International Journal of Machine Tools and Manufacture*, vol. 154, p. 103568, Jul. 2020. https://doi.org/10.1016/j.ijmachtools.2020.103568.

[4] M. Can, S. Koluaçik, E. Bahçe, H. Gökçe, and F. S. Tecellioglu, "Investigation of thermal damage in bone drilling: Hybrid processing method and pathological evaluation of existing methods," *Journal of the Mechanical Behavior of Biomedical Materials*, vol. 126, p. 105030, Feb. 2022. https://doi.org/10.1016/j.jmbbm.2021.105030.

[5] S. Li, L. Shu, T. Kizaki, W. Bai, M. Terashima, and N. Sugita, "Cortical bone drilling: A time series experimental analysis of thermal characteristics," *Journal of Manufacturing Processes*, vol. 64, pp. 606–619, Apr. 2021. https://doi.org/10.1016/j.jmapro.2021.01.046.

[6] H. Tian, X. Dang, D. Meng, B. Tian, and J. Li, "Influence of drilling parameters on bone drilling force and temperature by FE simulation and parameters optimization based Taguchi method," *Alexandria Engineering Journal*, vol. 75, pp. 115–126, Jul. 2023. https://doi.org/10.1016/j.aej.2023.05.048.

[7] G. Singh, A. Babbar, V. Jain, and D. Gupta, "Comparative statement for diametric delamination in drilling of cortical bone with conventional and ultrasonic assisted drilling techniques," *Journal of Orthopaedics*, vol. 25, pp. 53–58, May 2021. https://doi.org/10.1016/j.jor.2021.03.017.

[8] M. F. A. Akhbar and A. R. Yusoff, "Drilling of bone: thermal osteonecrosis regions induced by drilling parameters," *Biomed. Phys. Eng. Express*, vol. 5, no. 6, p. 065003, Sep. 2019.

[9] S. Akula, S. N. Nayak, G. Bolar, and V. Managuli, "Comparison of conventional drilling and helical milling for hole making in Ti6Al4V titanium alloy under sustainable dry condition," *Manufacturing Rev.*, vol. 8, p. 12, 2021. https://doi.org/10.1051/mfreview/2021010.

[10] J. C. Roukema and Y. Altintas, "Generalized modeling of drilling vibrations. Part I: Time domain model of drilling kinematics, dynamics and hole formation," *International Journal of Machine Tools and Manufacture*, vol. 47, no. 9, pp. 1455–1473, Jul. 2007.

[11] R. Hussein, Y. Landon, and A. C. Araujo, *BONE PROPERTIES - A REVIEW FOR DRILLING APPLICATION*. 2023. doi: 10.26678/ABCM.COBEF2023.COF23-0563.

[12] W.-C. Chen, "Some experimental investigations in the drilling of carbon fiber-reinforced plastic (CFRP) composite laminates," *International Journal of Machine Tools and Manufacture*, vol. 37, no. 8, pp. 1097–1108, Aug. 1997.

[13] J. P. Davim, J. Campos Rubio, and A. Abrao, "A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates," *Composites Science and Technology - COMPOSITES SCI TECHNOL*, vol. 67, pp. 1939–1945, Jul. 2007.
technique results in reduction of forces and elimination of delamination. Following the tendency of orbital drilling technique for mechanical damage reduction, an optimization strategy for the cutting parameters of orbital drilling is among the work perspectives of this study.

## References

[1] Z. Jia, C. Zhang, F. Wang, R. Fu, and C. Chen, "An investigation of the effects of step drill geometry on drilling induced delamination and burr of Ti/CFRP stacks," *Composite Structures*, vol. 235, p. 111786, Mar. 2020. https://doi.org/10.1016/j.compstruct.2019.111786.

[2] Z. Liao, D. A. Axinte, and D. Gao, "A novel cutting tool design to avoid surface damage in bone machining," *International Journal of Machine Tools and Manufacture*, vol. 116, pp. 52–59, May 2017. https://doi.org/10.1016/j.ijmachtools.2017.01.003.

[3] L. Shu et al., "A novel self-centring drill bit design for low-trauma bone drilling," *International Journal of Machine Tools and Manufacture*, vol. 154, p. 103568, Jul. 2020. https://doi.org/10.1016/j.ijmachtools.2020.103568.

[4] M. Can, S. Koluaçik, E. Bahçe, H. Gokce, and F. S. Tecellioglu, "Investigation of thermal damage in bone drilling: Hybrid processing method and pathological evaluation of existing methods," *Journal of the Mechanical Behavior of Biomedical Materials*, vol. 126, p. 105030, Feb. 2022. https://doi.org/10.1016/j.jmbbm.2021.105030.

[5] S. Li, L. Shu, T. Kizaki, W. Bai, M. Terashima, and N. Sugita, "Cortical bone drilling: A time series experimental analysis of thermal characteristics," *Journal of Manufacturing Processes*, vol. 64, pp. 606–619, Apr. 2021. https://doi.org/10.1016/j.jmapro.2021.01.046.

[6] H. Tian, X. Dang, D. Meng, B. Tian, and J. Li, "Influence of drilling parameters on bone drilling force and temperature by FE simulation and parameters optimization based Taguchi method," *Alexandria Engineering Journal*, vol. 75, pp. 115–126, Jul. 2023. https://doi.org/10.1016/j.aej.2023.05.048.

[7] G. Singh, A. Babbar, V. Jain, and D. Gupta, "Comparative statement for diametric delamination in drilling of cortical bone with conventional and ultrasonic assisted drilling techniques," *Journal of Orthopaedics*, vol. 25, pp. 53–58, May 2021. https://doi.org/10.1016/j.jor.2021.03.017.

[8] M. F. A. Akhbar and A. R. Yusoff, "Drilling of bone: thermal osteonecrosis regions induced by drilling parameters," *Biomed. Phys. Eng. Express*, vol. 5, no. 6, p. 065003, Sep. 2019.

[9] S. Akula, S. N. Nayak, G. Bolar, and V. Managuli, "Comparison of conventional drilling and helical milling for hole making in Ti6Al4V titanium alloy under sustainable dry condition," *Manufacturing Rev.*, vol. 8, p. 12, 2021. https://doi.org/10.1051/mfreview/2021010.

[10] J. C. Roukema and Y. Altintas, "Generalized modeling of drilling vibrations. Part I: Time domain model of drilling kinematics, dynamics and hole formation," *International Journal of Machine Tools and Manufacture*, vol. 47, no. 9, pp. 1455–1473, Jul. 2007.

[11] R. Hussein, Y. Landon, and A. C. Araujo, *BONE PROPERTIES - A REVIEW FOR DRILLING APPLICATION*. 2023. doi: 10.26678/ABCM.COBEF2023.COF23-0563.

[12] W.-C. Chen, "Some experimental investigations in the drilling of carbon fiber-reinforced plastic (CFRP) composite laminates," *International Journal of Machine Tools and Manufacture*, vol. 37, no. 8, pp. 1097–1108, Aug. 1997.

[13] J. P. Davim, J. Campos Rubio, and A. Abrao, "A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates," *Composites Science and Technology - COMPOSITES SCI TECHNOL*, vol. 67, pp. 1939–1945, Jul. 2007.
# Micro-milling and micro-EDM residual stresses evaluation on post-processed 3D printed samples

HIKARU Abe<fnref n="1" />,ª, BINTI JUANIH Iman Farhana<fnref n="1" />,ᵇ, QUARTO Mariangela<fnref n="2" />,ᶜ, ABENI Andrea<fnref n="3" />,ᵈ, ATTANASIO Aldo<fnref n="3" />,ᵉ, D'URSO Gianluca<fnref n="2" />,ᵉ, GINESTRA Paola<fnref n="3" />,⁹* and MATSUMURA Takashi<fnref n="1" />,ʰ

**Keywords:** Micro Milling, Micro-EDM, Residual Stress, 3D Printed Metals

**Abstract.** Micromachining is essential for producing microscopic components or obtaining complicated details in high-precision devices. Micromachining on 3D-printed metal samples is a cutting-edge, interdisciplinary field that blends additive manufacturing with accurate subtractive machining processes at the microscale. This method aims to precisely remove material from 3D-Printed metal items to obtain complicated geometries with strict tolerances, and high-end surface finishes. The surface integrity is not only determined by the surface roughness, but it depends also on the residual stresses caused by the thermo-mechanical loads during the Additive Manufacturing and the Micromachining processes. This paper aims to analyze the residual stresses measurements after Micromachining with both Micro-Milling and Micro-Electrical Discharge Machining of 3D-Printed metal samples. 17-4 PH stainless steel 3D printed samples were produced with Laser Powder Bed Fusion. The analysis of the residual stresses was carried out by producing microchannels of different sizes (200 µm and 800 µm) to also investigate the scale effect.

**Introduction**

Micro machining of 3D printed metals represents a critical issue in the context of precision manufacturing, allowing the fabrication of precise components at microscopic scales. This process involves the use of specialized machining techniques to finish, shape, or modify Additive manufacturing (AM) metal parts with utmost precision and accuracy. The advent of 3D printing technologies has revolutionised manufacturing by enabling the creation of complex geometries that were previously challenging or impossible to produce using conventional machining methods. However, despite the complexity and precision achievable through AM, the printed components often require further finishing operations and customization, specifically in industries such as aerospace, medical devices, or electronics, where micron-level accuracy is the target. Micromachining techniques encompass a range of processes to operate at extremely high resolutions. These processes are used to achieve complex features, precise surface finishes, and tight tolerances on 3D printed metal parts. The challenge in Micromachining lies not only in the small-scale operations but also in addressing the inherent characteristics of 3D printed metals. Residual stresses, anisotropic properties, and surface roughness resulting from the AM process can influence the machinability of these materials [1], requiring specialized strategies and tooling. In

<figure><img src="image_1767.png" /></figure>

<footnote n="1">¹Department of Mechanical Engineering, Tokyo Denki University, 5 Senjyu Asahi-cho, Adachi-ku, Tokyo, 120-8551, Japan</footnote>

<footnote n="2">²Department of Management, Information and Production Engineering, University of Bergamo, Via Pasubio 7/b, 24044, Dalmine, Italy</footnote>

<footnote n="3">³Department of Mechanical and Industrial Engineering, Università degli Studi di Brescia, Via Branze 38, 25123 Italy</footnote>

<footnote>ª22kmk03@ms.dendai.ac.jp, ᵇ23kmk03@ms.dendai.ac.jp, ᶜmariangela.quarto@unibg.it, ᵈandrea.abeni@unibs.it, ᵉaldo.attanasio@unibs.it, ᵇgianluca.durso@unibg.it, ᵈpaola.ginestra@unibs.it, ᵇtmatsumu@cck.dendai.ac.jp</footnote>
# Micro-milling and micro-EDM residual stresses evaluation on post-processed 3D printed samples

HIKARU Abe<fnref n="1" />,ª, BINTI JUANIH Iman Farhana<fnref n="1" />,ᵇ, QUARTO Mariangela<fnref n="2" />,ᶜ, ABENI Andrea<fnref n="3" />,ᵈ, ATTANASIO Aldo<fnref n="3" />,ᵉ, D'URSO Gianluca<fnref n="2" />,ᵉ, GINESTRA Paola<fnref n="3" />,⁹* and MATSUMURA Takashi<fnref n="1" />,ʰ

**Keywords:** Micro Milling, Micro-EDM, Residual Stress, 3D Printed Metals

**Abstract.** Micromachining is essential for producing microscopic components or obtaining complicated details in high-precision devices. Micromachining on 3D-printed metal samples is a cutting-edge, interdisciplinary field that blends additive manufacturing with accurate subtractive machining processes at the microscale. This method aims to precisely remove material from 3D-Printed metal items to obtain complicated geometries with strict tolerances, and high-end surface finishes. The surface integrity is not only determined by the surface roughness, but it depends also on the residual stresses caused by the thermo-mechanical loads during the Additive Manufacturing and the Micromachining processes. This paper aims to analyze the residual stresses measurements after Micromachining with both Micro-Milling and Micro-Electrical Discharge Machining of 3D-Printed metal samples. 17-4 PH stainless steel 3D printed samples were produced with Laser Powder Bed Fusion. The analysis of the residual stresses was carried out by producing microchannels of different sizes (200 µm and 800 µm) to also investigate the scale effect.

**Introduction**

Micro machining of 3D printed metals represents a critical issue in the context of precision manufacturing, allowing the fabrication of precise components at microscopic scales. This process involves the use of specialized machining techniques to finish, shape, or modify Additive manufacturing (AM) metal parts with utmost precision and accuracy. The advent of 3D printing technologies has revolutionised manufacturing by enabling the creation of complex geometries that were previously challenging or impossible to produce using conventional machining methods. However, despite the complexity and precision achievable through AM, the printed components often require further finishing operations and customization, specifically in industries such as aerospace, medical devices, or electronics, where micron-level accuracy is the target. Micromachining techniques encompass a range of processes to operate at extremely high resolutions. These processes are used to achieve complex features, precise surface finishes, and tight tolerances on 3D printed metal parts. The challenge in Micromachining lies not only in the small-scale operations but also in addressing the inherent characteristics of 3D printed metals. Residual stresses, anisotropic properties, and surface roughness resulting from the AM process can influence the machinability of these materials [1], requiring specialized strategies and tooling. In

<figure><img src="image_1765.png" /></figure>

<footnote n="1">¹Department of Mechanical Engineering, Tokyo Denki University, 5 Senjyu Asahi-cho, Adachi-ku, Tokyo, 120-8551, Japan</footnote>

<footnote n="2">²Department of Management, Information and Production Engineering, University of Bergamo, Via Pasubio 7/b, 24044, Dalmine, Italy</footnote>

<footnote n="3">³Department of Mechanical and Industrial Engineering, Università degli Studi di Brescia, Via Branze 38, 25123 Italy</footnote>

<footnote>ª22kmk03@ms.dendai.ac.jp, ᵇ23kmk03@ms.dendai.ac.jp, ᶜmariangela.quarto@unibg.it, ᵈandrea.abeni@unibs.it, ᵉaldo.attanasio@unibs.it, ᵇgianluca.durso@unibg.it, ᵈpaola.ginestra@unibs.it, ᵇtmatsumu@cck.dendai.ac.jp</footnote>
addition to conventional subtractive methods, such as Micro-Milling, non-traditional Micromachining approaches like laser machining, electrochemical machining, or Micro-Electrical Discharge Machining are also employed to achieve high precision on complex geometries and high surface finishing without introducing excessive mechanical stress [2].

Residual stresses in 3D printed metals represent internal stresses within a material that remain after the manufacturing process. They emerge due to elastic and plastic strain, thermal gradients, solidification, and phase transformations inherent in the AM process. When metals are 3D printed, each layer is sequentially deposited and rapidly cooled, causing uneven thermal contraction and solidification. This non-uniform cooling leads to localized contractions, creating internal stresses within the material. Additionally, the repeated heating and cooling cycles during the printing process can induce phase transformations, further contributing to residual stresses. These residual stresses can significantly impact the properties and performance of 3D printed components. They might cause warping, cracking, or distortion in the printed parts, reducing their structural integrity and dimensional accuracy. Moreover, residual stress tends to be compressive near the center of part and tensile near the surfaces [3]. In fact, compressive residual stresses can play a beneficial role in reinforcing the longevity and durability of components, particularly in metallurgy and manufacturing [4]. The evaluation and management of residual stresses in 3D printed metals are critical to ensure the reliability and quality of the final products. Various techniques, such as simulation methods, X-ray diffraction, neutron diffraction, and strain gauge methods, are employed to measure and understand these internal stresses [5]. Mitigation strategies, including post-processing treatments like heat annealing, are often implemented to reduce residual stresses and enhance the overall performance and durability of 3D printed metal components.

The evaluation of residual stresses on 3D printed metals processed by Micromachining techniques [6] can be crucial both for the final application and for the understanding of the micro mechanisms that occur during the material removal process [7] on metals produced by AM. This paper aims to evaluate Residual Stresses (RS) on machined samples produced by Laser Powder Bed Fusion (LPBF). The samples have been machined with Micro-Milling (MM) and Micro-Electrical Discharge Machining (micro-EDM) and the residual stresses have been analysed on processed samples.

## Materials and Methods

**Additive Manufacturing:** The AM samples had a square base measuring 25 x 25 mm and a height of 2.2 mm, designed as parallelepipeds. For the LPBF process, in order to prevent contaminants, the samples were produced in a Nitrogen atmosphere with a regulated O₂ level of less than 0.1 vol.%, using the commercial LPBF machine ProX 100 (3D System, Rock Hill, USA). The parameters of LPBF were: laser power of 100W, scan speed of 300 mm/s, and layer thickness of 30 µm. 17-4 PH stainless steel (3D Systems, USA) powder has been used for the production and the samples have been machined as-built, without heat treatment.

**Micro-machining (MM):** The MM tests were performed on the ultra-precision five-axis nano-precision machining center KERN Pyramid Nano (Kern Micro Technik, Olympiastr. 2, D-82418 Murnau-Westried, Germany) with a spindle able to reach 50000 RPM, equipped with a Heidenhain iTCN 530 numeric control. The microfeatures were machined by tungsten carbide tools with titanium nitride coating with a nominal diameter of 200 µm (RIME HM79/02) and 800 µm (RIME HM79/02). Microchannels of 200 µm and 800 µm width and a constant length of 5 mm were machined in a single pass. The depth of the channels was 20 µm for the 200 µm channels and 50 µm for the 800 µm channels. The Material Removal Rate (MRR) was calculated as the product of the feed rate and the section of the machined channels, to compare MM with micro-EDM. With a tool diameter of 200 µm, the feed per tooth was kept at 1 µm/tooth, the cutting speed was set at 22 m/min and the MRR was 0.28 mm³/min. With a tool diameter of 800 µm, the feed per tooth was kept at 3 µm/tooth, the cutting speed was set at 40 m/min and the MRR was 3.82 mm³/min.
addition to conventional subtractive methods, such as Micro-Milling, non-traditional Micromachining approaches like laser machining, electrochemical machining, or Micro-Electrical Discharge Machining are also employed to achieve high precision on complex geometries and high surface finishing without introducing excessive mechanical stress [2].

Residual stresses in 3D printed metals represent internal stresses within a material that remain after the manufacturing process. They emerge due to elastic and plastic strain, thermal gradients, solidification, and phase transformations inherent in the AM process. When metals are 3D printed, each layer is sequentially deposited and rapidly cooled, causing uneven thermal contraction and solidification. This non-uniform cooling leads to localized contractions, creating internal stresses within the material. Additionally, the repeated heating and cooling cycles during the printing process can induce phase transformations, further contributing to residual stresses. These residual stresses can significantly impact the properties and performance of 3D printed components. They might cause warping, cracking, or distortion in the printed parts, reducing their structural integrity and dimensional accuracy. Moreover, residual stress tends to be compressive near the center of part and tensile near the surfaces [3]. In fact, compressive residual stresses can play a beneficial role in reinforcing the longevity and durability of components, particularly in metallurgy and manufacturing [4]. The evaluation and management of residual stresses in 3D printed metals are critical to ensure the reliability and quality of the final products. Various techniques, such as simulation methods, X-ray diffraction, neutron diffraction, and strain gauge methods, are employed to measure and understand these internal stresses [5]. Mitigation strategies, including post-processing treatments like heat annealing, are often implemented to reduce residual stresses and enhance the overall performance and durability of 3D printed metal components.

The evaluation of residual stresses on 3D printed metals processed by Micromachining techniques [6] can be crucial both for the final application and for the understanding of the micro mechanisms that occur during the material removal process [7] on metals produced by AM. This paper aims to evaluate Residual Stresses (RS) on machined samples produced by Laser Powder Bed Fusion (LPBF). The samples have been machined with Micro-Milling (MM) and Micro-Electrical Discharge Machining (micro-EDM) and the residual stresses have been analysed on processed samples.

## Materials and Methods

**Additive Manufacturing:** The AM samples had a square base measuring 25 x 25 mm and a height of 2.2 mm, designed as parallelepipeds. For the LPBF process, in order to prevent contaminants, the samples were produced in a Nitrogen atmosphere with a regulated O₂ level of less than 0.1 vol.%, using the commercial LPBF machine ProX 100 (3D System, Rock Hill, USA). The parameters of LPBF were: laser power of 100W, scan speed of 300 mm/s, and layer thickness of 30 µm. 17-4 PH stainless steel (3D Systems, USA) powder has been used for the production and the samples have been machined as-built, without heat treatment.

**Micro-machining (MM):** The MM tests were performed on the ultra-precision five-axis nano-precision machining center KERN Pyramid Nano (Kern Micro Technik, Olympiastr. 2, D-82418 Murnau-Westried, Germany) with a spindle able to reach 50000 RPM, equipped with a Heidenhain iTCN 530 numeric control. The microfeatures were machined by tungsten carbide tools with titanium nitride coating with a nominal diameter of 200 µm (RIME HM79/02) and 800 µm (RIME HM79/02). Microchannels of 200 µm and 800 µm width and a constant length of 5 mm were machined in a single pass. The depth of the channels was 20 µm for the 200 µm channels and 50 µm for the 800 µm channels. The Material Removal Rate (MRR) was calculated as the product of the feed rate and the section of the machined channels, to compare MM with micro-EDM. With a tool diameter of 200 µm, the feed per tooth was kept at 1 µm/tooth, the cutting speed was set at 22 m/min and the MRR was 0.28 mm³/min. With a tool diameter of 800 µm, the feed per tooth was kept at 3 µm/tooth, the cutting speed was set at 40 m/min and the MRR was 3.82 mm³/min.
Micro-EDM: tests were performed on Sarix SX-200 micro-EDM, with CAM software to define the electrode paths. Two cylindrical tungsten carbide electrodes were chosen with 200 µm and 800 µm diameters. The current peak and width, in relation to electrode, materials, and geometry, determined the MRR. A medium pulse shape, a peak current index of 100, a voltage of 100 V, and a width of 4 µs were set for all the tests. The average value of the MRR for the 200 µm channels was 0.00169 mm³/min while the MRR for the 800 µm channels was 0.01074 mm³/min.

The channels were machined on four pockets, each with a depth of 0.2 mm on the top surface of the AM samples. The sample surfaces were pre-machined using a four-flute flat-bottom end mill with a nominal diameter of 3 mm. Further machining and pre-machining details are reported in [6]. The MM and micro-EDM cutting conditions have been chosen on the basis of previous analyses and researches on conventional and AM metals [1, 2, 6].

Residual stresses and FIB evaluation: The residual stresses were evaluated with a µ-X360s Residual stresses analyzer (© Pustec Industrial Co.,Ltd, Japan). For each sample (Fig. 1), the residual stress of each point parallel to the feed direction is measured one time in four different directions (90° intervals). Table 1 reports the parameters used for the evaluation of the RS.

<table><caption>Table 1. Parameters used for the RS analysis with µ-X360s.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Collimator type</td><td>1 mm diameter</td></tr><tr><td>X-ray target material</td><td>Cr</td></tr><tr><td>X-ray tube voltage (kV)</td><td>30</td></tr><tr><td>X-ray tube beam current (mA)</td><td>1.0</td></tr><tr><td>X-ray incidence angle (°)</td><td>35</td></tr><tr><td>Spectra count time (s)</td><td>90 (depending on conditions)</td></tr></tbody></table>

<figure><img src="image_1768.png" /><figcaption>Figure 1. An example of the RS measurement at 0° degrees: EDM processed samples (A) and MM samples (B). The images shown are for illustration purposes only.</figcaption></figure>

The Focused Ion Beam system used in this work to observe the samples was the Hitachi FIB-SEM system (SMI2050-D, Hitachi High Tech, Tokyo, Japan) at 30 kV. The samples are sequentially milled using an ion beam perpendicular to the specimen while imaging the exposed surface using an electron beam. This approach allows larger scale nano-structures to be
Micro-EDM: tests were performed on Sarix SX-200 micro-EDM, with CAM software to define the electrode paths. Two cylindrical tungsten carbide electrodes were chosen with 200 µm and 800 µm diameters. The current peak and width, in relation to electrode, materials, and geometry, determined the MRR. A medium pulse shape, a peak current index of 100, a voltage of 100 V, and a width of 4 µs were set for all the tests. The average value of the MRR for the 200 µm channels was 0.00169 mm³/min while the MRR for the 800 µm channels was 0.01074 mm³/min.

The channels were machined on four pockets, each with a depth of 0.2 mm on the top surface of the AM samples. The sample surfaces were pre-machined using a four-flute flat-bottom end mill with a nominal diameter of 3 mm. Further machining and pre-machining details are reported in [6]. The MM and micro-EDM cutting conditions have been chosen on the basis of previous analyses and researches on conventional and AM metals [1, 2, 6].

Residual stresses and FIB evaluation: The residual stresses were evaluated with a µ-X360s Residual stresses analyzer (© Pustec Industrial Co.,Ltd, Japan). For each sample (Fig. 1), the residual stress of each point parallel to the feed direction is measured one time in four different directions (90° intervals). Table 1 reports the parameters used for the evaluation of the RS.

<table><caption>Table 1. Parameters used for the RS analysis with µ-X360s.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Collimator type</td><td>1 mm diameter</td></tr><tr><td>X-ray target material</td><td>Cr</td></tr><tr><td>X-ray tube voltage (kV)</td><td>30</td></tr><tr><td>X-ray tube beam current (mA)</td><td>1.0</td></tr><tr><td>X-ray incidence angle (°)</td><td>35</td></tr><tr><td>Spectra count time (s)</td><td>90 (depending on conditions)</td></tr></tbody></table>

<figure><img src="image_1766.png" /><figcaption>Figure 1. An example of the RS measurement at 0° degrees: EDM processed samples (A) and MM samples (B). The images shown are for illustration purposes only.</figcaption></figure>

The Focused Ion Beam system used in this work to observe the samples was the Hitachi FIB-SEM system (SMI2050-D, Hitachi High Tech, Tokyo, Japan) at 30 kV. The samples are sequentially milled using an ion beam perpendicular to the specimen while imaging the exposed surface using an electron beam. This approach allows larger scale nano-structures to be
characterized by SEM. The process is destructive, since the specimen is being sequentially milled away after each image is collected. The collected series of images is then reconstructed to a 3D volume by registering the image stack and removing artifacts.

Each analysis was carried out on three replicas with three measurements repetitions.

## Results

The RS measured in this study are not including only the residual stresses of the pure channel, but also part of the residual stresses around the channels, as indicated by the yellow circles in Fig. 2, which highlight the measurement area. Anyway, considering the intensity of the X-ray as a Gaussian distribution, the values of RS are estimated as that of the center of groove, due to the higher intensity of the X-ray around the center of the measured area. For reference, the maximum X-ray penetration depth in the RS measurement is about 10 µm, but the actual penetration depth is shallower due to the inclination of the device.

Regarding micro-EDM, due to the metallurgical changes and the action of high temperatures, compressive RS appears (Fig. 2). Specifically, micro-EDM is characterized by an increase of the temperature in a very localized area: the area surrounding the processed zone does not undergo this rapid increase in temperature and therefore the melted material tends to resolidify in a constrained area surrounded by lower temperatures which can generate compressive RS. Isotropic (bidirectional) residual stress is observed. That is associated with the uniform distribution of the individual discharges in micro-EDM. With the increment in the discharge energy (it corresponds to the increment of MRR), the residual stress also increases. This may be attributed to the increase in thickness of the defective white layer, in which the RS is a maximum. The RS measured for the 0.8 mm channels tends to be more tensile than the RS on the 0.2 mm channels (as shown in the example of Fig. 1 A and reported in literature [8]). However, observing the 0.2 mm channels measurement area, it is worth noticing that is more influenced by the surrounding surface state that has been previously machined.

<figure><img src="image_1769.png" /><figcaption>Figure 2. RS measurements at four degrees on the micro-EDM processed samples.</figcaption></figure>