<figure><img src="image_679.png" /><figcaption>Fig. 7: OM-images of composite rod cross sections at microscope magnification of 1000x</figcaption></figure>

In order to investigate if during ball milling or extrusion processing oxidation of aluminum powder or formation of Al₄C₃ phase might have occurred, XRD phase analysis (measured with Bruker D8 advanced diffractometer) was conducted exemplarily. XRD was conducted on the sample with the highest strength (0.25% AA-GNPs, 1h, 300rpm) as well as on the sample with highest overall microhardness (1.0% AA-GNPs, 4h, 300rpm). The diffraction patterns are given in Fig. 9. In case of the sample with highest microhardness besides aluminum no other peaks were detected. In case of the specimen with highest microhardness peaks of aluminum oxide (cubic Al₂O₃) were found. Hence, it seems that at least in this case during HSBM for 1h with significant heat development in atmospheric environment, oxidation of aluminum powder occurred to some extent. The Al₄C₃-phase did not appear in the patterns and thus, did not seem to have formed during processing.
<figure><img src="image_669.png" /><figcaption>Fig. 8: OM-overview images of composite rod cross sections at microscope magnification of 50x</figcaption></figure>

<figure><img src="image_670.png" /><figcaption>Fig. 9: XRD pattern for extruded rod materials of 0.25% AA-GNPs ball milled for 1h at 300rpm and 1.0% AA-GNP milled 4h at 300rpm</figcaption></figure>
<figure><img src="image_680.png" /><figcaption>Fig. 8: OM-overview images of composite rod cross sections at microscope magnification of 50x</figcaption></figure>

<figure><img src="image_681.png" /><figcaption>Fig. 9: XRD pattern for extruded rod materials of 0.25% AA-GNPs ball milled for 1h at 300rpm and 1.0% AA-GNP milled 4h at 300rpm</figcaption></figure>
## Conclusions

In this study a powder metallurgic approach of ball milling and indirect extrusion processing was performed to investigate the effects of processing parameters on the mechanical properties of extruded aluminum-graphene-composites. The following conclusions can be drawn:

* HSBM increased extrusion loads in indirect extrusion by about 100% compared to LSBM.

* HSBM resulted in bad rod surface quality (massive surface cracking).

* HSBM increased TYS by 67% and UTS by 95% but decreased fracture strain significantly.

* Higher graphene contents did not improve strengths but decreased fracture strain.

* HSBM increased microhardness by 167% for pure Al and 210% (1% AA-GNP-composite).

* Graphene agglomerates were found in all extruded rod cross sections.

* No significant differences in mechanical composite properties between commercial AA-graphene and self-synthesized OAEG were found.

## Acknowledgements

The authors like to thank Zeynep Alici (chair of advanced ceramics, TU Berlin) for synthesis of OAEG for this study as well as Delf Kober (chair of advanced ceramics, TU Berlin) for conducting XRD phase analysis. This project was founded by the German Research Foundation (DFG) (grant number 426183735).

## References

[1] K.S. Novoselov et al., Electric field effect in atomically thin carbon films, Science 306 (2004) 666-669. https://doi.org/10.1126/science.1102896

[2] C. Lee et al., Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene, Science 321, (2008) 385-388. https://doi.org/10.1126/science.1157996

[3] F. Chen et al., Graphene-Reinforced Aluminum Matrix Composites: A Review of Synthesis Methods and Properties, The Journal of The Minerals, Metals & Materials Society (TMS) 70 (2018) 837-845. https://doi.org/10.1007/s11837-018-2810-7

[4] M. Negendank et al., Effects of disc milling and indirect extrusion processing on mechanical properties of aluminum-graphene-composites with commercial GNPs, Materials Research Proceedings 28 (2023) 455-466. https://doi.org/10.21741/9781644902479-50

[5] Y. Jiang et al., Tailoring the structure and mechanical properties of graphene nanosheet/aluminum composites by flake powder metallurgy via shift-speed ball milling, Composites Part A 111 (2018) 73-82. https://doi.org/10.1016/j.compositesa.2018.05.022

[6] R. Xu et al., Balanced strength and ductility in CNT/Al composites achieved by flake powder metallurgy via shift-speed ball milling, Composites Part A 96 (2017) 57-66. https://doi.org/10.1016/j.compositesa.2017.02.017

[7] H. Zhang et al., Enhanced mechanical properties of Al5083 alloy with graphene nanoplates prepared by ball milling and hot extrusion, Mat. Sci. & Eng. A 658 (2016) 8-15. http://dx.doi.org/10.1016/j.msea.2016.01.076

[8] Zheng Zhong et al., High-content graphene nanoplatelet reinforced aluminum composites produced by ball milling and hot extrusion, Science China Technological Sciences 63 (2020) 1426-1435. https://doi.org/10.1007/s11431-020-1670-4

[9] M. Negendank et al., Extrusion and characterization of aluminum/graphene composites, Proceedings of the 24th Int. Conf. on Material Forming (2021). https://doi.org/10.25518/esaform21.3714
## Conclusions

In this study a powder metallurgic approach of ball milling and indirect extrusion processing was performed to investigate the effects of processing parameters on the mechanical properties of extruded aluminum-graphene-composites. The following conclusions can be drawn:

* HSBM increased extrusion loads in indirect extrusion by about 100% compared to LSBM.

* HSBM resulted in bad rod surface quality (massive surface cracking).

* HSBM increased TYS by 67% and UTS by 95% but decreased fracture strain significantly.

* Higher graphene contents did not improve strengths but decreased fracture strain.

* HSBM increased microhardness by 167% for pure Al and 210% (1% AA-GNP-composite).

* Graphene agglomerates were found in all extruded rod cross sections.

* No significant differences in mechanical composite properties between commercial AA-graphene and self-synthesized OAEG were found.

## Acknowledgements

The authors like to thank Zeynep Alici (chair of advanced ceramics, TU Berlin) for synthesis of OAEG for this study as well as Delf Kober (chair of advanced ceramics, TU Berlin) for conducting XRD phase analysis. This project was founded by the German Research Foundation (DFG) (grant number 426183735).

## References

[1] K.S. Novoselov et al., Electric field effect in atomically thin carbon films, Science 306 (2004) 666-669. https://doi.org/10.1126/science.1102896

[2] C. Lee et al., Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene, Science 321, (2008) 385-388. https://doi.org/10.1126/science.1157996

[3] F. Chen et al., Graphene-Reinforced Aluminum Matrix Composites: A Review of Synthesis Methods and Properties, The Journal of The Minerals, Metals & Materials Society (TMS) 70 (2018) 837-845. https://doi.org/10.1007/s11837-018-2810-7

[4] M. Negendank et al., Effects of disc milling and indirect extrusion processing on mechanical properties of aluminum-graphene-composites with commercial GNPs, Materials Research Proceedings 28 (2023) 455-466. https://doi.org/10.21741/9781644902479-50

[5] Y. Jiang et al., Tailoring the structure and mechanical properties of graphene nanosheet/aluminum composites by flake powder metallurgy via shift-speed ball milling, Composites Part A 111 (2018) 73-82. https://doi.org/10.1016/j.compositesa.2018.05.022

[6] R. Xu et al., Balanced strength and ductility in CNT/Al composites achieved by flake powder metallurgy via shift-speed ball milling, Composites Part A 96 (2017) 57-66. https://doi.org/10.1016/j.compositesa.2017.02.017

[7] H. Zhang et al., Enhanced mechanical properties of Al5083 alloy with graphene nanoplates prepared by ball milling and hot extrusion, Mat. Sci. & Eng. A 658 (2016) 8-15. http://dx.doi.org/10.1016/j.msea.2016.01.076

[8] Zheng Zhong et al., High-content graphene nanoplatelet reinforced aluminum composites produced by ball milling and hot extrusion, Science China Technological Sciences 63 (2020) 1426-1435. https://doi.org/10.1007/s11431-020-1670-4

[9] M. Negendank et al., Extrusion and characterization of aluminum/graphene composites, Proceedings of the 24th Int. Conf. on Material Forming (2021). https://doi.org/10.25518/esaform21.3714
# Analysis of mechanical properties of stainless steel small diameter cold drawn wires

VAISSETTE Julien¹,ᵃ, MABRU Catherine¹,ᵇ and PAREDES Manuel¹,ᶜ*

¹ICA, Université de Toulouse, UPS, INSA, ISAE-SUPAERO, MINES-ALBI, CNRS, 3 rue Caroline Aigle, 31400 Toulouse, France

ᵃvaissett@insa-toulouse.fr, ᵇcatherine.mabru@isae-supaero.fr, ᶜparedes@insa-toulouse.fr

**Keywords:** Residual Stress, Cold Drawing, Characterization, Small Diameter Wire

**Abstract.** Wire suppliers commonly only commit to the chemical composition and minimum mechanical resistance of their wires in their material certificates. Unfortunately, the cold drawing process induces residual stresses that potentially have a strong impact on the overall behavior of the wire. An explanation of the difference between the industrial ‘tensile modulus’ and ‘flexural modulus’ whereas the material has a unique ‘Young Modulus’ is proposed by mean of FEA and experimental tests. Moreover, in order to better estimate the complex material state of small wires (with a diameter smaller than one millimeter) a novel experimental approach has been developed to estimate the residual stress.

## Introduction

Small cold drawn wires are extensively used in mechanical systems. Some material with structural effects can exhibit flexural modulus and tensile modulus that can differ from the Young Modulus such as composite materials [1].

The elongation of grains and reduction in diameter during drawing impart a needle-like shape to the wire's grains, resulting in pronounced microstructural anisotropy between cross-sections and longitudinal views, as clearly shown in Fig. 1 [2]. This inherent anisotropy suggests potential differences in mechanical properties, specifically in flexural and tensile moduli.

<figure><img src="image_671.png" /><figcaption>Figure 1: Anisotropy between the cross-sectional area (a) and the longitudinal cross-section (b) of a drawn wire [2].</figcaption></figure>

Moreover, the drawing process can induce axial residual stresses in the wire [3], creating a complex stress state within the material as shown in Fig. 2.

<figure><img src="image_672.png" /></figure>
# Analysis of mechanical properties of stainless steel small diameter cold drawn wires

VAISSETTE Julien¹,ᵃ, MABRU Catherine¹,ᵇ and PAREDES Manuel¹,ᶜ*

¹ICA, Université de Toulouse, UPS, INSA, ISAE-SUPAERO, MINES-ALBI, CNRS, 3 rue Caroline Aigle, 31400 Toulouse, France

ᵃvaissett@insa-toulouse.fr, ᵇcatherine.mabru@isae-supaero.fr, ᶜparedes@insa-toulouse.fr

**Keywords:** Residual Stress, Cold Drawing, Characterization, Small Diameter Wire

**Abstract.** Wire suppliers commonly only commit to the chemical composition and minimum mechanical resistance of their wires in their material certificates. Unfortunately, the cold drawing process induces residual stresses that potentially have a strong impact on the overall behavior of the wire. An explanation of the difference between the industrial ‘tensile modulus’ and ‘flexural modulus’ whereas the material has a unique ‘Young Modulus’ is proposed by mean of FEA and experimental tests. Moreover, in order to better estimate the complex material state of small wires (with a diameter smaller than one millimeter) a novel experimental approach has been developed to estimate the residual stress.

**Introduction**

Small cold drawn wires are extensively used in mechanical systems. Some material with structural effects can exhibit flexural modulus and tensile modulus that can differ from the Young Modulus such as composite materials [1].

The elongation of grains and reduction in diameter during drawing impart a needle-like shape to the wire's grains, resulting in pronounced microstructural anisotropy between cross-sections and longitudinal views, as clearly shown in Fig. 1 [2]. This inherent anisotropy suggests potential differences in mechanical properties, specifically in flexural and tensile moduli.

<figure><img src="image_682.png" /><figcaption>Figure 1: Anisotropy between the cross-sectional area (a) and the longitudinal cross-section (b) of a drawn wire [2].</figcaption></figure>

Moreover, the drawing process can induce axial residual stresses in the wire [3], creating a complex stress state within the material as shown in Fig. 2.

<figure><img src="image_683.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
<figure><img src="image_673.png" /><figcaption>Figure 2: Gradient of residual axial stress after drawing [3].</figcaption></figure>

Detailed studies using X-rays and Neutron diffraction reveal surface tension and central compression in drawn wires post-drawing [4] (see Fig. 3). Other methods of characterization of residual stresses exist (incremental drilling method [5] or the Sachs method [6] for instance), they are all difficult to apply on wires with small diameters: for diameters inferior to 2 mm, longitudinal drilling in the wire is extremely difficult.

<figure><img src="image_674.png" /><figcaption>Figure 3: Evolution of axial stress after drawing [4].</figcaption></figure>

These residual stresses, combined with grain deformation, potentially impact the structural behavior of small cold drawn wires. In order to show why manufacturers and engineers commonly use tensile and flexural moduli instead of a unique Young Modulus, a finite element study that highlights the behavior of a small cold drawn wire in a bending test is first presented. Then a novel approach to estimate the residual stress is described. Afterwards, conclusions are presented.

The aim of this work is to gain a better understanding of residual stresses in drawn wires, for use in spring production. In so doing, this work delves into an area hitherto little studied by spring manufacturers, to improve the control of dimensional tolerances of their springs.

## Study of the bending properties

Many mechanical components as springs use small mechanical drawn wires. Small drawn wires undergo manufacturing processes that alter their mechanical properties. While manufacturers often designate distinct elastic moduli for tension and bending in wires smaller than one millimeter, scientific consensus recognizes the Young modulus as the fundamental measure. This finite element study aims to elucidate the drawing process and bending mechanisms, explaining why wires might exhibit lower stiffness than anticipated via the common Young Modulus, leading to the prevalent use of a specialized elastic modulus in bending.

The model first simulates an approximate drawing process to modify the material state and include residual stress into the wire. Compared to the real process, in the simulation the wire is drawn in one step and without temperature effect. The numerical simulation model involves
<figure><img src="image_684.png" /><figcaption>Figure 2: Gradient of residual axial stress after drawing [3].</figcaption></figure>

Detailed studies using X-rays and Neutron diffraction reveal surface tension and central compression in drawn wires post-drawing [4] (see Fig. 3). Other methods of characterization of residual stresses exist (incremental drilling method [5] or the Sachs method [6] for instance), they are all difficult to apply on wires with small diameters: for diameters inferior to 2 mm, longitudinal drilling in the wire is extremely difficult.

<figure><img src="image_685.png" /><figcaption>Figure 3: Evolution of axial stress after drawing [4].</figcaption></figure>

These residual stresses, combined with grain deformation, potentially impact the structural behavior of small cold drawn wires. In order to show why manufacturers and engineers commonly use tensile and flexural moduli instead of a unique Young Modulus, a finite element study that highlights the behavior of a small cold drawn wire in a bending test is first presented. Then a novel approach to estimate the residual stress is described. Afterwards, conclusions are presented.

The aim of this work is to gain a better understanding of residual stresses in drawn wires, for use in spring production. In so doing, this work delves into an area hitherto little studied by spring manufacturers, to improve the control of dimensional tolerances of their springs.

## Study of the bending properties

Many mechanical components as springs use small mechanical drawn wires. Small drawn wires undergo manufacturing processes that alter their mechanical properties. While manufacturers often designate distinct elastic moduli for tension and bending in wires smaller than one millimeter, scientific consensus recognizes the Young modulus as the fundamental measure. This finite element study aims to elucidate the drawing process and bending mechanisms, explaining why wires might exhibit lower stiffness than anticipated via the common Young Modulus, leading to the prevalent use of a specialized elastic modulus in bending.

The model first simulates an approximate drawing process to modify the material state and include residual stress into the wire. Compared to the real process, in the simulation the wire is drawn in one step and without temperature effect. The numerical simulation model involves
forcing a 0.84 mm diameter wire through a 0.80 mm diameter die. As for the bending test, the center distance between the fixed supports is 22.3 mm, their diameter is 3 mm and the diameter of the mobile support is also 3 mm.

The FEA is performed with Abaqus Standard (implicit), C3D20R elements (A 20-node quadratic brick, reduced integration) and 12 elements in diameter. A plane symmetry is exploited to decrease the computing time. The mesh is illustrated in Fig. 4 and the material properties are presented in Table 1. The simulation exploits multilinear kinematics hardening in order to simulate the Bauschinger effect (not the same behavior in compression and in tension) [6].

<figure><img src="image_675.png" /><figcaption>Figure 4: Finite element model.</figcaption></figure>

<table><caption>Table 1: Material properties</caption><thead><tr><th>Stress (MPa)</th><th>Plastic strain</th><th>Young Modulus (GPa)</th><th>Poisson coefficient</th></tr></thead><tbody><tr><td>1000</td><td>0</td><td rowspan="4">165</td><td rowspan="4">0,28</td></tr><tr><td>2030</td><td>0,0072</td></tr><tr><td>2070</td><td>0,0108</td></tr><tr><td>2080</td><td>3</td></tr></tbody></table>

The common formula of mechanics of materials (Euler-Bernoulli equation for beam bending) is exploited on the linear behavior of the load/deflection curve to identify the flexural modulus $E_f$. As expected, the obtained flexural modulus without drawing is equal to the Young Modulus $E_f = 165$ GPa = E. Moreover, the flexural modulus after drawing is lower than the Young Modulus $E_f = 122$ GPa < E. Nevertheless, the bending curve has a linear rate that corresponds to a lower value.
forcing a 0.84 mm diameter wire through a 0.80 mm diameter die. As for the bending test, the center distance between the fixed supports is 22.3 mm, their diameter is 3 mm and the diameter of the mobile support is also 3 mm.

The FEA is performed with Abaqus Standard (implicit), C3D20R elements (A 20-node quadratic brick, reduced integration) and 12 elements in diameter. A plane symmetry is exploited to decrease the computing time. The mesh is illustrated in Fig. 4 and the material properties are presented in Table 1. The simulation exploits multilinear kinematics hardening in order to simulate the Bauschinger effect (not the same behavior in compression and in tension) [6].

<figure><img src="image_686.png" /><figcaption>Figure 4: Finite element model.</figcaption></figure>

<table><caption>Table 1: Material properties</caption><thead><tr><td>Stress (MPa)</td><td>Plastic strain</td><td>Young Modulus (GPa)</td><td>Poisson coefficient</td></tr></thead><tbody><tr><td>1000</td><td>0</td><td rowspan="4">165</td><td rowspan="4">0,28</td></tr><tr><td>2030</td><td>0,0072</td></tr><tr><td>2070</td><td>0,0108</td></tr><tr><td>2080</td><td>3</td></tr></tbody></table>

The common formula of mechanics of materials (Euler-Bernoulli equation for beam bending) is exploited on the linear behavior of the load/deflection curve to identify the flexural modulus $E_f$. As expected, the obtained flexural modulus without drawing is equal to the Young Modulus $E_f = 165$ GPa = E. Moreover, the flexural modulus after drawing is lower than the Young Modulus $E_f = 122$ GPa < E. Nevertheless, the bending curve has a linear rate that corresponds to a lower value.
<figure><img src="image_676.png" /><figcaption>Figure 5: Evolution of plastic strain in a bending test after drawing.</figcaption></figure>

In order to explain this phenomenon, the evolution of plastic strain on a section all along the test has been plotted (see Fig. 5). The initial state after drawing is set as a reference. In a common bending test one half of the wire is in tension, the other one is in compression and the center is neutral. When the bending test begins, the wire is first in an elastic state and there is no plastic strain. Then, at a given level of external load, plastic strain appears at the external surface and progresses to the center of the wire. We can see this evolution on the upper part of Fig. 5 which is dedicated to the half of the wire in tension (positive values of plastic strain). Node 1 is at the external surface and after about 1.2 seconds of simulation time, first yielding occurs and the plastic strain progresses towards the center (node 7).

In a bending test of a material supposed to be linear elastic, homogeneous and isotropic, the behavior is the same in the compression state of the wire but it is not the case here. As shown in Fig. 5, because of the drawing process (and thanks to the multilinear kinematic hardening simulation that considers the Bauschinger effect), early plastic strain directly appears at the beginning of the test in the part of the wire in a compression state (negative values of plastic strain). This area is highlighted by an arrow in Fig. 5 (nodes 8 to 12 are involved, except node 13 at the surface which is in a preliminary tension state due to the prestress). Thus, at the beginning of the test, even when the global bending behavior of the wire is linear, a part of the section is plastically deformed. We know that the material in increasing plastic state has a lower stiffness. As a consequence, the whole stiffness of the section is decreased and that explains why the calculated flexural modulus is lower than the Young modulus. From a scientific point of view, a small cold drawn wire is no more a uniform material but could be considered as a complex structure.

## Estimate of the residual stress

The previous FEA simulates a simplified drawing process. This simulation enables to highlight the axial residual stress after drawing (Fig. 6). As previously shown, the external surface is in tension and the core is in a compression state.
<figure><img src="image_687.png" /><figcaption>Figure 5: Evolution of plastic strain in a bending test after drawing.</figcaption></figure>

In order to explain this phenomenon, the evolution of plastic strain on a section all along the test has been plotted (see Fig. 5). The initial state after drawing is set as a reference. In a common bending test one half of the wire is in tension, the other one is in compression and the center is neutral. When the bending test begins, the wire is first in an elastic state and there is no plastic strain. Then, at a given level of external load, plastic strain appears at the external surface and progresses to the center of the wire. We can see this evolution on the upper part of Fig. 5 which is dedicated to the half of the wire in tension (positive values of plastic strain). Node 1 is at the external surface and after about 1.2 seconds of simulation time, first yielding occurs and the plastic strain progresses towards the center (node 7).

In a bending test of a material supposed to be linear elastic, homogeneous and isotropic, the behavior is the same in the compression state of the wire but it is not the case here. As shown in Fig. 5, because of the drawing process (and thanks to the multilinear kinematic hardening simulation that considers the Bauschinger effect), early plastic strain directly appears at the beginning of the test in the part of the wire in a compression state (negative values of plastic strain). This area is highlighted by an arrow in Fig. 5 (nodes 8 to 12 are involved, except node 13 at the surface which is in a preliminary tension state due to the prestress). Thus, at the beginning of the test, even when the global bending behavior of the wire is linear, a part of the section is plastically deformed. We know that the material in increasing plastic state has a lower stiffness. As a consequence, the whole stiffness of the section is decreased and that explains why the calculated flexural modulus is lower than the Young modulus. From a scientific point of view, a small cold drawn wire is no more a uniform material but could be considered as a complex structure.

## Estimate of the residual stress

The previous FEA simulates a simplified drawing process. This simulation enables to highlight the axial residual stress after drawing (Fig. 6). As previously shown, the external surface is in tension and the core is in a compression state.
<figure><img src="image_677.png" /><figcaption>Figure 6: Axial residual stress state in the section of a 0.8mm diameter wire (FEA).</figcaption></figure>

Unfortunately, the common approaches to evaluate the residual stress are not directly available for small wires. For that reason, we propose a new approach based on material removal. The experimental approach [8] consists in removing a part of the material by coating the wire in a soluble resin and then gently polishing the sample in order to make a flat cut parallel to the wire axis. After resin dissolving, the new curvature is measured as described in Fig. 7.

<figure><img src="image_678.png" /><figcaption>Figure 7: Experimental steps of the matter removal method for small wires.</figcaption></figure>

The curvature measurement is not trivial when dealing with small wires. In our case, curvature measurements were performed on Alicona InfiniteFocusSL, an optical 3D measurement device. Once the curvature is measured, the next step consists in an analytical approach to estimate the residual stress state. This approach has been inspired by the work of Han et al. for ultra-thin plate-shaped optical parts [9]. The idea is to propose an analytical formula for the residual stress shape and then to find the parameters values that will induce the experimental curvature previously found.

Before material removal, the residual stress induces a global equilibrium in the wire. After matter removal the initial equilibrium vanishes and a new equilibrium is found that induces a
<figure><img src="image_688.png" /><figcaption>Figure 6: Axial residual stress state in the section of a 0.8mm diameter wire (FEA).</figcaption></figure>

Unfortunately, the common approaches to evaluate the residual stress are not directly available for small wires. For that reason, we propose a new approach based on material removal. The experimental approach [8] consists in removing a part of the material by coating the wire in a soluble resin and then gently polishing the sample in order to make a flat cut parallel to the wire axis. After resin dissolving, the new curvature is measured as described in Fig. 7.

<figure><img src="image_689.png" /><figcaption>Figure 7: Experimental steps of the matter removal method for small wires.</figcaption></figure>

The curvature measurement is not trivial when dealing with small wires. In our case, curvature measurements were performed on Alicona InfiniteFocusSL, an optical 3D measurement device. Once the curvature is measured, the next step consists in an analytical approach to estimate the residual stress state. This approach has been inspired by the work of Han et al. for ultra-thin plate-shaped optical parts [9]. The idea is to propose an analytical formula for the residual stress shape and then to find the parameters values that will induce the experimental curvature previously found.

Before material removal, the residual stress induces a global equilibrium in the wire. After matter removal the initial equilibrium vanishes and a new equilibrium is found that induces a
curvature of the wire. This curvature can be linked to the bending momentum M<sub>b</sub> induced by the removed residual stress.

$$M_b = \frac{a}{b^2} 2EI \qquad (1)$$

Where a is the maximal radial displacement, b is the mean axial distance, I is the area moment of inertia after matter removal and E the Young modulus.

<figure><img src="image_679.png" /><figcaption>Figure 8: wire profile after matter removal.</figcaption></figure>

The next step involves evaluating the residual stress in the wire. First of all, a residual stress profile is proposed that can fit the profile shown in Fig. 6.

$$\sigma_0(r) = A \left[ -\cos \left( \pi \left( \frac{|r|}{R} \right)^n \right) + B \right] \qquad (2)$$

with:

A, n, B: parameters to be defined

R: wire radius

r: radial coordinate, between -R and R.

In order to find the unknown parameters, we need to study the wire equilibrium before and after matter removal. We know that before matter removal the wire is in equilibrium. Thus, the following equation is satisfied:

$$\iint \sigma_0(r) dS = 0 \qquad (3)$$

We also know that the removal of a part of the wire induces M<sub>b</sub>. Thus:

$$M_b = \int y' \sigma_0(r) dS \qquad (4)$$

Where y' is the coordinate in the wire normal to the cut as illustrated in Fig. 9
curvature of the wire. This curvature can be linked to the bending momentum M<sub>b</sub> induced by the removed residual stress.

$$M_b = \frac{a}{b^2} 2EI \quad (1)$$

Where a is the maximal radial displacement, b is the mean axial distance, I is the area moment of inertia after matter removal and E the Young modulus.

<figure><img src="image_690.png" /><figcaption>Figure 8: wire profile after matter removal.</figcaption></figure>

The next step involves evaluating the residual stress in the wire. First of all, a residual stress profile is proposed that can fit the profile shown in Fig. 6.

$$\sigma_0(r) = A \left[ -\cos \left( \pi \left( \frac{|r|}{R} \right)^n \right) + B \right] \quad (2)$$

with:

A, n, B: parameters to be defined

R: wire radius

r: radial coordinate, between -R and R.

In order to find the unknown parameters, we need to study the wire equilibrium before and after matter removal. We know that before matter removal the wire is in equilibrium. Thus, the following equation is satisfied:

$$\iint \sigma_0(r) dS = 0 \quad (3)$$

We also know that the removal of a part of the wire induces M<sub>b</sub>. Thus:

$$M_b = \int y' \sigma_0(r) dS \quad (4)$$

Where y' is the coordinate in the wire normal to the cut as illustrated in Fig. 9
<figure><img src="image_680.png" /><figcaption>Figure 9: wire section after matter removal.</figcaption></figure>

We consequently have 2 equations to be satisfied but 3 parameters to be found. In order to solve this problem, we chose to fix the radial coordinate where the radial stress is null (0.3mm in the case illustrated in Fig. 6). The parameters values are finally determined using a short optimization algorithm.

In an application example, a stainless-steel AISI 302 wire (0.8mm diameter, E=185 GPa) has been tested. The experimental measurements induce a= 0.078mm, b=5mm and $y_{max}$ = 0.23mm (see Fig.11). Thus $I'=0.01167mm^4$ and $M_b$ = 13.47 N.mm. The solving process enables to estimate the equation for the axial residual stress and illustrated in Fig. 10:

$$ \sigma_0(r) = 865 \left[ -\cos\left(\pi\left(\frac{|r|}{0.4}\right)^{2.93}\right) + 0.216 \right] \quad (4) $$

<figure><img src="image_681.png" /><figcaption>Figure 10: Evaluation of the axial residual stress in the wire.</figcaption></figure>
<figure><img src="image_691.png" /><figcaption>Figure 9: wire section after matter removal.</figcaption></figure>

We consequently have 2 equations to be satisfied but 3 parameters to be found. In order to solve this problem, we chose to fix the radial coordinate where the radial stress is null (0.3mm in the case illustrated in Fig. 6). The parameters values are finally determined using a short optimization algorithm.

In an application example, a stainless-steel AISI 302 wire (0.8mm diameter, E=185 GPa) has been tested. The experimental measurements induce a= 0.078mm, b=5mm and $y_{max}$ = 0.23mm (see Fig.11). Thus $I'=0.01167mm^4$ and $M_b$ = 13.47 N.mm. The solving process enables to estimate the equation for the axial residual stress and illustrated in Fig. 10:

$$ \sigma_0(r) = 865 \left[ -\cos\left(\pi\left(\frac{|r|}{0.4}\right)^{2.93}\right) + 0.216 \right] \quad (4) $$

<figure><img src="image_692.png" /><figcaption>Figure 10: Evaluation of the axial residual stress in the wire.</figcaption></figure>
<figure><img src="image_682.png" /><figcaption>Figure 11: Radial displacement measured after the removal of a 0.17 mm layer, on a 0.8 mm diameter wire.</figcaption></figure>

It is important to point out that the analytical estimate of residual stress intensity seen in Fig. 10 cannot be directly compared with that obtained after numerical simulation of a wire simplified drawing operation in Fig. 6. Indeed, the latter simulation is not quantitatively objective, since it studied residual stress intensity after a wire drawing operation not fully representative of the 0.8 mm diameter wire drawn here. The two curves can therefore only be compared qualitatively.

## Conclusions

The aim of this paper was to delve into the mechanics involved in both the drawing process and bending to clarify why a wire might exhibit a softer behavior than anticipated by its Young modulus. Additionally, it sought to estimate the residual axial stresses.

A simplified Finite Element Analysis of the drawing process helped gauge the material's state upon delivery to manufacturers. Subsequently, the simulation extended to a bending test. In-depth simulation analysis revealed that the loss of stiffness of the linear behavior was due to substantial plastic deformation during drawing, particularly an inner portion of the wire undergoing direct plastic compression upon the onset of the bending test. This localized plasticity beneath the surface, despite the wire's overall linear behavior, contributed to a decreased overall stiffness, necessitating the use of a specific elastic modulus in bending for a consistent equivalence.

Furthermore, an innovative experimental method was developed to characterize residual stresses, tailored for small wires (diameter less than 1 mm). The method involved coating the wire in a soluble resin, followed by gentle polishing to create a flat cut parallel to the wire axis. Similar to other residual characterization methods, the goal was to observe the deformed shape resulting from the release of residual stress post-polishing. An analytical approach was employed to estimate the magnitude of the residual stress causing the curvature change.

Experiments performed on a 0.8 mm diameter AISI 302 wire revealed remarkably high residual stress intensity, underscoring the necessity for a thorough characterization of residual stress profiles in cold-drawn wires. However, further research is required to refine the initial estimate of the radius where axial residual stress is null.

## References

[1] T.-I. Lee, C. Kim, M. S. Kim, T.-S. Kim, Flexural and tensile moduli of flexible FR4 substrates, Polym. Test., 53 (2016) 70-76. https://doi.org/10.1016/j.polymertesting.2016.05.012
<figure><img src="image_693.png" /><figcaption>Figure 11: Radial displacement measured after the removal of a 0.17 mm layer, on a 0.8 mm diameter wire.</figcaption></figure>

It is important to point out that the analytical estimate of residual stress intensity seen in Fig. 10 cannot be directly compared with that obtained after numerical simulation of a wire simplified drawing operation in Fig. 6. Indeed, the latter simulation is not quantitatively objective, since it studied residual stress intensity after a wire drawing operation not fully representative of the 0.8 mm diameter wire drawn here. The two curves can therefore only be compared qualitatively.

## Conclusions

The aim of this paper was to delve into the mechanics involved in both the drawing process and bending to clarify why a wire might exhibit a softer behavior than anticipated by its Young modulus. Additionally, it sought to estimate the residual axial stresses.

A simplified Finite Element Analysis of the drawing process helped gauge the material's state upon delivery to manufacturers. Subsequently, the simulation extended to a bending test. In-depth simulation analysis revealed that the loss of stiffness of the linear behavior was due to substantial plastic deformation during drawing, particularly an inner portion of the wire undergoing direct plastic compression upon the onset of the bending test. This localized plasticity beneath the surface, despite the wire's overall linear behavior, contributed to a decreased overall stiffness, necessitating the use of a specific elastic modulus in bending for a consistent equivalence.

Furthermore, an innovative experimental method was developed to characterize residual stresses, tailored for small wires (diameter less than 1 mm). The method involved coating the wire in a soluble resin, followed by gentle polishing to create a flat cut parallel to the wire axis. Similar to other residual characterization methods, the goal was to observe the deformed shape resulting from the release of residual stress post-polishing. An analytical approach was employed to estimate the magnitude of the residual stress causing the curvature change.

Experiments performed on a 0.8 mm diameter AISI 302 wire revealed remarkably high residual stress intensity, underscoring the necessity for a thorough characterization of residual stress profiles in cold-drawn wires. However, further research is required to refine the initial estimate of the radius where axial residual stress is null.

## References

[1] T.-I. Lee, C. Kim, M. S. Kim, T.-S. Kim, Flexural and tensile moduli of flexible FR4 substrates, Polym. Test., 53 (2016) 70-76. https://doi.org/10.1016/j.polymertesting.2016.05.012
[2] J. Atienza, J. Ruiz-Hervias, M. Elices, The Role of Residual Stresses in the Performance and Durability of Prestressing Steel Wires, Exp. Mech., 52, (2012) 881-893.
https://doi.org/10.1007/s11340-012-9597-1

[3] S.-K. Lee, H. H., K. M., B. M., Influence of Process Parameters on Residual Stress and Reducing Residual Stress in Drawn Wire, Trans. Mater. Process., 14 (2005) 704-711.
https://doi.org/10.5228/KSPP.2005.14.8.704

[4] J. M. Atienza, J. Ruiz-Hervías, M. Martinez-Perez, F. Mompeán, M. García-Hernández, M. Elices, Residual stresses in cold drawn pearlitic rods, 52-4 (2005) 305-309.
https://doi.org/10.1016/J.SCRIPTAMAT.2004.10.010

[5] G. Montay, O. Sicot, A. Maras, et al. Two Dimensions Residual Stresses Analysis Through Incremental Groove Machining Combined with Electronic Speckle Pattern Interferometry. Exp Mech 49 (2009) 459-469.
https://doi.org/10.1007/s11340-008-9151-3

[6] A. P. Parker, A critical examination of Sachs' material-removal method for determination of residual stress, J. of Pressure Vessel Technology, 126 (2004) 234-236.
https://doi.org/10.1115/1.1689357

[7] W. M. Stobbs, S. Paetke, The Bauschinger effect in cold drawn patented wire, Acta Metall., 33-5 (1985) 777-783.
https://doi.org/10.1016/0001-6160(85)90101-4

[8] Julien Vaïssette, Manuel Paredes, Catherine Mabru, A residual stress characterization method by matter removal of a small diameter wire, 11th International Conference on Residual Stresses - Nancy - France - 27-30th March 2022

[9] X. Han, Z. Jin, Q. Mu, L. Niu, P. Zhou, Internal stress and deformation analysis of ultra-thin plate-shaped optical parts in thinning process, Opt. Express, 27-19 (2019) 27202-27214.
https://doi.org/10.1364/OE.27.027202
[2] J. Atienza, J. Ruiz-Hervias, M. Elices, The Role of Residual Stresses in the Performance and Durability of Prestressing Steel Wires, Exp. Mech., 52, (2012) 881-893.
https://doi.org/10.1007/s11340-012-9597-1

[3] S.-K. Lee, H. H., K. M., B. M., Influence of Process Parameters on Residual Stress and Reducing Residual Stress in Drawn Wire, Trans. Mater. Process., 14 (2005) 704-711.
https://doi.org/10.5228/KSPP.2005.14.8.704

[4] J. M. Atienza, J. Ruiz-Hervías, M. Martinez-Perez, F. Mompeán, M. García-Hernández, M. Elices, Residual stresses in cold drawn pearlitic rods, 52-4 (2005) 305-309.
https://doi.org/10.1016/J.SCRIPTAMAT.2004.10.010

[5] G. Montay, O. Sicot, A. Maras, et al. Two Dimensions Residual Stresses Analysis Through Incremental Groove Machining Combined with Electronic Speckle Pattern Interferometry. Exp Mech 49 (2009) 459-469.
https://doi.org/10.1007/s11340-008-9151-3

[6] A. P. Parker, A critical examination of Sachs' material-removal method for determination of residual stress, J. of Pressure Vessel Technology, 126 (2004) 234-236.
https://doi.org/10.1115/1.1689357

[7] W. M. Stobbs, S. Paetke, The Bauschinger effect in cold drawn patented wire, Acta Metall., 33-5 (1985) 777-783.
https://doi.org/10.1016/0001-6160(85)90101-4

[8] Julien Vaïssette, Manuel Paredes, Catherine Mabru, A residual stress characterization method by matter removal of a small diameter wire, 11th International Conference on Residual Stresses - Nancy - France - 27-30th March 2022

[9] X. Han, Z. Jin, Q. Mu, L. Niu, P. Zhou, Internal stress and deformation analysis of ultra-thin plate-shaped optical parts in thinning process, Opt. Express, 27-19 (2019) 27202-27214.
https://doi.org/10.1364/OE.27.027202
# Characterization and modelling of grain growth in Zr-Nb alloys: Niobium concentration influence

HAHN Pauline<fnref n="1" />,<fnref n="2" />,ᵃ*, GAILLAC Alexis<fnref n="2" />, FLIPON Baptiste<fnref n="1" />, BIGNON Madeleine<fnref n="1" />, BOZZOLO Nathalie<fnref n="1" /> and BERNACKI Marc<fnref n="1" />

**Keywords:** Zr-Nb Alloys, Grain Growth, Solute Drag, Numerical Simulation

**Abstract.** Zr-Nb alloys are mainly used in the nuclear energy industry. A study was carried out with the aim of understanding the physical phenomena involved during the hot extrusion stage and modeling these phenomena with numerical simulations. However, it is first essential to determine the material parameters, including reduced mobility during pure grain growth. Experimental and numerical campaigns are therefore carried out to define this parameter, material and model dependent. Consequently, this study focuses exclusively on the influence of niobium content on the grain growth kinetics of Zr-Nb alloy.

**Introduction**

In nuclear fuel assemblies, zirconium alloys are mainly used due to their low neutron capture cross section. These alloys fulfill the high requirements of nuclear applications regarding good mechanical properties and corrosion resistance in relation to their in-use environment (mainly cladding tubes and structural components of fuel assemblies located in the nuclear power plant core) [1]. The material properties are determined by their microstructure, which evolves during the manufacturing process. In this work, the interest is focused on the hot extrusion stage to study recrystallization (ReX) and grain growth (GG) kinetics in Zr-Nb alloys. In the literature, alloying elements, including niobium, have been reported to strongly impact ReX and related phenomena [2]. The commercial Zr-1Nb alloy showed some difficulty in recrystallizing after hot extrusion. In order to understand the physical phenomena involved, a first stage consists of studying and quantifying the influence of niobium on GG kinetics on Zr-Nb alloy, composed of an α matrix (hexagonal close-packed) and of a β phase (body-centered cubic) considered as second phase particles population [3]. Experimental heat treatments were done to first obtain a fully recrystallized microstructure and then carry out annealing at different temperatures. Alloying elements such as niobium can segregate at grain boundaries (GB). During GG, if the diffusion coefficient of these segregated elements in the matrix is slower than the GB migration velocity, these elements will have an impact on the GB migration kinetics, known as solute drag effect [4]. Moreover, samples with 1% of niobium are analyzed to identify the Smith Zener pinning pressure associated with second phase particles [4,5,6]. Experimental characterizations were performed to be able to model the GG kinetic with predictive full-field simulations based on a level-set description of the GB [7].

**Materials and methods**

The Zr-1Nb alloy contains several alloying elements, of which niobium is the main element at 1% by weight. To determine GG kinetics, a single-phase material (without second phase particles) must be considered. Consequently, small ingots of approximately 1.7 kg were produced with different niobium contents in order to identify the influence of niobium in solid solution on GG kinetics. It was considered 0 - 0.2 - 0.4% of niobium by mass. 0.4% of niobium corresponds to the

<figure><img src="image_683.png" /></figure>

<footnote n="1">¹Mines Paris – PSL, CNRS, CEMEF, UMR 7635, France</footnote>

<footnote n="2">²Framatome – Components Research Center, France</footnote>

<footnote>ᵃpauline.hahn@minesparis.psl.eu</footnote>
# Characterization and modelling of grain growth in Zr-Nb alloys: Niobium concentration influence

HAHN Pauline<fnref n="1" />,<fnref n="2" />,ᵃ*, GAILLAC Alexis<fnref n="2" />, FLIPON Baptiste<fnref n="1" />, BIGNON Madeleine<fnref n="1" />, BOZZOLO Nathalie<fnref n="1" /> and BERNACKI Marc<fnref n="1" />

**Keywords:** Zr-Nb Alloys, Grain Growth, Solute Drag, Numerical Simulation

**Abstract.** Zr-Nb alloys are mainly used in the nuclear energy industry. A study was carried out with the aim of understanding the physical phenomena involved during the hot extrusion stage and modeling these phenomena with numerical simulations. However, it is first essential to determine the material parameters, including reduced mobility during pure grain growth. Experimental and numerical campaigns are therefore carried out to define this parameter, material and model dependent. Consequently, this study focuses exclusively on the influence of niobium content on the grain growth kinetics of Zr-Nb alloy.

**Introduction**

In nuclear fuel assemblies, zirconium alloys are mainly used due to their low neutron capture cross section. These alloys fulfill the high requirements of nuclear applications regarding good mechanical properties and corrosion resistance in relation to their in-use environment (mainly cladding tubes and structural components of fuel assemblies located in the nuclear power plant core) [1]. The material properties are determined by their microstructure, which evolves during the manufacturing process. In this work, the interest is focused on the hot extrusion stage to study recrystallization (ReX) and grain growth (GG) kinetics in Zr-Nb alloys. In the literature, alloying elements, including niobium, have been reported to strongly impact ReX and related phenomena [2]. The commercial Zr-1Nb alloy showed some difficulty in recrystallizing after hot extrusion. In order to understand the physical phenomena involved, a first stage consists of studying and quantifying the influence of niobium on GG kinetics on Zr-Nb alloy, composed of an α matrix (hexagonal close-packed) and of a β phase (body-centered cubic) considered as second phase particles population [3]. Experimental heat treatments were done to first obtain a fully recrystallized microstructure and then carry out annealing at different temperatures. Alloying elements such as niobium can segregate at grain boundaries (GB). During GG, if the diffusion coefficient of these segregated elements in the matrix is slower than the GB migration velocity, these elements will have an impact on the GB migration kinetics, known as solute drag effect [4]. Moreover, samples with 1% of niobium are analyzed to identify the Smith Zener pinning pressure associated with second phase particles [4,5,6]. Experimental characterizations were performed to be able to model the GG kinetic with predictive full-field simulations based on a level-set description of the GB [7].

**Materials and methods**

The Zr-1Nb alloy contains several alloying elements, of which niobium is the main element at 1% by weight. To determine GG kinetics, a single-phase material (without second phase particles) must be considered. Consequently, small ingots of approximately 1.7 kg were produced with different niobium contents in order to identify the influence of niobium in solid solution on GG kinetics. It was considered 0 - 0.2 - 0.4% of niobium by mass. 0.4% of niobium corresponds to the

<figure><img src="image_694.png" /></figure>

<footnote n="1">¹Mines Paris – PSL, CNRS, CEMEF, UMR 7635, France</footnote>

<footnote n="2">²Framatome – Components Research Center, France</footnote>

<footnote>ᵃpauline.hahn@minesparis.psl.eu</footnote>
solubility limit of niobium in the α phase at 580°C. An ingot with 1% of niobium was also produced to take into account the impact of second phase particles on GG kinetics. Each ingot contains around 1400 ppm of oxygen and 300 ppm of iron. Fig. 1 illustrates the manufacturing process used to obtain sheets of about 3mm thick and from which 5x5mm samples are taken. It should be noted that the reduction ratio of the last cold rolling pass is 50%.

The last heat treatment duration at 580°C in Fig. 1 was adapted for each niobium content in order to begin GG heat treatments from a fully recrystallized state. Static recrystallization kinetics are also influenced by the niobium content. According to microhardness tests on a Zr-1Nb sheet

<figure><img src="image_684.png" /><figcaption>Fig. 1. Sheet metal manufacturing process for this study.</figcaption></figure>

presented by Tian et al. [8], 140 minutes are required to obtain a fully recrystallized material after heat treatment at 580°C. The reduction rate of this sheet is 50%, identical to the reduction rate of the samples in this study. An experimental matrix was defined to track the evolution of the recrystallized fraction between the initial state and 150 minutes of heat treatment at 580°C for each niobium content. ReX and GG annealing temperatures were fixed to avoid moving into the two-phase α + β domain. This temperature is around 600°C. 580°C allows an overshoot margin for the furnace during the beginning of the heat treatment. After defining the fully recrystallized metallurgical state as a function of niobium content, GG annealings were carried out. This curvature flow mechanism is thermally activated and driven by the minimization of the total surface energy of the system, leading to a decrease of the GB network length [4]. This experimental and numerical campaign will highlight the influence of niobium on GG kinetics. The material parameters dependent on the numerical model used will also be determined, in particular the GB mobility. Therefore, 3 heat treatment temperatures are considered, and for each temperature at least 3 durations to use Burke and Turnbull's law [9]. The experimental matrix was adapted following the first results. A complete matrix has been defined for the Zr-0Nb samples with at least 3 heat treatment durations at 530°C, 550°C and 580°C. Results from samples with 0.2% and 0.4% of niobium will be used to identify the niobium solute drag effect on grain growth. The solute drag effect involves the impact of some alloying elements in solid solution on the velocity of GB migration [4]. Research by Zahler et al. [10] highlights this phenomenon.
solubility limit of niobium in the α phase at 580°C. An ingot with 1% of niobium was also produced to take into account the impact of second phase particles on GG kinetics. Each ingot contains around 1400 ppm of oxygen and 300 ppm of iron. Fig. 1 illustrates the manufacturing process used to obtain sheets of about 3mm thick and from which 5x5mm samples are taken. It should be noted that the reduction ratio of the last cold rolling pass is 50%.

The last heat treatment duration at 580°C in Fig. 1 was adapted for each niobium content in order to begin GG heat treatments from a fully recrystallized state. Static recrystallization kinetics are also influenced by the niobium content. According to microhardness tests on a Zr-1Nb sheet

<figure><img src="image_695.png" /><figcaption>Fig. 1. Sheet metal manufacturing process for this study.</figcaption></figure>

presented by Tian et al. [8], 140 minutes are required to obtain a fully recrystallized material after heat treatment at 580°C. The reduction rate of this sheet is 50%, identical to the reduction rate of the samples in this study. An experimental matrix was defined to track the evolution of the recrystallized fraction between the initial state and 150 minutes of heat treatment at 580°C for each niobium content. ReX and GG annealing temperatures were fixed to avoid moving into the two-phase α + β domain. This temperature is around 600°C. 580°C allows an overshoot margin for the furnace during the beginning of the heat treatment. After defining the fully recrystallized metallurgical state as a function of niobium content, GG annealings were carried out. This curvature flow mechanism is thermally activated and driven by the minimization of the total surface energy of the system, leading to a decrease of the GB network length [4]. This experimental and numerical campaign will highlight the influence of niobium on GG kinetics. The material parameters dependent on the numerical model used will also be determined, in particular the GB mobility. Therefore, 3 heat treatment temperatures are considered, and for each temperature at least 3 durations to use Burke and Turnbull's law [9]. The experimental matrix was adapted following the first results. A complete matrix has been defined for the Zr-0Nb samples with at least 3 heat treatment durations at 530°C, 550°C and 580°C. Results from samples with 0.2% and 0.4% of niobium will be used to identify the niobium solute drag effect on grain growth. The solute drag effect involves the impact of some alloying elements in solid solution on the velocity of GB migration [4]. Research by Zahler et al. [10] highlights this phenomenon.
## Experimental procedure

In order to determine the recrystallized fraction in the samples resulting from the ReX heat treatment, EBSD analysis was selected. To obtain statistically significant data, it was decided to use a view field of 500 µm wide and 375 µm high with a measurement step of 500 nm. The recrystallized fraction criterion was then defined with a GAKAM (grain average kernel average misorientation) value of less than 0.4° to consider a recrystallized grain. To characterize grain size after GG annealing, optical microscope images were taken under polarized light. This method was chosen to save time compared with EBSD analysis. During the final metallographic preparation stage, a thin oxide layer is created on the surface. Under polarized light, the surface is reflected in different ways depending on the crystallographic orientation of the grains. A view field of 330 µm wide and 250 µm high is considered. In order to be able to discretise the different crystallographic orientations during image post-processing, 3 images were taken of the same area with a different polarization, as shown in Fig. 2. In addition, in order to be representative of the sample, 3 observation zones are also recommended. The article by Flipon et al. [11] explains the method used via the Fiji software [12]. A step in the Fiji image post-processing macro also recorded information on the area of each grain captured. Grains without a closed outline are not taken into account. The equivalent circle diameter (ECD) can therefore be calculated. The ECD for a grain is calculated using Eq. 1.

<figure><img src="image_685.png" /><figcaption>Fig. 2. Images of the same area with a different polarisation. View field of 330 µm wide.</figcaption></figure>

$$ECD_i = 2\sqrt{\frac{A_i}{\pi}} \qquad (1)$$

with $A_i$ the area of a grain. Subsequently, an average ECD value can be calculated from the individual ECDs as Eq. 2.

$$\overline{ECD} = \frac{1}{N} \sum_{i=1}^{N} ECD_{i} \qquad (2)$$

with N the total number of grains studied. A comparison was carried out between the average grain size obtained using EBSD maps (reference values) and the average grain size obtained using this method of optical observation and post-processing. The results are similar. The number of grains taken into account in this analysis (at least 1500 grains) gives statistical results for the average grain size of the sample.

## Numerical simulations

The microstructural evolution associated with ReX and GG can be modeled in several ways. In our case, full-field simulations based on a level-set (LS) description of the GB were chosen [7]. These simulations are highly accurate, providing an explicit microstructure description of local phenomena. Using the experimental results of this campaign, it is possible to model the GG
## Experimental procedure

In order to determine the recrystallized fraction in the samples resulting from the ReX heat treatment, EBSD analysis was selected. To obtain statistically significant data, it was decided to use a view field of 500 µm wide and 375 µm high with a measurement step of 500 nm. The recrystallized fraction criterion was then defined with a GAKAM (grain average kernel average misorientation) value of less than 0.4° to consider a recrystallized grain. To characterize grain size after GG annealing, optical microscope images were taken under polarized light. This method was chosen to save time compared with EBSD analysis. During the final metallographic preparation stage, a thin oxide layer is created on the surface. Under polarized light, the surface is reflected in different ways depending on the crystallographic orientation of the grains. A view field of 330 µm wide and 250 µm high is considered. In order to be able to discretise the different crystallographic orientations during image post-processing, 3 images were taken of the same area with a different polarization, as shown in Fig. 2. In addition, in order to be representative of the sample, 3 observation zones are also recommended. The article by Flipon et al. [11] explains the method used via the Fiji software [12]. A step in the Fiji image post-processing macro also recorded information on the area of each grain captured. Grains without a closed outline are not taken into account. The equivalent circle diameter (ECD) can therefore be calculated. The ECD for a grain is calculated using Eq. 1.

<figure><img src="image_696.png" /><figcaption>Fig. 2. Images of the same area with a different polarisation. View field of 330 µm wide.</figcaption></figure>

$$ECD_i = 2 \sqrt{\frac{A_i}{\pi}} \qquad (1)$$

with $A_i$ the area of a grain. Subsequently, an average ECD value can be calculated from the individual ECDs as Eq. 2.

$$\overline{ECD} = \frac{1}{N} \sum_{i=1}^{N} ECD_{i} \qquad (2)$$

with N the total number of grains studied. A comparison was carried out between the average grain size obtained using EBSD maps (reference values) and the average grain size obtained using this method of optical observation and post-processing. The results are similar. The number of grains taken into account in this analysis (at least 1500 grains) gives statistical results for the average grain size of the sample.

## Numerical simulations

The microstructural evolution associated with ReX and GG can be modeled in several ways. In our case, full-field simulations based on a level-set (LS) description of the GB were chosen [7]. These simulations are highly accurate, providing an explicit microstructure description of local phenomena. Using the experimental results of this campaign, it is possible to model the GG
kinetics in the DIGIMU software [13]. The microstructure was generated by immersing an experimental image with around 900 grains. The GB migration velocity is expressed in Eq. 3 [4].

$$ \vec{v} = MP\vec{n} \qquad (3) $$

with M the grain boundary mobility (M = 1.7 × 10⁻³ mm⁴/J.s for simulation with 0% of niobium at 580°C), P the sum of the driving pressures applied on the GB and n the unit normal exiting at the GB. In the case of GG, the term P in Eq. 4 refers only to the capillary pressure term P_c [4] and can be considered as a first approximation.

$$ P = P_c = -\gamma\kappa \qquad (4) $$

where γ corresponds to the GB energy (in our case, values taken from the literature [14] for Zircaloy-4) and κ is the mean GB curvature (trace of the curvature tensor in 3D). Then, GB mobility can be expressed as follows in Eq. 5 using an Arrhenius law.

$$ M = M_0 \exp \left( \frac{-Q}{RT} \right) \qquad (5) $$

with M₀ the preexponential constant factor, Q the mobility activation energy, R the perfect gas constant and T the absolute temperature. The model selected for the solute drag effect for 0.2% and 0.4% of niobium will be described in more details in the following section.

## Results and discussion

By post-processing images taken with an optical microscope under polarized light, a grain size distribution was obtained for each sample (heat treatment time and niobium content) and an average grain size in diameter (ECD) was determined. Fig. 3 illustrates the different kinetics obtained as a function of niobium content with a heat treatment temperature of 580°C. GG kinetics decreased with increasing niobium content. A significant difference was observed between 0.2% and 0.4% of niobium. Furthermore, with 0% of niobium, the kinetics of GG are still low compared with other materials, due to the low heat treatment temperature for GG. However, to remain in the single-phase range, the heat treatment temperature cannot be higher. A threshold is quickly reached after 4 to 8 hours of heat treatment, even with 0% of niobium. The presence of other elements, such as oxygen in solid solution, as well as the presence of iron, could be the reason for this plateau. A parallel investigation is underway on pure zirconium samples in order to detect the possible solute drag effect of oxygen. The results for different heat treatment temperatures for 0% of niobium are exhibited in Fig. 4. This graph shows how GG kinetics decrease with decreasing temperature. These data will be used to estimate a reduced mobility value for each temperature using Burke and Turnbull's law [9].
kinetics in the DIGIMU software [13]. The microstructure was generated by immersing an experimental image with around 900 grains. The GB migration velocity is expressed in Eq. 3 [4].

$$ \vec{v} = MP\vec{n} \qquad (3) $$

with M the grain boundary mobility (M = 1.7 × 10⁻³ mm⁴/J.s for simulation with 0% of niobium at 580°C), P the sum of the driving pressures applied on the GB and n the unit normal exiting at the GB. In the case of GG, the term P in Eq. 4 refers only to the capillary pressure term P_c [4] and can be considered as a first approximation.

$$ P = P_c = -\gamma\kappa \qquad (4) $$

where γ corresponds to the GB energy (in our case, values taken from the literature [14] for Zircaloy-4) and κ is the mean GB curvature (trace of the curvature tensor in 3D). Then, GB mobility can be expressed as follows in Eq. 5 using an Arrhenius law.

$$ M = M_0 \exp \left( \frac{-Q}{RT} \right) \qquad (5) $$

with M₀ the preexponential constant factor, Q the mobility activation energy, R the perfect gas constant and T the absolute temperature. The model selected for the solute drag effect for 0.2% and 0.4% of niobium will be described in more details in the following section.

## Results and discussion

By post-processing images taken with an optical microscope under polarized light, a grain size distribution was obtained for each sample (heat treatment time and niobium content) and an average grain size in diameter (ECD) was determined. Fig. 3 illustrates the different kinetics obtained as a function of niobium content with a heat treatment temperature of 580°C. GG kinetics decreased with increasing niobium content. A significant difference was observed between 0.2% and 0.4% of niobium. Furthermore, with 0% of niobium, the kinetics of GG are still low compared with other materials, due to the low heat treatment temperature for GG. However, to remain in the single-phase range, the heat treatment temperature cannot be higher. A threshold is quickly reached after 4 to 8 hours of heat treatment, even with 0% of niobium. The presence of other elements, such as oxygen in solid solution, as well as the presence of iron, could be the reason for this plateau. A parallel investigation is underway on pure zirconium samples in order to detect the possible solute drag effect of oxygen. The results for different heat treatment temperatures for 0% of niobium are exhibited in Fig. 4. This graph shows how GG kinetics decrease with decreasing temperature. These data will be used to estimate a reduced mobility value for each temperature using Burke and Turnbull's law [9].
<figure><img src="image_686.png" /><figcaption>Fig. 3. ECD evolution as a function of heat treatment time and niobium content at 580°C.</figcaption></figure>

<figure><img src="image_687.png" /><figcaption>Fig. 4. ECD evolution as a function of different heat treatment temperature for 0% of niobium.</figcaption></figure>

Given the threshold observed on the experimental data at 580°C with 0% of niobium (see Fig. 4), the reduced mobility was estimated only on the first experimental points, from the initial state to 8 hours of heat treatment. Full-field numerical simulations were carried out with the estimated value of reduced mobility. This value was subsequently readjusted following the initial simulation results because of its dependence on the material and the numerical model used. The simulation results following readjustment of the parameter are shown in Fig. 5. Considering the low mobility value and the grain size heterogeneity, it is necessary to immerse an experimental image to obtain GG kinetics coherent with the experimental results.
<figure><img src="image_697.png" /><figcaption>Fig. 3. ECD evolution as a function of heat treatment time and niobium content at 580°C.</figcaption></figure>

<figure><img src="image_698.png" /><figcaption>Fig. 4. ECD evolution as a function of different heat treatment temperature for 0% of niobium.</figcaption></figure>

Given the threshold observed on the experimental data at 580°C with 0% of niobium (see Fig. 4), the reduced mobility was estimated only on the first experimental points, from the initial state to 8 hours of heat treatment. Full-field numerical simulations were carried out with the estimated value of reduced mobility. This value was subsequently readjusted following the initial simulation results because of its dependence on the material and the numerical model used. The simulation results following readjustment of the parameter are shown in Fig. 5. Considering the low mobility value and the grain size heterogeneity, it is necessary to immerse an experimental image to obtain GG kinetics coherent with the experimental results.
<figure><img src="image_688.png" /><figcaption>Fig. 5. Grain growth kinetics comparison between experimental and simulation data for Zr-0Nb at 580°C.</figcaption></figure>

Consequently, around 900 grains are taken into account, which does not provide a representative picture of the microstructure evolution. In the simulations, the average grain size is calculated in the same way as the experimental data, considering only “closed” grains. With a solute drag effect (results with 0.2 and 0.4% niobium), Burke and Turnbull's law [9] cannot be predictive of GG kinetics. At mesoscopic scale, the solute drag effect is usually incorporated into the mobility parameter by modifying the apparent activation energy or by applying an additional driving pressure known as solute pressure. In the literature, full-field simulations based on a level-set description of GB has not been found to model solute drag effect in metallic materials. The Cahn-Lücke-Stüwe model [15,16,17] was firstly adopted to represent the solute drag effect in Zr-Nb alloys to improve the representativity of full-field simulations [18]. The principle consists in adding, in the convective part of the level-set (LS) transport equations, a velocity-dependent pinning pressure $P_i$ to the GB migration velocity as described in Eq. 6 and Eq. 7.

$$P = P_c + P_i \qquad (6)$$

$$P_i = \frac{\alpha \nu c_0}{1 + \beta^2 \nu^2} \qquad (7)$$

with $\nu$ the GB velocity, $c_0$ the concentration of alloying element, $\alpha$ and $\beta$ material parameters to be identified. The pinning pressure (cf. Eq. 7) can thus be added to the global equation (cf. Eq. 3) to define the GB migration velocity taking into account the solute drag effect [15]. For this reason, the Cahn-Lücke-Stüwe (CLS) model will be used to add a pinning pressure (homogeneous in all the material) in order to reduce the GB migration velocity. At the same time, a solute drag effect due to oxygen is probably present in samples with 0% of niobium as illustrated in Fig. 5 and will be confirmed by comparing GG results with pure zirconium.

## Summary

In this work, the characterization and modelling of GG kinetics in Zr-Nb alloys were studied. The influence of niobium content was highlighted based on experimental investigations. The conclusions and prospects of this study are summarised as follows:
<figure><img src="image_699.png" /><figcaption>Fig. 5. Grain growth kinetics comparison between experimental and simulation data for Zr-0Nb at 580°C.</figcaption></figure>

Consequently, around 900 grains are taken into account, which does not provide a representative picture of the microstructure evolution. In the simulations, the average grain size is calculated in the same way as the experimental data, considering only “closed” grains. With a solute drag effect (results with 0.2 and 0.4% niobium), Burke and Turnbull's law [9] cannot be predictive of GG kinetics. At mesoscopic scale, the solute drag effect is usually incorporated into the mobility parameter by modifying the apparent activation energy or by applying an additional driving pressure known as solute pressure. In the literature, full-field simulations based on a level-set description of GB has not been found to model solute drag effect in metallic materials. The Cahn-Lücke-Stüwe model [15,16,17] was firstly adopted to represent the solute drag effect in Zr-Nb alloys to improve the representativity of full-field simulations [18]. The principle consists in adding, in the convective part of the level-set (LS) transport equations, a velocity-dependent pinning pressure $P_i$ to the GB migration velocity as described in Eq. 6 and Eq. 7.

$$P = P_c + P_i \qquad (6)$$

$$P_i = \frac{\alpha \nu c_0}{1 + \beta^2 \nu^2} \qquad (7)$$

with $\nu$ the GB velocity, $c_0$ the concentration of alloying element, $\alpha$ and $\beta$ material parameters to be identified. The pinning pressure (cf. Eq. 7) can thus be added to the global equation (cf. Eq. 3) to define the GB migration velocity taking into account the solute drag effect [15]. For this reason, the Cahn-Lücke-Stüwe (CLS) model will be used to add a pinning pressure (homogeneous in all the material) in order to reduce the GB migration velocity. At the same time, a solute drag effect due to oxygen is probably present in samples with 0% of niobium as illustrated in Fig. 5 and will be confirmed by comparing GG results with pure zirconium.

## Summary

In this work, the characterization and modelling of GG kinetics in Zr-Nb alloys were studied. The influence of niobium content was highlighted based on experimental investigations. The conclusions and prospects of this study are summarised as follows:
* Numerical simulations for grain growth were conducted and reduced mobility was identified for 0% of niobium content.

* The influence of niobium content has been observed on ReX and GG kinetics.

* Oxygen content in solid solution seems to influence GB migration velocity.

* In future work, CLS model will be adapted to take into account the impact of the solute drag effect due to niobium in solid solution.

* Second phase particles will be included in numerical simulations to compare with experimental results with 1% of niobium.

## References

[1] D. O. Northwood, The development and applications of zirconium alloys, Materials & Design, 6 (1985) 58-70. https://doi-org/10.1016/0261-3069(85)90165-7

[2] H. Yang, J. Shen, and Y. Matsukawa, Effects of alloying elements (Sn, Nb, Cr and Mo) on the microstructure and mechanical properties of zirconium alloys, Journal of Nuclear Science and Technology, 52 (2015) 1162-1173. https://doi.org/10.1080/00223131.2014.996622

[3] A. Gaillac, Mise en forme des alliages de zirconium et de hafnium, Techniques de l'Ingénieur, M3190(V1) (2018). https://doi.org/10.51257/a-v1-m3190

[4] J. Humphreys, G. S. Rohrer, and A. Rollett, Recrystallization and Related Annealing Phenomena, Materials Science and Engineering, Elsevier, third edition, 2017.

[5] C. S. Smith, Introduction to grains, phases, and interfaces – an interpretation of microstructure, Transaction metallurgy Society of AIME, 175 (1948) 15-51.

[6] C. Zener, Theory of growth of spherical precipitates from solid solution, Journal of Applied Physics, 20 (1949) 950-953.

[7] M. Bernacki, R. E. Logé, and T. Coupez, Level set framework for the finite-element modelling of recrystallization and grain growth in polycrystalline materials, Scripta Materialia, 64 (2011) 525-528. https://doi.org/10.1016/j.scriptamat.2010.11.032

[8] H. Tian, X. Wang, W. Gong, J. Zhou, and H. Zhang, Recrystallization behavior of cold-rolled Zr-1Nb alloy, Journal of Nuclear Materials, 456 (2015) 321-328. https://doi.org/10.1016/j.jnucmat.2014.09.084

[9] J. E. Burke and D. Turnbull, Recrystallization and grain growth, Progress in Metal Physics, 3 (1952) 220-292. https://doi.org/10.1016/0502-8205(52)90009-9

[10] M. P. Zahler, S. M. Kraschewski, H. Störmer, D. Gerthsen, M. Bäurer, and W. Rheinheimer, Grain growth and segregation in Fe-doped SrTiO3: Experimental evidence for solute drag, Journal of the European Ceramic Society, 43 (2023) 1613-1624. https://doi.org/10.1016/j.jeurceramsoc.2022.11.074

[11] B. Flipon, V. Grand, B. Murgas, A. Gaillac, A. Nicolaï, N. Bozzolo, and M. Bernacki, Grain size characterization in metallic alloys using different microscopy and post-processing techniques, Materials Characterization, 174 (2021) 1-14. https://doi.org/10.1016/j.matchar.2021.110977

[12] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D. James White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona, Fiji: an open-source platform for biological-image analysis, Nature Methods, 9 (2012) 676-682. https://doi.org/10.1038/nmeth.2019

[13] Transvalor, DIGIMU Software, Read on https://www.transvalor.com/en/digimu the 10 December 2023.
* Numerical simulations for grain growth were conducted and reduced mobility was identified for 0% of niobium content.

* The influence of niobium content has been observed on ReX and GG kinetics.

* Oxygen content in solid solution seems to influence GB migration velocity.

* In future work, CLS model will be adapted to take into account the impact of the solute drag effect due to niobium in solid solution.

* Second phase particles will be included in numerical simulations to compare with experimental results with 1% of niobium.

## References

[1] D. O. Northwood, The development and applications of zirconium alloys, Materials & Design, 6 (1985) 58-70. https://doi-org/10.1016/0261-3069(85)90165-7

[2] H. Yang, J. Shen, and Y. Matsukawa, Effects of alloying elements (Sn, Nb, Cr and Mo) on the microstructure and mechanical properties of zirconium alloys, Journal of Nuclear Science and Technology, 52 (2015) 1162-1173. https://doi.org/10.1080/00223131.2014.996622

[3] A. Gaillac, Mise en forme des alliages de zirconium et de hafnium, Techniques de l'Ingénieur, M3190(V1) (2018). https://doi.org/10.51257/a-v1-m3190

[4] J. Humphreys, G. S. Rohrer, and A. Rollett, Recrystallization and Related Annealing Phenomena, Materials Science and Engineering, Elsevier, third edition, 2017.

[5] C. S. Smith, Introduction to grains, phases, and interfaces – an interpretation of microstructure, Transaction metallurgy Society of AIME, 175 (1948) 15-51.

[6] C. Zener, Theory of growth of spherical precipitates from solid solution, Journal of Applied Physics, 20 (1949) 950-953.

[7] M. Bernacki, R. E. Logé, and T. Coupez, Level set framework for the finite-element modelling of recrystallization and grain growth in polycrystalline materials, Scripta Materialia, 64 (2011) 525-528. https://doi.org/10.1016/j.scriptamat.2010.11.032

[8] H. Tian, X. Wang, W. Gong, J. Zhou, and H. Zhang, Recrystallization behavior of cold-rolled Zr-1Nb alloy, Journal of Nuclear Materials, 456 (2015) 321-328. https://doi.org/10.1016/j.jnucmat.2014.09.084

[9] J. E. Burke and D. Turnbull, Recrystallization and grain growth, Progress in Metal Physics, 3 (1952) 220-292. https://doi.org/10.1016/0502-8205(52)90009-9

[10] M. P. Zahler, S. M. Kraschewski, H. Störmer, D. Gerthsen, M. Bäurer, and W. Rheinheimer, Grain growth and segregation in Fe-doped SrTiO3: Experimental evidence for solute drag, Journal of the European Ceramic Society, 43 (2023) 1613-1624. https://doi.org/10.1016/j.jeurceramsoc.2022.11.074

[11] B. Flipon, V. Grand, B. Murgas, A. Gaillac, A. Nicolaï, N. Bozzolo, and M. Bernacki, Grain size characterization in metallic alloys using different microscopy and post-processing techniques, Materials Characterization, 174 (2021) 1-14. https://doi.org/10.1016/j.matchar.2021.110977

[12] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D. James White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona, Fiji: an open-source platform for biological-image analysis, Nature Methods, 9 (2012) 676-682. https://doi.org/10.1038/nmeth.2019

[13] Transvalor, DIGIMU Software, Read on https://www.transvalor.com/en/digimu the 10 December 2023.
[14] J. W. C. Dunlop, Y. J. M. Bréchet, L. Legras, and H. S. Zurob, Modelling isothermal and non-isothermal recrystallisation kinetics: Application to Zircaloy-4, Journal of Nuclear Materials, 366 (2007) 178-186. https://doi.org/10.1016/j.jnucmat.2006.12.074

[15] J. W. Cahn, The impurity-drag effect in grain boundary motion, Acta Metallurgica, 10 (1962) 789-798. https://doi.org/10.1002/9781118788295.ch16

[16] K. Lücke and H. P. Stüwe, Recovery and Recrystallization of Metals, Interscience by L. Himmel edition, 1963.

[17] K. Lücke and H. P. Stüwe, On the theory of impurity controlled grain boundary motion, Acta Metallurgica, 19 (1971) 1087-1099. https://doi.org/10.1016/0001-6160(71)90041-1

[18] J. Furstoss, C. Petit, A. Tommasi, C. Ganino, D. Muñoz, M. Bernacki, On the role of solute drag in reconciling laboratory and natural constraints on olivine grain growth kinetics, Geophysical Journal International, 224 (2021) 1360-1370. https://doi.org/10.1093/gji/ggaa520
[14] J. W. C. Dunlop, Y. J. M. Bréchet, L. Legras, and H. S. Zurob, Modelling isothermal and non-isothermal recrystallisation kinetics: Application to Zircaloy-4, Journal of Nuclear Materials, 366 (2007) 178-186. https://doi.org/10.1016/j.jnucmat.2006.12.074

[15] J. W. Cahn, The impurity-drag effect in grain boundary motion, Acta Metallurgica, 10 (1962) 789-798. https://doi.org/10.1002/9781118788295.ch16

[16] K. Lücke and H. P. Stüwe, Recovery and Recrystallization of Metals, Interscience by L. Himmel edition, 1963.

[17] K. Lücke and H. P. Stüwe, On the theory of impurity controlled grain boundary motion, Acta Metallurgica, 19 (1971) 1087-1099. https://doi.org/10.1016/0001-6160(71)90041-1

[18] J. Furstoss, C. Petit, A. Tommasi, C. Ganino, D. Muñoz, M. Bernacki, On the role of solute drag in reconciling laboratory and natural constraints on olivine grain growth kinetics, Geophysical Journal International, 224 (2021) 1360-1370. https://doi.org/10.1093/gji/ggaa520
# Investigation of heat transfer dependencies in quenching of extrusion profiles based on experiment and FEM simulation

KNIAZKIN Ivan¹,ª*, KRYLOV Vladimir¹,ᵇ, SHITIKOV Andrei¹,ᶜ, KULAKOV Ivan¹,ᵈ, BIBA Nikolay¹,ᵉ

¹Micas Simulations Limited, Temple Court, 107 Oxford Road, Oxford, OX4 2ER, United Kingdom

ªivanknjazkin@gmail.com, ᵇkrylov@qform3d.com, ᶜa.a.sh.ggml@gmail.com, ᵈkulakov@qform3d.com, ᶜnick@qform3d.com

**Keywords:** Quenching, Aluminium, Qform, Extrusion, Heat Transfer, Spray Cooling, Cooling Rate

**Abstract.** The objective of this study is to establish a comprehensive method, devoid of computational fluid dynamics (CFD) calculations, for determining the surface heat transfer coefficient during spray cooling. This technique enables the derivation of the heat transfer coefficient's dependence on the temperature of the cooled surface by utilizing known data regarding the number, relative positioning, and catalog characteristics of nozzles. Additionally, adjustable parameters of the quenching system such as the pressure and flow rates of water and air are considered in the methodology. Implemented within the commercial finite element (FE) package QForm UK, this technique is validated using independent experimental data that faithfully replicates real production conditions encountered during the quenching of extruded profiles made from aluminum alloys.

## Introduction

Controlling extrusion parameters is crucial to produce precise, high-quality aluminum profiles with specific dimensions and mechanical properties. The goal is to create defect-free profiles within tight tolerances, ensuring acceptable material properties and increased productivity. Post-extrusion treatments like quenching and aging are essential for achieving the desired product strength, especially since most profiles are made from heat-treatable aluminum. At the same time, quenching itself may cause some shape and surface imperfections, especially in case of water-based methods [1].

The choice of quenching method significantly affects the final properties of profiles. Here, advanced cooling simulations play a vital role in optimizing quenching processes, contributing to improved profile uniformity and superior mechanical properties. However, to effectively utilize the simulation engine and derive meaningful results, it is imperative to incorporate reliable inputs crucial for simulating the quenching process. Among these inputs, the heat transfer coefficient plays a pivotal role in defining the temperature evolution within the profile during cooling. The dependency of the heat transfer coefficient on the surface temperature of the profile and liquid flux density profoundly influences the accuracy of simulation and the subsequent applicability of the software in an industrial setting.

Integrating simulation techniques with practical applications enhances the precision and efficiency of manufacturing heat-treatable aluminum alloy profiles [2]. Numerous studies have focused on the quenching of aluminum profiles [3]-[5]. However, the literature concerning numerical simulations of quenching processes in profile extrusion is notably limited, with only a few works available [1, 6, 7].

<figure><img src="image_689.png" /></figure>
# Investigation of heat transfer dependencies in quenching of extrusion profiles based on experiment and FEM simulation

KNIAZKIN Ivan¹,ª*, KRYLOV Vladimir¹,ᵇ, SHITIKOV Andrei¹,ᶜ, KULAKOV Ivan¹,ᵈ, BIBA Nikolay¹,ᵉ

¹Micas Simulations Limited, Temple Court, 107 Oxford Road, Oxford, OX4 2ER, United Kingdom

ªivanknjazkin@gmail.com, ᵇkrylov@qform3d.com, ᶜa.a.sh.ggml@gmail.com, ᵈkulakov@qform3d.com, ᶜnick@qform3d.com

**Keywords:** Quenching, Aluminium, Qform, Extrusion, Heat Transfer, Spray Cooling, Cooling Rate

**Abstract.** The objective of this study is to establish a comprehensive method, devoid of computational fluid dynamics (CFD) calculations, for determining the surface heat transfer coefficient during spray cooling. This technique enables the derivation of the heat transfer coefficient's dependence on the temperature of the cooled surface by utilizing known data regarding the number, relative positioning, and catalog characteristics of nozzles. Additionally, adjustable parameters of the quenching system such as the pressure and flow rates of water and air are considered in the methodology. Implemented within the commercial finite element (FE) package QForm UK, this technique is validated using independent experimental data that faithfully replicates real production conditions encountered during the quenching of extruded profiles made from aluminum alloys.

## Introduction

Controlling extrusion parameters is crucial to produce precise, high-quality aluminum profiles with specific dimensions and mechanical properties. The goal is to create defect-free profiles within tight tolerances, ensuring acceptable material properties and increased productivity. Post-extrusion treatments like quenching and aging are essential for achieving the desired product strength, especially since most profiles are made from heat-treatable aluminum. At the same time, quenching itself may cause some shape and surface imperfections, especially in case of water-based methods [1].

The choice of quenching method significantly affects the final properties of profiles. Here, advanced cooling simulations play a vital role in optimizing quenching processes, contributing to improved profile uniformity and superior mechanical properties. However, to effectively utilize the simulation engine and derive meaningful results, it is imperative to incorporate reliable inputs crucial for simulating the quenching process. Among these inputs, the heat transfer coefficient plays a pivotal role in defining the temperature evolution within the profile during cooling. The dependency of the heat transfer coefficient on the surface temperature of the profile and liquid flux density profoundly influences the accuracy of simulation and the subsequent applicability of the software in an industrial setting.

Integrating simulation techniques with practical applications enhances the precision and efficiency of manufacturing heat-treatable aluminum alloy profiles [2]. Numerous studies have focused on the quenching of aluminum profiles [3]-[5]. However, the literature concerning numerical simulations of quenching processes in profile extrusion is notably limited, with only a few works available [1, 6, 7].

<figure><img src="image_700.png" /></figure>
## Description of the Case Study

Researchers in [8] conducted an experiment to track temperature changes along a treated profile during cooling. They used a spray cooling device on a separate roller conveyor system, eliminating the need for attachment to the press line. To replicate industrial cooling conditions, an electric resistance furnace and forced air circulation mechanism were set up (Fig. 1).

The cooling system included water and air elements, housed in a frame with two holders, each equipped with 30 nozzles. These nozzles were evenly spaced at 56 mm intervals in a vertical arrangement (Fig. 2). Maintaining a fixed distance of 100 mm between the nozzle exit and the profile surface, water and air pressures were controlled within a range of ±0.05 bar from their set values.

<figure><img src="image_690.png" /><figcaption>Fig. 1. Water-air cooling installation: 1 – furnace; 2 – roller conveyor; 3 – frame; 4 - holder with nozzles; 5 – guides; 6 - sample with installed thermocouples [8]</figcaption></figure>

The profiles used in the experiment were EN AW-6060 aluminum alloy with a 20x40 mm rectangular cross-section. Samples were moved using a cable wound around a drum connected to a motor shaft, allowing rotational speeds from 7 to 84 rpm and linear velocities from 2 to 24 m/min, covering typical extrusion speeds.

In this study, the one of the main focuses was on simulating the described process. The aim was to determine convective boundary conditions in such a way that the computed cooling curve aligns with the curve obtained from a thermocouple placed at the center of the sample during cooling in laboratory tests conducted under industrial conditions.

<figure><img src="image_691.png" /><figcaption>Fig. 2. Rescaled scheme of the investigated cooling process [8]</figcaption></figure>
## Description of the Case Study

Researchers in [8] conducted an experiment to track temperature changes along a treated profile during cooling. They used a spray cooling device on a separate roller conveyor system, eliminating the need for attachment to the press line. To replicate industrial cooling conditions, an electric resistance furnace and forced air circulation mechanism were set up (Fig. 1).

The cooling system included water and air elements, housed in a frame with two holders, each equipped with 30 nozzles. These nozzles were evenly spaced at 56 mm intervals in a vertical arrangement (Fig. 2). Maintaining a fixed distance of 100 mm between the nozzle exit and the profile surface, water and air pressures were controlled within a range of ±0.05 bar from their set values.

<figure><img src="image_701.png" /><figcaption>Fig. 1. Water-air cooling installation: 1 – furnace; 2 – roller conveyor; 3 – frame; 4 - holder with nozzles; 5 – guides; 6 - sample with installed thermocouples [8]</figcaption></figure>

The profiles used in the experiment were EN AW-6060 aluminum alloy with a 20x40 mm rectangular cross-section. Samples were moved using a cable wound around a drum connected to a motor shaft, allowing rotational speeds from 7 to 84 rpm and linear velocities from 2 to 24 m/min, covering typical extrusion speeds.

In this study, the one of the main focuses was on simulating the described process. The aim was to determine convective boundary conditions in such a way that the computed cooling curve aligns with the curve obtained from a thermocouple placed at the center of the sample during cooling in laboratory tests conducted under industrial conditions.

<figure><img src="image_702.png" /><figcaption>Fig. 2. Rescaled scheme of the investigated cooling process [8]</figcaption></figure>
## Description of the Parameters and Algorithms Used in the Simulation

The simulation was performed using a specialized module in the QForm UK FEM software, designed specifically for simulating quenching with various conditions and cooling systems, including water spraying. The technological parameters used in simulation are detailed in Table 1.

<table><caption>Table 1. Technological parameters of the process used in simulation</caption><thead><tr><td>Mode name</td><td>p<sub>water</sub> [bar]</td><td>p<sub>air</sub> [bar]</td><td>V<sub>profile</sub> [m/min]</td><td>T<sub>initial</sub> [°C]</td></tr></thead><tbody><tr><td>W3A3-3</td><td>3</td><td>3</td><td>3</td><td>545</td></tr><tr><td>W5A3-3</td><td>5</td><td>3</td><td>3</td><td>539</td></tr><tr><td>W5A3-6</td><td>5</td><td>3</td><td>6</td><td>539</td></tr></tbody></table>

The study detailed in article [8] utilized full-cone water-air nozzles, specifically SU12 from Spraying Systems Co. These nozzles consist of two components: the fluid cap 2850 and the air cap 73160. As outlined in the catalog [9], the spray angle falls within the range of 12-15 degrees (considered equal to 15 degrees for simulation), with the orifice equivalent diameter measuring 0.71 millimeters (converted from inches).

During the cooling process using sprayers, the efficiency of heat transfer is contingent upon the distance between the surface undergoing cooling and the nozzle of the sprayer (Fig. 3). The liquid flux density diminishes along the spray axis of the spraying jet, decreasing from its maximum value ($q_{max}$) at the nozzle exit to zero (Fig. 3b). Additionally, an uneven distribution is observed in the cross-section of the jet, wherein the highest flux density in a specific section of the jet, defined by z coordinate, decreases from $q_{max}$ at the center to zero at the boundary (Fig. 3a). To incorporate the spatial dependency of the heat transfer coefficient, relative distribution functions $\eta_r$ and $\eta_z$ are introduced in this study.

<figure><img src="image_692.png" /><figcaption>Fig. 3. Distribution of heat transfer intensity: a – in the cross-section; b – along the spraying jet by distance from nozzle</figcaption></figure>
## Description of the Parameters and Algorithms Used in the Simulation

The simulation was performed using a specialized module in the QForm UK FEM software, designed specifically for simulating quenching with various conditions and cooling systems, including water spraying. The technological parameters used in simulation are detailed in Table 1.

<table><caption>Table 1. Technological parameters of the process used in simulation</caption><thead><tr><td>Mode name</td><td>p<sub>water</sub> [bar]</td><td>p<sub>air</sub> [bar]</td><td>V<sub>profile</sub> [m/min]</td><td>T<sub>initial</sub> [°C]</td></tr></thead><tbody><tr><td>W3A3-3</td><td>3</td><td>3</td><td>3</td><td>545</td></tr><tr><td>W5A3-3</td><td>5</td><td>3</td><td>3</td><td>539</td></tr><tr><td>W5A3-6</td><td>5</td><td>3</td><td>6</td><td>539</td></tr></tbody></table>

The study detailed in article [8] utilized full-cone water-air nozzles, specifically SU12 from Spraying Systems Co. These nozzles consist of two components: the fluid cap 2850 and the air cap 73160. As outlined in the catalog [9], the spray angle falls within the range of 12-15 degrees (considered equal to 15 degrees for simulation), with the orifice equivalent diameter measuring 0.71 millimeters (converted from inches).

During the cooling process using sprayers, the efficiency of heat transfer is contingent upon the distance between the surface undergoing cooling and the nozzle of the sprayer (Fig. 3). The liquid flux density diminishes along the spray axis of the spraying jet, decreasing from its maximum value ($q_{max}$) at the nozzle exit to zero (Fig. 3b). Additionally, an uneven distribution is observed in the cross-section of the jet, wherein the highest flux density in a specific section of the jet, defined by z coordinate, decreases from $q_{max}$ at the center to zero at the boundary (Fig. 3a). To incorporate the spatial dependency of the heat transfer coefficient, relative distribution functions $\eta_r$ and $\eta_z$ are introduced in this study.

<figure><img src="image_703.png" /><figcaption>Fig. 3. Distribution of heat transfer intensity: a – in the cross-section; b – along the spraying jet by distance from nozzle</figcaption></figure>
Relative liquid flux in spray cross cut. In QForm UK, it is assumed that relative liquid flux in spray cross cut is expressed using the following dependence [10, 11]:

$$ \eta_r(r) = \frac{q_r^z}{q_{max|z=0}^z} = \left[ 1 - \left( \frac{r}{r_{max}} \right)^{\frac{3}{2}} \right]^2 \quad (1) $$

Here r – relative distance from axis of symmetry of the jet to the cross-section point, $r_{max}$ – maximum radius of the cross section of the jet. Graphical representation of this dependence is presented in Fig. 3a.

The average value of the relative distribution function across the cross section of the jet can be calculated using the following formula:

$$ \bar{\eta}_r = \frac{2\pi \int_0^{r_{max}} \eta_r(r) r dr}{\pi r_{max}^2} \quad (2) $$

Relative liquid flux by distance from nozzle. Relative liquid flux by distance from nozzle can be described with the following system of equations:

$$ \eta_z(z) = \frac{q_{max}^z}{q_{max|z=0}^z} = \begin{cases} \eta_{z0}, & z \le z_0 \\ \eta_{z0} - \frac{\eta_{z0} - \eta_{z1}}{z_1 - z_0} \cdot (z - z_0), & z_0 < z \le z_1 \\ \eta_{z1} \cdot \left(\frac{z_1}{z}\right)^n, & z_1 < z \end{cases} \quad (3) $$

z – distance from the nozzle exit to the cross section of the jet, the section from $z_0$ to $z_1$ indicates the linear velocity decrease, $> z_1$ – section of nonlinear decrease in jet velocity [10, 11]. Graphical representation of this dependence is shown in the Fig. 3b.

Thus, an expression can be derived to determine the surface density of the liquid flux at any specified point within the local coordinate system of the spraying jet:

$$ q(z,r) = \frac{4 \cdot Q}{\pi \cdot d(z)^2} \cdot \eta_z(z) \cdot \frac{\eta_r(r)}{\bar{\eta}_r} \quad (4) $$

The variable $Q = f(p_w, p_a)$ represents the dependence of liquid flow rate through the nozzle on water pressure and air pressure, as provided in the catalog [9].

## Derivation of Heat Transfer Relationships

Regrettably, in the laboratory setup described in [8], all sprayers are positioned equidistant from the surface of the cooled profile. Consequently, it is not feasible to verify the parameters of the function $\eta_z$ based on the data provided in [8]. Therefore, in subsequent calculations, it is assumed that $\eta_z$ remains constant at a value of 1.

In a general context, the heat transfer coefficient is contingent on both the liquid flux density and the temperature of the cooled surface. Utilizing the insights from reference [12], the following expression can be proposed to represent the dependence of the surface heat transfer coefficient on liquid flux density and surface temperature:

$$ \overline{HTC}(q, T) = \overline{HTC}(q) \cdot \xi(T) \quad (5) $$

In this equation, $\overline{HTC}(q)$ denotes the mean heat transfer coefficient averaged over total surface temperature range. The parameter $\xi$ represents the ratio of the heat transfer coefficient at a specific temperature to the average heat transfer coefficient across the entire temperature range as a function of the surface temperature [12]. Its graphical depiction is presented in Fig 4.
Relative liquid flux in spray cross cut. In QForm UK, it is assumed that relative liquid flux in spray cross cut is expressed using the following dependence [10, 11]:

$$ \eta_r(r) = \frac{q_r^z}{q_{max|z=0}^z} = \left[ 1 - \left( \frac{r}{r_{max}} \right)^{\frac{3}{2}} \right]^2 \quad (1) $$

Here $r$ – relative distance from axis of symmetry of the jet to the cross-section point, $r_{max}$ – maximum radius of the cross section of the jet. Graphical representation of this dependence is presented in Fig. 3a.

The average value of the relative distribution function across the cross section of the jet can be calculated using the following formula:

$$ \bar{\eta}_r = \frac{2\pi \int_0^{r_{max}} \eta_r(r) r dr}{\pi r_{max}^2} \quad (2) $$

Relative liquid flux by distance from nozzle. Relative liquid flux by distance from nozzle can be described with the following system of equations:

$$ \eta_z(z) = \frac{q_{max}^z}{q_{max|z=0}^z} = \begin{cases} \eta_{z0}, & z \le z_0 \\ \eta_{z0} - \frac{\eta_{z0} - \eta_{z1}}{z_1 - z_0} \cdot (z - z_0), & z_0 < z \le z_1 \\ \eta_{z1} \cdot \left(\frac{z_1}{z}\right)^n, & z_1 < z \end{cases} \quad (3) $$

$z$ – distance from the nozzle exit to the cross section of the jet, the section from $z_0$ to $z_1$ indicates the linear velocity decrease, $> z_1$ – section of nonlinear decrease in jet velocity [10, 11]. Graphical representation of this dependence is shown in the Fig. 3b.

Thus, an expression can be derived to determine the surface density of the liquid flux at any specified point within the local coordinate system of the spraying jet:

$$ q(z,r) = \frac{4 \cdot Q}{\pi \cdot d(z)^2} \cdot \eta_z(z) \cdot \frac{\eta_r(r)}{\bar{\eta}_r} \quad (4) $$

The variable $Q = f(p_w, p_a)$ represents the dependence of liquid flow rate through the nozzle on water pressure and air pressure, as provided in the catalog [9].

## Derivation of Heat Transfer Relationships

Regrettably, in the laboratory setup described in [8], all sprayers are positioned equidistant from the surface of the cooled profile. Consequently, it is not feasible to verify the parameters of the function $\eta_z$ based on the data provided in [8]. Therefore, in subsequent calculations, it is assumed that $\eta_z$ remains constant at a value of 1.

In a general context, the heat transfer coefficient is contingent on both the liquid flux density and the temperature of the cooled surface. Utilizing the insights from reference [12], the following expression can be proposed to represent the dependence of the surface heat transfer coefficient on liquid flux density and surface temperature:

$$ \overline{HTC}(q, T) = \overline{HTC}(q) \cdot \xi(T) \quad (5) $$

In this equation, $\overline{HTC}(q)$ denotes the mean heat transfer coefficient averaged over total surface temperature range. The parameter $\xi$ represents the ratio of the heat transfer coefficient at a specific temperature to the average heat transfer coefficient across the entire temperature range as a function of the surface temperature [12]. Its graphical depiction is presented in Fig 4.
<figure><img src="image_693.png" /><figcaption>Fig. 4. Dependence of the relative heat transfer coefficient on temperature for the investigated spray parameters</figcaption></figure>

Derivation of the Dependence of HTC on Liquid Flux Density. To establish a dependency in the form of $\overline{HTC} = f(q)$, the surface liquid f density values were determined for each combination of cooling modes explored in the study outlined in the work [8] in the following form:

$$q_{w_i a_i} = \frac{4Q_{w_i A_i}}{\pi d(z^*)^2} \qquad (6)$$

Here, $Q_{w_i A_i}$ – liquid flow rate for the certain cooling mode, $d = f(z)$ represents the cross-sectional diameter of the jet at a given distance $z$ from the nozzle, and $z^* = 100$ mm signifies the distance from the nozzle to the cooled surface in the experimental setup depicted in Fig. 2.

Unfortunately, the catalog [9] for the Spayer Systems nozzle model used in experiment lacks the required water flow rates for modes W1A2 and W5A3, essential for establishing the functional relationship, along with mode W3A3. To address this gap in information, a regression was constructed using available values, followed by extrapolation. Table 2 provides the derived values for the liquid flux density for the relevant combinations.

<table><caption>Table 2. Liquid flux density for different cooling modes</caption><tr><td>Cooling mode</td><td>W1A2</td><td>W3A3</td><td>W5A3</td></tr><tr><td>q, L/(min·m²)</td><td>124</td><td>415</td><td>967</td></tr></table>

Hence, to characterize the dependence of the $\overline{HTC}$ on the determined liquid flux density values, a power-law regression was employed:

$$\overline{HTC}(q) = 0.425q^{0.533} + 1.456 \qquad (7)$$
<figure><img src="image_704.png" /><figcaption>Fig. 4. Dependence of the relative heat transfer coefficient on temperature for the investigated spray parameters</figcaption></figure>

Derivation of the Dependence of HTC on Liquid Flux Density. To establish a dependency in the form of $\overline{HTC} = f(q)$, the surface liquid f density values were determined for each combination of cooling modes explored in the study outlined in the work [8] in the following form:

$$q_{w_i a_i} = \frac{4Q_{w_i A_i}}{\pi d(z^*)^2} \qquad (6)$$

Here, $Q_{w_i A_i}$ – liquid flow rate for the certain cooling mode, $d = f(z)$ represents the cross-sectional diameter of the jet at a given distance $z$ from the nozzle, and $z^* = 100$ mm signifies the distance from the nozzle to the cooled surface in the experimental setup depicted in Fig. 2.

Unfortunately, the catalog [9] for the Spayer Systems nozzle model used in experiment lacks the required water flow rates for modes W1A2 and W5A3, essential for establishing the functional relationship, along with mode W3A3. To address this gap in information, a regression was constructed using available values, followed by extrapolation. Table 2 provides the derived values for the liquid flux density for the relevant combinations.

<table><caption>Table 2. Liquid flux density for different cooling modes</caption><tr><td>Cooling mode</td><td>W1A2</td><td>W3A3</td><td>W5A3</td></tr><tr><td>q, L/(min·m²)</td><td>124</td><td>415</td><td>967</td></tr></table>

Hence, to characterize the dependence of the $\overline{HTC}$ on the determined liquid flux density values, a power-law regression was employed:

$$\overline{HTC}(q) = 0.425q^{0.533} + 1.456 \qquad (7)$$
This dependence is graphically represented on Fig. 5.

<figure><img src="image_694.png" /><figcaption>Fig. 5. Dependence of the heat transfer coefficient averaged over total temperature range on liquid flux</figcaption></figure>

Fig. 6 represents the resulting dependence of heat transfer coefficient on temperature which was additionally extrapolated to the working temperature of extrusion process.

<figure><img src="image_695.png" /><figcaption>Fig. 6. Dependence of the heat transfer coefficient on temperature</figcaption></figure>

## Results and Discussion

Fig. 7 presents the simulation outcomes, illustrating different phases of the process and providing a visual representation of the simulation setup along with the arrangement of the nozzles. The initial temperatures of the profile were set at 545°C, 539°C, and 539°C, respectively, in alignment with the data reported in [8]. This accounts for the temperature variation as the sample progresses from the furnace to the table immediately before entering the designated quenching station.
This dependence is graphically represented on Fig. 5.

<figure><img src="image_705.png" /><figcaption>Fig. 5. Dependence of the heat transfer coefficient averaged over total temperature range on liquid flux</figcaption></figure>

Fig. 6 represents the resulting dependence of heat transfer coefficient on temperature which was additionally extrapolated to the working temperature of extrusion process.

<figure><img src="image_706.png" /><figcaption>Fig. 6. Dependence of the heat transfer coefficient on temperature</figcaption></figure>

## Results and Discussion

Fig. 7 presents the simulation outcomes, illustrating different phases of the process and providing a visual representation of the simulation setup along with the arrangement of the nozzles. The initial temperatures of the profile were set at 545°C, 539°C, and 539°C, respectively, in alignment with the data reported in [8]. This accounts for the temperature variation as the sample progresses from the furnace to the table immediately before entering the designated quenching station.
<figure><img src="image_696.png" /><figcaption>Fig. 7. Temperature distribution at different stages of W3A3-3 simulation process and nozzles alignment. Stages from left to right: beginning, intermediate, final.</figcaption></figure>

The temperature changes for the W3A3-3 mode (indicating a profile speed of 3 m/min) were closely observed at the thermocouple location shown in Fig. 2. Fig. 8a demonstrates that the simulation matches well with the actual experimental results. At a profile speed of 3 m/min, the sample spent about 33 seconds in the quenching system, reducing to around 16.5 seconds at a speed of 6 m/min. However, these results specifically refer to the effective cooling range, focusing on the times when significant temperature changes occurred. Nonetheless, it is worthwhile to plot time with the zero point corresponding to the moment when the sample initiates movement from the oven. This allows for the consideration of temperature drops attributed to internal heat exchange, motion velocity, and cooling intensity before the region of the sample with the thermocouple reaches the quenching box.

<figure><img src="image_697.png" /><figcaption>Fig. 8. Comparison of simulation and experiment for W3A3-3 condition: a – temperature-time dependence; b – diagonal graph</figcaption></figure>

To assess simulation accuracy, a statistical analysis was conducted to measure the difference between the simulation and actual values. Figure 8b illustrates this evaluation with a diagonal
<figure><img src="image_707.png" /><figcaption>Fig. 7. Temperature distribution at different stages of W3A3-3 simulation process and nozzles alignment. Stages from left to right: beginning, intermediate, final.</figcaption></figure>

The temperature changes for the W3A3-3 mode (indicating a profile speed of 3 m/min) were closely observed at the thermocouple location shown in Fig. 2. Fig. 8a demonstrates that the simulation matches well with the actual experimental results. At a profile speed of 3 m/min, the sample spent about 33 seconds in the quenching system, reducing to around 16.5 seconds at a speed of 6 m/min. However, these results specifically refer to the effective cooling range, focusing on the times when significant temperature changes occurred. Nonetheless, it is worthwhile to plot time with the zero point corresponding to the moment when the sample initiates movement from the oven. This allows for the consideration of temperature drops attributed to internal heat exchange, motion velocity, and cooling intensity before the region of the sample with the thermocouple reaches the quenching box.

<figure><img src="image_708.png" /><figcaption>Fig. 8. Comparison of simulation and experiment for W3A3-3 condition: a – temperature-time dependence; b – diagonal graph</figcaption></figure>

To assess simulation accuracy, a statistical analysis was conducted to measure the difference between the simulation and actual values. Figure 8b illustrates this evaluation with a diagonal
graph, showing that all data points fall within the ±15% deviation range from the reference value, confirming the simulation's high accuracy.

The same analysis was applied to the W5A3-3 mode (Fig. 9) and the W5A3-6 mode (Fig. 10), revealing similarly reliable temperature predictions over time. However, there were variations in effective cooling times among these modes. The W3A3-3 mode had the longest effective cooling time range of about 10.3 seconds, the W5A3-6 mode had the shortest range of around 8.2 seconds, and the W5A3-3 mode fell in between with an effective cooling time range of about 9.2 seconds.

<figure><img src="image_698.png" /><figcaption>Fig. 9. Comparison of simulation and experiment for W5A3-3 condition: a – temperature-time dependence; b – diagonal graph</figcaption></figure>

It's important to highlight that even when doubling the profile velocity, the slight variation in effective cooling time emphasizes the significant impact of profile thickness on temperature distribution within the profile.

<figure><img src="image_699.png" /><figcaption>Fig. 10. Comparison of simulation and experiment for W5A3-6 condition: a – temperature-time dependence; b – diagonal graph</figcaption></figure>

To enhance the evaluation of simulation accuracy and provide a clear visual representation, a frequency plot depicting point scatters between the experiment and simulation for various
graph, showing that all data points fall within the ±15% deviation range from the reference value, confirming the simulation's high accuracy.

The same analysis was applied to the W5A3-3 mode (Fig. 9) and the W5A3-6 mode (Fig. 10), revealing similarly reliable temperature predictions over time. However, there were variations in effective cooling times among these modes. The W3A3-3 mode had the longest effective cooling time range of about 10.3 seconds, the W5A3-6 mode had the shortest range of around 8.2 seconds, and the W5A3-3 mode fell in between with an effective cooling time range of about 9.2 seconds.

<figure><img src="image_709.png" /><figcaption>Fig. 9. Comparison of simulation and experiment for W5A3-3 condition: a – temperature-time dependence; b – diagonal graph</figcaption></figure>

It's important to highlight that even when doubling the profile velocity, the slight variation in effective cooling time emphasizes the significant impact of profile thickness on temperature distribution within the profile.

<figure><img src="image_710.png" /><figcaption>Fig. 10. Comparison of simulation and experiment for W5A3-6 condition: a – temperature-time dependence; b – diagonal graph</figcaption></figure>

To enhance the evaluation of simulation accuracy and provide a clear visual representation, a frequency plot depicting point scatters between the experiment and simulation for various
temperature intervals has been generated for each tested mode (Fig. 11). The histogram illustrates the percentage of measurement points relative to the total number of points, considering the difference between the experiment and simulation within specific temperature difference intervals. Across all tested modes, the majority of points are concentrated in the central row, characterized by the smallest differences. This observation underscores the accuracy of the established relationships between heat transfer coefficient, temperature, and liquid flux density.

<figure><img src="image_700.png" /><figcaption>Fig. 11. Frequency histogram of the difference between simulation and experiment for different modes: a – W3A3-3; b – W5A3-3; c – W5A3-6</figcaption></figure>

## Summary

This study focused on simulating the cooling processes of extruded profiles, specifically examining the accuracy of numerical simulation in quenching. Employing the QForm UK software, the authors showcased temperature tracing results during quenching, comparing them with laboratory tests.
temperature intervals has been generated for each tested mode (Fig. 11). The histogram illustrates the percentage of measurement points relative to the total number of points, considering the difference between the experiment and simulation within specific temperature difference intervals. Across all tested modes, the majority of points are concentrated in the central row, characterized by the smallest differences. This observation underscores the accuracy of the established relationships between heat transfer coefficient, temperature, and liquid flux density.

<figure><img src="image_711.png" /><figcaption>Fig. 11. Frequency histogram of the difference between simulation and experiment for different modes: a – W3A3-3; b – W5A3-3; c – W5A3-6</figcaption></figure>

## Summary

This study focused on simulating the cooling processes of extruded profiles, specifically examining the accuracy of numerical simulation in quenching. Employing the QForm UK software, the authors showcased temperature tracing results during quenching, comparing them with laboratory tests.
To enable precise simulation, fundamental dependencies among the heat transfer coefficient, surface temperature, and liquid flux density were established. This involved processing experimental data obtained from laboratory tests closely mirroring industrial conditions.

In conclusion, the study highlights the heightened accuracy of the derived dependencies. This newfound precision allows for the utilization of simulation in optimizing industrial quenching processes for aluminum profiles where water-spray cooling is used.

## References

[1] Bikass, S., B. Andersson, A. Pilipenko, and H. P. Langtangen. Simulation of the distortion mechanisms due to non-uniform cooling in the aluminum extrusion process. International journal of thermal sciences 52 (2012) 50-58. https://doi.org/10.1016/j.ijthermalsci.2011.06.002

[2] Negozio, M., Segatori, A., Pelaccia, R., Reggiani, B., & Donati, L. (2024). Experimental investigation and numerical prediction of the peripheral coarse grain (PCG) evolution during the extrusion of different AA6082 aluminum alloy profiles. Materials Characterization, 113723. https://doi.org/10.1016/j.matchar.2024.113723

[3] Tiryakioglu, M., and G. Totten. Quenching aluminum components in water: Problems and alternatives. In Heat Treating: Proceedings of the 18 th Conference, October 12 th-15 th. 1998.

[4] WILLIAMS William M., SANDNES Lise, MA Jun, TRONVOLL Sigmund Arntsønn, WELO Torgeir. Yield stress and work hardening behavior of extruded AA6082 profiles under different homogenisation and extrusion conditions. Materials Research Proceedings 28 (2023) 467-476. https://doi.org/10.21741/9781644902479-51

[5] Osten, Julia, Benjamin Milkereit, Michael Reich, Bin Yang, Armin Springer, Karina Nowak, and Olaf Kessler. Development of precipitation hardening parameters for high strength alloy AA 7068. Materials 13(4) (2020) 918. https://doi.org/10.3390/ma13040918

[6] Mei, R. B., L. Bao, C. S. Li, J. K. Wang, and X. H. Liu. FE analysis of 6063 aluminium profiles with complex cross-section during online quenching processes. Mechanics 21(2) (2015) 99-106. https://doi.org/10.5755/j01.mech.21.2.11733

[7] Kniazkin I., Kulakov I., Shitikov A., Raedt H.-W. Quenching of extruded profiles – finite element simulation as a tool to minimise profile distortion. Aluminium Extrusion Industry, April (2022) 34-35.

[8] Golovko, A. N., D. Rodman, F. Nürnberger, M. Schaper, Ya V. Frolov, and S. M. Beliaiev. Investigation of the Water-Air Cooling Process of the Thick-Walled Extruded Profile Made of Alloy En Aw-6060 on the Output Table. Metallurgical & Mining Industry 4(2) (2012).

[9] Information on https://www.spray.com/-/media/dam/industrial/usa/sales-material/catalog/cat76a-aaMetric.pdf

[10] Abramovich, G. N., Girshovich, T. A., Krasheninnikov, S. I., Sekundov, A. N., and I. P Smirnova. The theory of turbulent jets. Moscow Izdatel Nauka, 1984.

[11] Schlichting, H., Gersten, K. Turbulent Free Shear Flows. In: Boundary-Layer Theory. Springer, Berlin, Heidelberg, 2017. https://doi.org/10.1007/978-3-662-52919-5_22

[12] Golovko, O., I. Frolov, D. Rodman, F. Nürnberger, O. Grydin, and M. Schaper. Spray cooling of extruded EN AW-6082 aluminium alloy sheets: spatial heat transfer coefficients. Forschung im Ingenieurwesen 3(78) (2014) 131-137. https://doi.org/10.1007/s10010-014-0181-y
To enable precise simulation, fundamental dependencies among the heat transfer coefficient, surface temperature, and liquid flux density were established. This involved processing experimental data obtained from laboratory tests closely mirroring industrial conditions.

In conclusion, the study highlights the heightened accuracy of the derived dependencies. This newfound precision allows for the utilization of simulation in optimizing industrial quenching processes for aluminum profiles where water-spray cooling is used.

## References

[1] Bikass, S., B. Andersson, A. Pilipenko, and H. P. Langtangen. Simulation of the distortion mechanisms due to non-uniform cooling in the aluminum extrusion process. International journal of thermal sciences 52 (2012) 50-58. https://doi.org/10.1016/j.ijthermalsci.2011.06.002

[2] Negozio, M., Segatori, A., Pelaccia, R., Reggiani, B., & Donati, L. (2024). Experimental investigation and numerical prediction of the peripheral coarse grain (PCG) evolution during the extrusion of different AA6082 aluminum alloy profiles. Materials Characterization, 113723. https://doi.org/10.1016/j.matchar.2024.113723

[3] Tiryakioglu, M., and G. Totten. Quenching aluminum components in water: Problems and alternatives. In Heat Treating: Proceedings of the 18 th Conference, October 12 th-15 th. 1998.

[4] WILLIAMS William M., SANDNES Lise, MA Jun, TRONVOLL Sigmund Arntsønn, WELO Torgeir. Yield stress and work hardening behavior of extruded AA6082 profiles under different homogenisation and extrusion conditions. Materials Research Proceedings 28 (2023) 467-476. https://doi.org/10.21741/9781644902479-51

[5] Osten, Julia, Benjamin Milkereit, Michael Reich, Bin Yang, Armin Springer, Karina Nowak, and Olaf Kessler. Development of precipitation hardening parameters for high strength alloy AA 7068. Materials 13(4) (2020) 918. https://doi.org/10.3390/ma13040918

[6] Mei, R. B., L. Bao, C. S. Li, J. K. Wang, and X. H. Liu. FE analysis of 6063 aluminium profiles with complex cross-section during online quenching processes. Mechanics 21(2) (2015) 99-106. https://doi.org/10.5755/j01.mech.21.2.11733

[7] Kniazkin I., Kulakov I., Shitikov A., Raedt H.-W. Quenching of extruded profiles – finite element simulation as a tool to minimise profile distortion. Aluminium Extrusion Industry, April (2022) 34-35.

[8] Golovko, A. N., D. Rodman, F. Nürnberger, M. Schaper, Ya V. Frolov, and S. M. Beliaiev. Investigation of the Water-Air Cooling Process of the Thick-Walled Extruded Profile Made of Alloy En Aw-6060 on the Output Table. Metallurgical & Mining Industry 4(2) (2012).

[9] Information on https://www.spray.com/-/media/dam/industrial/usa/sales-material/catalog/cat76a-aaMetric.pdf

[10] Abramovich, G. N., Girshovich, T. A., Krasheninnikov, S. I., Sekundov, A. N., and I. P Smirnova. The theory of turbulent jets. Moscow Izdatel Nauka, 1984.

[11] Schlichting, H., Gersten, K. Turbulent Free Shear Flows. In: Boundary-Layer Theory. Springer, Berlin, Heidelberg, 2017. https://doi.org/10.1007/978-3-662-52919-5_22

[12] Golovko, O., I. Frolov, D. Rodman, F. Nürnberger, O. Grydin, and M. Schaper. Spray cooling of extruded EN AW-6082 aluminium alloy sheets: spatial heat transfer coefficients. Forschung im Ingenieurwesen 3(78) (2014) 131-137. https://doi.org/10.1007/s10010-014-0181-y
# Heterostructures produced through severe plastic deformation of multilayered systems: Steel-Ti and steel-Mg

PABICH Bartłomiej¹,ª*, MAJTA Janusz¹,ᵇ, KWIECIEŃ Marcin¹,ᶜ,
CICHOCKI Kamil¹,ᵈ and LISIECKI Łukasz¹,ᵉ

¹AGH University of Krakow, Poland

ªbpabich@agh.edu.pl, ᵇmajta@agh.edu.pl, ᶜmkwiecie@agh.edu.pl, ᵈcichocki@agh.edu.pl,

ᵉlisiecki@agh.edu.pl

**Keywords:** Heterostructured Materials, Multilayer Composite, Multi-Stage Wire Drawing

**Abstract.** The presented research encompasses the processes of manufacturing incoherent metal-to-metal composite materials based on a multi-layered system. The primary aim of the present research is to achieve heterostructured materials with small "hard" phase particles within the "soft" phase matrix. Particles of the "hard" material can be obtained as a result of the loss of their continuity, for example, due to severe plastic deformation effects, such as accumulative roll bonding or multi-stage deep wire drawing processes. The presented research results constitute the first stage of implementing the aforementioned research plan. Basic relationships between the parameters of the deformation process and microstructural changes, as well as rheological properties of the investigated components of multilayer systems, i.e., microalloyed steel, Ti, and Mg, were determined. For this purpose, channel die compression tests and multi-stage deep wire drawing (DWD) experiments were conducted. The microstructural examinations and mechanical properties of the produced materials have indicated directions for further research.

**Introduction**

Continuous development and the increasing demand for new structural materials necessitate the design of materials that exhibit both high strength and good ductility. Since the mechanical properties of metals and alloys are determined by their deformation and strengthening mechanisms, it is essential to base the design of new structural materials on modifying and improving these mechanisms. Therefore, the proposal to combine materials with different workabilities and structurally incoherent multilayered components, further varying in microstructure component morphology, enables the attainment of an attractive combination of mechanical properties [1,2].

The main focus of the present study is to generally combine several mechanical and microstructural aspects to produce new structural multilayered materials, based on inhomogeneous deformation of incoherent metal-to-metal composites, i.e., microalloyed steel (bcc) and Ti or Mg (hcp), with specific properties. The presented results are part of the preliminary research for an ongoing research project on multilayer systems, with a specific focus on the phenomena occurring at the interface of the bcc and hcp phases. The influence of such a diverse microstructure is being investigated in terms of the following factors:

i) The level of strengthening achieved by refining the structures of the matrix (grains, pearlite clusters) and the second phase (particles of fragmented Ti or Mg layers) through compression tests and deep wire drawing (DWD).

ii) The introduction of new deformation mechanisms due to the presence of incoherent second-phase particles at the micro-meso scale.

iii) Changes in competitive processes such as fracture or recovery events, which are related to the structural refinement of multilayered, inhomogeneous metal-to-metal composites.

<figure><img src="image_701.png" /></figure>
# Heterostructures produced through severe plastic deformation of multilayered systems: Steel-Ti and steel-Mg

PABICH Bartłomiej¹,ª*, MAJTA Janusz¹,ᵇ, KWIECIEŃ Marcin¹,ᶜ,
CICHOCKI Kamil¹,ᵈ and LISIECKI Łukasz¹,ᵉ

¹AGH University of Krakow, Poland

ªbpabich@agh.edu.pl, ᵇmajta@agh.edu.pl, ᶜmkwiecie@agh.edu.pl, ᵈcichocki@agh.edu.pl,

ᵉlisiecki@agh.edu.pl

**Keywords:** Heterostructured Materials, Multilayer Composite, Multi-Stage Wire Drawing

**Abstract.** The presented research encompasses the processes of manufacturing incoherent metal-to-metal composite materials based on a multi-layered system. The primary aim of the present research is to achieve heterostructured materials with small "hard" phase particles within the "soft" phase matrix. Particles of the "hard" material can be obtained as a result of the loss of their continuity, for example, due to severe plastic deformation effects, such as accumulative roll bonding or multi-stage deep wire drawing processes. The presented research results constitute the first stage of implementing the aforementioned research plan. Basic relationships between the parameters of the deformation process and microstructural changes, as well as rheological properties of the investigated components of multilayer systems, i.e., microalloyed steel, Ti, and Mg, were determined. For this purpose, channel die compression tests and multi-stage deep wire drawing (DWD) experiments were conducted. The microstructural examinations and mechanical properties of the produced materials have indicated directions for further research.

**Introduction**

Continuous development and the increasing demand for new structural materials necessitate the design of materials that exhibit both high strength and good ductility. Since the mechanical properties of metals and alloys are determined by their deformation and strengthening mechanisms, it is essential to base the design of new structural materials on modifying and improving these mechanisms. Therefore, the proposal to combine materials with different workabilities and structurally incoherent multilayered components, further varying in microstructure component morphology, enables the attainment of an attractive combination of mechanical properties [1,2].

The main focus of the present study is to generally combine several mechanical and microstructural aspects to produce new structural multilayered materials, based on inhomogeneous deformation of incoherent metal-to-metal composites, i.e., microalloyed steel (bcc) and Ti or Mg (hcp), with specific properties. The presented results are part of the preliminary research for an ongoing research project on multilayer systems, with a specific focus on the phenomena occurring at the interface of the bcc and hcp phases. The influence of such a diverse microstructure is being investigated in terms of the following factors:

i) The level of strengthening achieved by refining the structures of the matrix (grains, pearlite clusters) and the second phase (particles of fragmented Ti or Mg layers) through compression tests and deep wire drawing (DWD).

ii) The introduction of new deformation mechanisms due to the presence of incoherent second-phase particles at the micro-meso scale.

iii) Changes in competitive processes such as fracture or recovery events, which are related to the structural refinement of multilayered, inhomogeneous metal-to-metal composites.

<figure><img src="image_712.png" /></figure>
The procedure involved preliminary research of the channel tests and multi-stage wire drawing of materials composed of layers (Fig. 1) with different workabilities at room temperature, including microalloyed steel, Ti alloys, and Mg alloys. Previous studies [3] demonstrated that microalloyed steel with a high Nb content is characterized by both high strength and good ductility[4-7]. More ductile matrix material, i.e., microalloyed steel, due to complex strengthening mechanisms (*Considere* criterion) during severe plastic deformation and due to accumulated deformation energy, dislocation density, undergoes significant microstructure refinement through *in situ* recrystallization.

The presented results are part of a project in which various combinations of steel/Ti alloy/steel and steel/Mg alloy/steel systems underwent deformation in channel tests [8]. Additionally, the drawing process of the tube/rod system was carried out. The discussion of the results is based on microhardness measurements, tensile tests, optical microscopy, and analysis using the Electron Backscatter Diffraction (EBSD) technique to assess microstructural changes. Plastometric tests and microstructural investigations are complemented by representations of mechanical states in the examined multi-layered systems.

## Experimental

In the present studies, two deformation processes were employed for multilayer systems composed of microalloyed steel and Ti (St/Ti) or Mg (St/Mg): channel test and deep wire drawing (DWD). Both methods of deformation application allow the accumulation of strong plastic deformation energy in the deformed materials, represented by the induced strengthening through an increased density of geometrically necessary dislocations (GND).

In Fig. 1, schematics, and real images of the research setups, as well as the concepts of the applied deformation processes, are presented. In the case of channel compression tests, packages consisting of two layers of microalloyed steel were compressed, with a layer of Ti or Mg between them (Fig. 1a). All samples had a length of 15mm and a width of 10mm. Two thicknesses of the steel layer were used, namely 4 mm and 2 mm, along with spacers of Ti or Mg with a thickness of 1 mm. These prepared packages underwent the compression process with two total strain magnitudes of 0.25 and 0.5 (Tab.1). For the DWD processes, the procedure involved wire drawing a system with a circular cross-section, constructed by placing a rod of Ti or Mg with a diameter of 3 mm in a hollowed microalloyed steel cylinder with a diameter of 6.5 mm (Fig. 1b). The multi-stage wire drawing process was carried out with a unit reduction of cross-sectional area equal to 15%, until the maximum possible reduction in the diameter of the investigated multilayer system was achieved. Additionally, wire drawing processes were carried out in the same multi-pass scheme using a “homogeneous” microalloyed steel rod with an initial diameter of 6.5mm. In this case, the wire drawing process was executed until achieving a diameter reduction to 1.96mm (Tab. 2). Some wires were drawn to a diameter of 0.75mm. Intermediate annealing was not applied between drawing operations. The resulting research materials underwent microstructural analysis and mechanical property testing, including hardness and tensile tests.
The procedure involved preliminary research of the channel tests and multi-stage wire drawing of materials composed of layers (Fig. 1) with different workabilities at room temperature, including microalloyed steel, Ti alloys, and Mg alloys. Previous studies [3] demonstrated that microalloyed steel with a high Nb content is characterized by both high strength and good ductility[4-7]. More ductile matrix material, i.e., microalloyed steel, due to complex strengthening mechanisms (*Considere* criterion) during severe plastic deformation and due to accumulated deformation energy, dislocation density, undergoes significant microstructure refinement through *in situ* recrystallization.

The presented results are part of a project in which various combinations of steel/Ti alloy/steel and steel/Mg alloy/steel systems underwent deformation in channel tests [8]. Additionally, the drawing process of the tube/rod system was carried out. The discussion of the results is based on microhardness measurements, tensile tests, optical microscopy, and analysis using the Electron Backscatter Diffraction (EBSD) technique to assess microstructural changes. Plastometric tests and microstructural investigations are complemented by representations of mechanical states in the examined multi-layered systems.

## Experimental

In the present studies, two deformation processes were employed for multilayer systems composed of microalloyed steel and Ti (St/Ti) or Mg (St/Mg): channel test and deep wire drawing (DWD). Both methods of deformation application allow the accumulation of strong plastic deformation energy in the deformed materials, represented by the induced strengthening through an increased density of geometrically necessary dislocations (GND).

In Fig. 1, schematics, and real images of the research setups, as well as the concepts of the applied deformation processes, are presented. In the case of channel compression tests, packages consisting of two layers of microalloyed steel were compressed, with a layer of Ti or Mg between them (Fig. 1a). All samples had a length of 15mm and a width of 10mm. Two thicknesses of the steel layer were used, namely 4 mm and 2 mm, along with spacers of Ti or Mg with a thickness of 1 mm. These prepared packages underwent the compression process with two total strain magnitudes of 0.25 and 0.5 (Tab.1). For the DWD processes, the procedure involved wire drawing a system with a circular cross-section, constructed by placing a rod of Ti or Mg with a diameter of 3 mm in a hollowed microalloyed steel cylinder with a diameter of 6.5 mm (Fig. 1b). The multi-stage wire drawing process was carried out with a unit reduction of cross-sectional area equal to 15%, until the maximum possible reduction in the diameter of the investigated multilayer system was achieved. Additionally, wire drawing processes were carried out in the same multi-pass scheme using a “homogeneous” microalloyed steel rod with an initial diameter of 6.5mm. In this case, the wire drawing process was executed until achieving a diameter reduction to 1.96mm (Tab. 2). Some wires were drawn to a diameter of 0.75mm. Intermediate annealing was not applied between drawing operations. The resulting research materials underwent microstructural analysis and mechanical property testing, including hardness and tensile tests.
<figure><img src="image_702.png" /><figcaption>Fig. 1. Schematic and real representations of the channel test (a) and the drawing process of the multilayered (tube/rod) composite (b).</figcaption></figure>

The research materials used had the following characteristics:

* *Microalloyed steel:* bcc, Shear Modulus: $G = 78$ GPa, Chemical composition, (wt%): 0.07C/0.29Si/1.36Mn/0.067Nb/0.03Ti/0.16Cu/0.009N//Fe - bal. Initial grain size: 22 µm.

* *Ti, grade 2:* hcp, Shear Modulus: $G = 43$ GPa Chemical composition, (wt%): 0.01Fe/0.03H/ 0.04O/0.01N/Ti-bal. Initial grain size: 28 µm.

* *Mg, AZ31:* hcp, Shear Modulus: $G = 17$ GPa Chemical composition, (wt%): 3.27Al/0.98Zn/0.28Mn/0.0027Fe/0.00078Ni/0.0005Cu/0.11Si/Mg-bal. Initial grain size: 20-25 µm.

<table><caption>Table 1. Dimensions of samples subjected to channel compression tests and their corresponding effective strains. The sample names contain information about the thickness of the steel layers.</caption><thead><tr><th>Package type</th><th>Initial package height, H<sub>0</sub> [mm]</th><th>Compressed package height, H<sub>f</sub> [mm]</th><th>Effective strain ε=ln(H<sub>0</sub>/H<sub>f</sub>)</th></tr></thead><tbody><tr><td>Idealized dimensions</td><td>5,00<br>9,00</td><td>3,89<br>5,46</td><td>0,25<br>0,50</td></tr><tr><td>St/Ti_2</td><td>5,01</td><td>3,92</td><td>0,245</td></tr><tr><td>St/Ti_2</td><td>5,02</td><td>3,10</td><td>0,482</td></tr><tr><td>St/Ti_4</td><td>8,98</td><td>6,90</td><td>0,263</td></tr><tr><td>St/Ti_4</td><td>8,95</td><td>5,38</td><td>0,509</td></tr><tr><td>St/Mg_2</td><td>5,08</td><td>3,85</td><td>0,277</td></tr><tr><td>St/Mg_2</td><td>5,10</td><td>3,18</td><td>0,472</td></tr><tr><td>St/Mg_4</td><td>9,09</td><td>6,91</td><td>0,274</td></tr><tr><td>St/Mg_4</td><td>9,06</td><td>5,58</td><td>0,485</td></tr></tbody></table>

<caption>Table 2. Diameters and magnitudes of deformations obtained in individual passes of wire drawing, as well as the magnitudes of total accumulated effective strains.</caption>
<figure><img src="image_713.png" /><figcaption>Fig. 1. Schematic and real representations of the channel test (a) and the drawing process of the multilayered (tube/rod) composite (b).</figcaption></figure>

The research materials used had the following characteristics:

* Microalloyed steel: bcc, Shear Modulus: G = 78 GPa, Chemical composition, (wt%): 0.07C/0.29Si/1.36Mn/0.067Nb/0.03Ti/0.16Cu/0.009N//Fe - bal. Initial grain size: 22 µm.

* Ti, grade 2: hcp, Shear Modulus: G = 43 GPa Chemical composition, (wt%): 0.01Fe/0.03H/ 0.04O/0.01N/Ti-bal. Initial grain size: 28 µm.

* Mg, AZ31: hcp, Shear Modulus: G = 17 GPa Chemical composition, (wt%): 3.27Al/0.98Zn/0.28Mn/0.0027Fe/0.00078Ni/0.0005Cu/0.11Si/Mg-bal. Initial grain size: 20-25 µm.

<table><caption>Table 1. Dimensions of samples subjected to channel compression tests and their corresponding effective strains. The sample names contain information about the thickness of the steel layers.</caption><thead><tr><th>Package type</th><th>Initial package height, H<sub>0</sub> [mm]</th><th>Compressed package height, H<sub>f</sub> [mm]</th><th>Effective strain ε=ln(H<sub>0</sub>/H<sub>f</sub>)</th></tr></thead><tbody><tr><td>Idealized dimensions</td><td>5,00<br>9,00</td><td>3,89<br>5,46</td><td>0,25<br>0,50</td></tr><tr><td>St/Ti_2</td><td>5,01</td><td>3,92</td><td>0,245</td></tr><tr><td>St/Ti_2</td><td>5,02</td><td>3,10</td><td>0,482</td></tr><tr><td>St/Ti_4</td><td>8,98</td><td>6,90</td><td>0,263</td></tr><tr><td>St/Ti_4</td><td>8,95</td><td>5,38</td><td>0,509</td></tr><tr><td>St/Mg_2</td><td>5,08</td><td>3,85</td><td>0,277</td></tr><tr><td>St/Mg_2</td><td>5,10</td><td>3,18</td><td>0,472</td></tr><tr><td>St/Mg_4</td><td>9,09</td><td>6,91</td><td>0,274</td></tr><tr><td>St/Mg_4</td><td>9,06</td><td>5,58</td><td>0,485</td></tr></tbody></table>

<caption>Table 2. Diameters and magnitudes of deformations obtained in individual passes of wire drawing, as well as the magnitudes of total accumulated effective strains.</caption>
<table><thead><tr><th>No. of pass</th><th>Diameter of the die, D<sub>f</sub> [mm]</th><th>Reduction of area RA/pass, [%]</th><th>Effective strain/pass ε=2ln(D<sub>o</sub>/D<sub>f</sub>)</th><th>Accumulated effective strain</th></tr></thead><tbody><tr><td>0</td><td>6,5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>6,2</td><td>4,6</td><td>0,09</td><td>0,09</td></tr><tr><td>2</td><td>5,9</td><td>4,8</td><td>0,10</td><td>0,19</td></tr><tr><td>3</td><td>5,6</td><td>5,1</td><td>0,10</td><td>0,30</td></tr><tr><td>4</td><td>5,4</td><td>3,6</td><td>0,07</td><td>0,37</td></tr><tr><td>5</td><td>5,2</td><td>3,7</td><td>0,08</td><td>0,45</td></tr><tr><td>6</td><td>5,0</td><td>3,8</td><td>0,08</td><td>0,52</td></tr><tr><td>7</td><td>4,8</td><td>4,0</td><td>0,08</td><td>0,61</td></tr><tr><td>8</td><td>4,6</td><td>4,2</td><td>0,09</td><td>0,69</td></tr><tr><td>9</td><td>4,3</td><td>6,5</td><td>0,13</td><td>0,83</td></tr><tr><td>10</td><td>4,0</td><td>7,0</td><td>0,14</td><td>0,97</td></tr><tr><td>11</td><td>3,6</td><td>10,0</td><td>0,21</td><td>1,18</td></tr><tr><td>12</td><td>3,2</td><td>11,1</td><td>0,24</td><td>1,42</td></tr><tr><td>13</td><td>2,8</td><td>12,5</td><td>0,27</td><td>1,68</td></tr><tr><td>14</td><td>2,5</td><td>10,7</td><td>0,23</td><td>1,91</td></tr><tr><td>15</td><td>2,18</td><td>12,8</td><td>0,27</td><td>2,18</td></tr><tr><td>16</td><td>1,96</td><td>10,1</td><td>0,21</td><td>2,40</td></tr></tbody></table>

## Results and Discussion

The objective of this research stage was to evaluate the intensity and inhomogeneity of work hardening, as well as microstructure evolution, in the examined multilayered materials (steel, Mg, Ti) under various conditions of deformation, including variations in layer thickness and layer placement location. It is expected that the results of the assessment of the behavior of the tested materials in multilayer systems in the macro-scale will allow for a more precise selection of conditions and the construction of metal-to-metal composites in further stages of the project, i.e., study at the meso and micro scales.

Channel compression test. The first stage of the research involved conducting channel testing to define the basic relationships between the fundamental characteristics of the cooperating layers in the examined layer packages and microstructural changes as well as mechanical properties. The packages subjected to the study varied in the thickness of the cooperating layers of steel, Ti or Mg, as well as in the magnitudes of the total deformation imposed during the compression process.

The diagram of the testing system, used in the studies of channel compression, is presented in Fig. 1. In Fig. 2, the relative hardness changes are presented, both in the layers of steel and in Ti or Mg, as representations of strain hardening in specimens subjected to channel die compression tests. Hardness measurements were conducted with loads of 0.1 kG. The measurement was performed for compressed layers in packages.
<table><thead><tr><th>No. of pass</th><th>Diameter of the die, D<sub>f</sub> [mm]</th><th>Reduction of area RA/pass, [%]</th><th>Effective strain/pass ε=2ln(D<sub>o</sub>/D<sub>f</sub>)</th><th>Accumulated effective strain</th></tr></thead><tbody><tr><td>0</td><td>6,5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>6,2</td><td>4,6</td><td>0,09</td><td>0,09</td></tr><tr><td>2</td><td>5,9</td><td>4,8</td><td>0,10</td><td>0,19</td></tr><tr><td>3</td><td>5,6</td><td>5,1</td><td>0,10</td><td>0,30</td></tr><tr><td>4</td><td>5,4</td><td>3,6</td><td>0,07</td><td>0,37</td></tr><tr><td>5</td><td>5,2</td><td>3,7</td><td>0,08</td><td>0,45</td></tr><tr><td>6</td><td>5,0</td><td>3,8</td><td>0,08</td><td>0,52</td></tr><tr><td>7</td><td>4,8</td><td>4,0</td><td>0,08</td><td>0,61</td></tr><tr><td>8</td><td>4,6</td><td>4,2</td><td>0,09</td><td>0,69</td></tr><tr><td>9</td><td>4,3</td><td>6,5</td><td>0,13</td><td>0,83</td></tr><tr><td>10</td><td>4,0</td><td>7,0</td><td>0,14</td><td>0,97</td></tr><tr><td>11</td><td>3,6</td><td>10,0</td><td>0,21</td><td>1,18</td></tr><tr><td>12</td><td>3,2</td><td>11,1</td><td>0,24</td><td>1,42</td></tr><tr><td>13</td><td>2,8</td><td>12,5</td><td>0,27</td><td>1,68</td></tr><tr><td>14</td><td>2,5</td><td>10,7</td><td>0,23</td><td>1,91</td></tr><tr><td>15</td><td>2,18</td><td>12,8</td><td>0,27</td><td>2,18</td></tr><tr><td>16</td><td>1,96</td><td>10,1</td><td>0,21</td><td>2,40</td></tr></tbody></table>

## Results and Discussion

The objective of this research stage was to evaluate the intensity and inhomogeneity of work hardening, as well as microstructure evolution, in the examined multilayered materials (steel, Mg, Ti) under various conditions of deformation, including variations in layer thickness and layer placement location. It is expected that the results of the assessment of the behavior of the tested materials in multilayer systems in the macro-scale will allow for a more precise selection of conditions and the construction of metal-to-metal composites in further stages of the project, i.e., study at the meso and micro scales.

Channel compression test. The first stage of the research involved conducting channel testing to define the basic relationships between the fundamental characteristics of the cooperating layers in the examined layer packages and microstructural changes as well as mechanical properties. The packages subjected to the study varied in the thickness of the cooperating layers of steel, Ti or Mg, as well as in the magnitudes of the total deformation imposed during the compression process.

The diagram of the testing system, used in the studies of channel compression, is presented in Fig. 1. In Fig. 2, the relative hardness changes are presented, both in the layers of steel and in Ti or Mg, as representations of strain hardening in specimens subjected to channel die compression tests. Hardness measurements were conducted with loads of 0.1 kG. The measurement was performed for compressed layers in packages.
<figure><img src="image_703.png" /></figure>

<figure><img src="image_704.png" /></figure>

<figure><img src="image_705.png" /></figure>

<figure><img src="image_706.png" /><figcaption>Fig. 2. Changes in the relative hardness (an average value in the layers) of steel (a), Ti and Mg layers (b), as a function of the applied effective strain. Effect of the steel layer's location in the pack (top, bottom) (c) and (d), during channel compression test. The thickness of steel layers are indicated in the legend as: St/Mg_2, St/Mg_4 or St/Ti_2, St/Ti_4 (2mm or 4mm respectively).</figcaption></figure>

From the presented results comparison, it can be observed that the location of the steel layers did not show a considerable influence on their work hardening (Fig. 2c,d). Nevertheless, for larger deformations, the upper layer tends to exhibit greater strengthening (Fig. 2c). Increased work hardening (relative hardness increase) was significantly observed in the Ti layers compared to Mg, especially when higher deformation was applied.

For small deformations and thicker layers of steel, stronger effects of work hardening (GND density) were observed in the Ti layer than in the steel (Fig. 3a). The analysis of the KAM (Kernel Average Misorientation) showed also a slightly greater increase in GND density in the upper steel layers near the Ti contact, even in the case of small thicknesses of steel layers (Fig. 3b). This shows that one can expect greater hardening effect in layers located farther from the axis of drawn inhomogeneous wires.

The results presented in Fig. 2 and Fig. 3 allow for the selection of appropriate thicknesses for individual layers in an inhomogeneous steel/Mg or steel/Ti drawing system, in order to optimize the plastic flow of the entire system during wire drawing processes in the next stage of research.
<figure><img src="image_714.png" /><figcaption>Fig. 2. Changes in the relative hardness (an average value in the layers) of steel (a), Ti and Mg layers (b), as a function of the applied effective strain. Effect of the steel layer's location in the pack (top, bottom) (c) and (d), during channel compression test. The thickness of steel layers are indicated in the legend as: St/Mg_2, St/Mg_4 or St/Ti_2, St/Ti_4 (2mm or 4mm respectively).</figcaption></figure>

From the presented results comparison, it can be observed that the location of the steel layers did not show a considerable influence on their work hardening (Fig. 2c,d). Nevertheless, for larger deformations, the upper layer tends to exhibit greater strengthening (Fig. 2c). Increased work hardening (relative hardness increase) was significantly observed in the Ti layers compared to Mg, especially when higher deformation was applied.

For small deformations and thicker layers of steel, stronger effects of work hardening (GND density) were observed in the Ti layer than in the steel (Fig. 3a). The analysis of the KAM (Kernel Average Misorientation) showed also a slightly greater increase in GND density in the upper steel layers near the Ti contact, even in the case of small thicknesses of steel layers (Fig. 3b). This shows that one can expect greater hardening effect in layers located farther from the axis of drawn inhomogeneous wires.

The results presented in Fig. 2 and Fig. 3 allow for the selection of appropriate thicknesses for individual layers in an inhomogeneous steel/Mg or steel/Ti drawing system, in order to optimize the plastic flow of the entire system during wire drawing processes in the next stage of research.
<figure><img src="image_707.png" /><figcaption>Fig. 3. Example of results from microstructural studies using EBSD analysis. KAM analysis revealed different increases in the density of GND (geometrically necessary dislocations) in the layers of steel and Ti as representations of differences in work hardening states –(a) and the variation in the degree of work hardening in the layers of steel due to their position in the stack – (b). St/Ti/St_4_0.25 denotes a system composed of steel and Ti with a steel layer thickness of 4mm deformed with an effective strain of 0.25.</figcaption></figure>

Deep Wire Drawing (DWD). The second part of the study focused on the processes of wire drawing heterogeneous systems, as presented in Fig. 1b. Inside the microalloyed steel wire rods, smaller rods with a diameter of 3mm were placed. The system with rods made of Ti or Mg was studied. Simultaneously, deep wire drawing processes of homogeneous wires from microalloyed steel were carried out. In this case, the results confirmed earlier research findings [9], which demonstrated that by using the applied DWD process, effects characteristic of Severe Plastic
<figure><img src="image_715.png" /><figcaption>Fig. 3. Example of results from microstructural studies using EBSD analysis. KAM analysis revealed different increases in the density of GND (geometrically necessary dislocations) in the layers of steel and Ti as representations of differences in work hardening states –(a) and the variation in the degree of work hardening in the layers of steel due to their position in the stack – (b). St/Ti/St_4_0.25 denotes a system composed of steel and Ti with a steel layer thickness of 4mm deformed with an effective strain of 0.25.</figcaption></figure>

Deep Wire Drawing (DWD). The second part of the study focused on the processes of wire drawing heterogeneous systems, as presented in Fig. 1b. Inside the microalloyed steel wire rods, smaller rods with a diameter of 3mm were placed. The system with rods made of Ti or Mg was studied. Simultaneously, deep wire drawing processes of homogeneous wires from microalloyed steel were carried out. In this case, the results confirmed earlier research findings [9], which demonstrated that by using the applied DWD process, effects characteristic of Severe Plastic
<figure><img src="image_708.png" /><figcaption>Fig. 4. Characteristics of the base material in a multilayer system – microalloyed steel, in terms of work hardening and the resulting changes in mechanical properties of wires after multi-stage drawing. Samples taken from “homogeneous” wires drawn to specified diameters, in range of 6.5mm-0.75mm – a) and comparison of the mechanical properties of “homogeneous” (St) samples with systems containing Ti (St/Ti) or Mg (St/Mg), as a function of the total effective strain after multistage wire drawing- b).</figcaption></figure>

Deformation (SPD) could be achieved. In Fig. 4a, the changes in mechanical properties and microstructure of the “homogeneous” wires after various diameter reductions obtained in multi-stage wire drawing without interoperative annealing are presented. As a result, an average grain size of approximately 300nm was achieved at a diameter of 0.75 mm. The presented results confirm the appropriateness of selecting microalloyed steel as the base material in heterogeneous systems, i.e., a material that will exhibit very good plasticity and susceptibility to recrystallization.

In Fig. 4b, the changes in mechanical properties corresponding to different total accumulated effective strain in drawn homogeneous wire rods and those with Ti or Mg rods for a strain of 1.96 (highlighted in the chart by a blue bar) are shown. The obtained results indicate significant differences in plastic properties (RA, EL) between the studied materials. The observed changes in mechanical properties significantly differ from those obtained for steel wires, specifically in terms of strength properties for the St/Mg system and plastic properties for the St/Ti system. Nevertheless, further research is necessary to draw general conclusions. Currently presented results of the study on rods subjected to the DWD process, both homogeneous steel and those reinforced with Ti or Mg rods, indicate significantly different microstructural changes in the tested materials (Fig. 5). It was demonstrated that in the St/Ti system (Fig. 5a, b), the development of the microstructure, due to the position of material layers in the cross-sectional view, shows similarity to the results obtained in channel compression tests. An important observation is also the limitation of the total possible deformation in the St/Mg system. It was noticed that at a package diameter of 2.5 mm, and even larger, typical effects of coherence loss occurred inside Mg rods during drawing processes (Fig. 5c). Hence, the conclusion is drawn that Mg is less suitable for building structurally heterogeneous systems where strong deformation accumulation is expected to produce fine reinforcing particles in the matrix of ultrafine-grained micro-alloyed steel.
<figure><img src="image_716.png" /><figcaption>Fig. 4. Characteristics of the base material in a multilayer system – microalloyed steel, in terms of work hardening and the resulting changes in mechanical properties of wires after multi-stage drawing. Samples taken from “homogeneous” wires drawn to specified diameters, in range of 6.5mm-0.75mm – a) and comparison of the mechanical properties of “homogeneous” (St) samples with systems containing Ti (St/Ti) or Mg (St/Mg), as a function of the total effective strain after multistage wire drawing- b).</figcaption></figure>

Deformation (SPD) could be achieved. In Fig. 4a, the changes in mechanical properties and microstructure of the “homogeneous” wires after various diameter reductions obtained in multi-stage wire drawing without interoperative annealing are presented. As a result, an average grain size of approximately 300nm was achieved at a diameter of 0.75 mm. The presented results confirm the appropriateness of selecting microalloyed steel as the base material in heterogeneous systems, i.e., a material that will exhibit very good plasticity and susceptibility to in situ recrystallization.

In Fig. 4b, the changes in mechanical properties corresponding to different total accumulated effective strain in drawn homogeneous wire rods and those with Ti or Mg rods for a strain of 1.96 (highlighted in the chart by a blue bar) are shown. The obtained results indicate significant differences in plastic properties (RA, EL) between the studied materials. The observed changes in mechanical properties significantly differ from those obtained for steel wires, specifically in terms of strength properties for the St/Mg system and plastic properties for the St/Ti system. Nevertheless, further research is necessary to draw general conclusions. Currently presented results of the study on rods subjected to the DWD process, both homogeneous steel and those reinforced with Ti or Mg rods, indicate significantly different microstructural changes in the tested materials (Fig. 5). It was demonstrated that in the St/Ti system (Fig. 5a, b), the development of the microstructure, due to the position of material layers in the cross-sectional view, shows similarity to the results obtained in channel compression tests. An important observation is also the limitation of the total possible deformation in the St/Mg system. It was noticed that at a package diameter of 2.5 mm, and even larger, typical effects of coherence loss occurred inside Mg rods during drawing processes (Fig. 5c). Hence, the conclusion is drawn that Mg is less suitable for building structurally heterogeneous systems where strong deformation accumulation is expected to produce fine reinforcing particles in the matrix of ultrafine-grained micro-alloyed steel.
<figure><img src="image_709.png" /><figcaption>Fig. 5. Microstructures after the DWD, deformed to a diameter of 2.5mm. Image of the base from microalloyed steel in the near-surface layer (a) and in the contact zone with the Ti rod (b). Typical image of material coherence loss subjected to wire drawing (St/Mg).</figcaption></figure>

**Summary**

The current work has revealed the complexity of the rheological properties and microstructure
development in the investigated multilayer systems. It has been demonstrated that the variation in
the thickness of the matrix layers, i.e., steel in the steel/Ti/steel and steel/Mg/steel multilayer
systems, has an influence on the plastic flow of the entire multilayer system. A direct consequence
of this situation is the inhomogeneity of work hardening in individual layers and the localization
of microstructure deformation. This is expected to result in the distribution of dislocation densities
(GND) and fracture micro-mechanisms, leading to the disintegration of the Ti and Mg layers into
incoherent particles with varying dispersion.

It has been shown that the experimental results of DWD correlate well with effects observed in
the channel tests regarding the mechanical response of the investigated multilayer systems. The
results of the research will be used to design the next stages of the study, i.e., the composition of
pipe-rod and pipe-pipe-rod systems, in order to produce new multilayer drawn products with
attractive combinations of strength and ductility.

**Acknowledgment**

Research financially supported by National Science Centre, Poland, Grant No 2022/45/B/ST8/01383

**References**

[1] X.L.Wu, Y.T. Zhu, Heterogeneous materials: a new class of materials with unprecedented mechanical properties, Mater Res Lett. 5 (2017) 527-532.
https://doi.org/10.1016/j.pmatsci.2022.101019

[2] Y. Kim, G.H. Gu, R. E. Kim, M. H. Seo, H. S. Kim, Deformation behavior of lightweight clad sheet: Experiment and modeling, Mater. Sci Eng A 852 (2022) 1-9.
https://doi.org/10.1016/j.msea.2022.143666

[3] J. Majta, M. Kwiecień, P. Lisiecka-Graca, S. Dymek, R. Błoniarz, C. Trujillo, K. Muszka, Microstructural effects and mechanical response of microalloyed ferrite and austenite subjected to metal forming at extremely different rates of deformation, Mater. Today Commun. 28 (2021) 102472. https://doi.org/10.1016/j.mtcomm.2021.102472

[4] A. J. Deardo, Niobium in modern steels, Int. Mater. Rev. 48 (2003) 371-400.
https://doi.org/10.1179/095066003225008833

[5] S. Shanmugam, N.K. Ramisetti, R.D.K. Misra, T. Mannering, D. Panda, S. Jansto, Effect of cooling rate on the microstructure and mechanical properties of Nb-microalloyed steels, Ater. Sci. Eng. 460-461 (2007) 335-343. https://doi.org/10.1016/j.msea.2007.01.054
<figure><img src="image_717.png" /><figcaption>Fig. 5. Microstructures after the DWD, deformed to a diameter of 2.5mm. Image of the base from microalloyed steel in the near-surface layer (a) and in the contact zone with the Ti rod (b). Typical image of material coherence loss subjected to wire drawing (St/Mg).</figcaption></figure>

**Summary**

The current work has revealed the complexity of the rheological properties and microstructure
development in the investigated multilayer systems. It has been demonstrated that the variation in
the thickness of the matrix layers, i.e., steel in the steel/Ti/steel and steel/Mg/steel multilayer
systems, has an influence on the plastic flow of the entire multilayer system. A direct consequence
of this situation is the inhomogeneity of work hardening in individual layers and the localization
of microstructure deformation. This is expected to result in the distribution of dislocation densities
(GND) and fracture micro-mechanisms, leading to the disintegration of the Ti and Mg layers into
incoherent particles with varying dispersion.

It has been shown that the experimental results of DWD correlate well with effects observed in
the channel tests regarding the mechanical response of the investigated multilayer systems. The
results of the research will be used to design the next stages of the study, i.e., the composition of
pipe-rod and pipe-pipe-rod systems, in order to produce new multilayer drawn products with
attractive combinations of strength and ductility.

**Acknowledgment**

Research financially supported by National Science Centre, Poland, Grant No 2022/45/B/ST8/01383

**References**

[1] X.L.Wu, Y.T. Zhu, Heterogeneous materials: a new class of materials with unprecedented mechanical properties, Mater Res Lett. 5 (2017) 527-532.
https://doi.org/10.1016/j.pmatsci.2022.101019

[2] Y. Kim, G.H. Gu, R. E. Kim, M. H. Seo, H. S. Kim, Deformation behavior of lightweight clad sheet: Experiment and modeling, Mater. Sci Eng A 852 (2022) 1-9.
https://doi.org/10.1016/j.msea.2022.143666

[3] J. Majta, M. Kwiecień, P. Lisiecka-Graca, S. Dymek, R. Błoniarz, C. Trujillo, K. Muszka, Microstructural effects and mechanical response of microalloyed ferrite and austenite subjected to metal forming at extremely different rates of deformation, Mater. Today Commun. 28 (2021) 102472. https://doi.org/10.1016/j.mtcomm.2021.102472

[4] A. J. Deardo, Niobium in modern steels, Int. Mater. Rev. 48 (2003) 371-400.
https://doi.org/10.1179/095066003225008833

[5] S. Shanmugam, N.K. Ramisetti, R.D.K. Misra, T. Mannering, D. Panda, S. Jansto, Effect of cooling rate on the microstructure and mechanical properties of Nb-microalloyed steels, Ater. Sci. Eng. 460-461 (2007) 335-343. https://doi.org/10.1016/j.msea.2007.01.054
[6] K. Kamibayashi, Y. Tanabe, Y. Takemoto, I. Shimizu, T. Senuma, Influence of Ti and Nb on the Strength-Ductility-Hole Expansion Ratio Balance of Hot-rolled Low-carbon High-strength Steel Sheets, ISIJ Int. 52 (2012) 151-157. https://doi.org/10.2355/isijinternational.52.151

[7] F. F. Conde, F. J. Pina, J. M. Giarola, G. S. Pereira, J. C. Francisco, J. A. Avila, W. W. Bose Filho, Microstructure and Mechanical Properties of Nb-API X70 Low Carbon Steel, Metallogr. Microstruct. Anal. 10 (2021) 430-440. https://doi.org/10.1007/s13632-021-00761-7

[8] J. Majta, B. Pabich, Effects of work hardening gradients in incoherent metal-to-metal composites, Materials Info 2023: 6th World Congress on Materials Science and Engineering: theme: “Materials innovation and impact”, Rome, Italy, September 18-19 2023.

[9] J. Majta, M. Kwiecien, P. Lisiecka-Graca, R. Bloniarz, E. Kopytek, K. Muszka, Precipitation -based Strengthening Mechanisms in Ultrafine-grained and Nano-Structures in Microalloyed Steel Wires, 11th International Conference on Advanced Computational Engineering and Experimenting ACE-X, Vienna, Austria, 3-6 July 2017.
[6] K. Kamibayashi, Y. Tanabe, Y. Takemoto, I. Shimizu, T. Senuma, Influence of Ti and Nb on the Strength-Ductility-Hole Expansion Ratio Balance of Hot-rolled Low-carbon High-strength Steel Sheets, ISIJ Int. 52 (2012) 151-157. https://doi.org/10.2355/isijinternational.52.151

[7] F. F. Conde, F. J. Pina, J. M. Giarola, G. S. Pereira, J. C. Francisco, J. A. Avila, W. W. Bose Filho, Microstructure and Mechanical Properties of Nb-API X70 Low Carbon Steel, Metallogr. Microstruct. Anal. 10 (2021) 430-440. https://doi.org/10.1007/s13632-021-00761-7

[8] J. Majta, B. Pabich, Effects of work hardening gradients in incoherent metal-to-metal composites, Materials Info 2023: 6th World Congress on Materials Science and Engineering: theme: “Materials innovation and impact”, Rome, Italy, September 18-19 2023.

[9] J. Majta, M. Kwiecien, P. Lisiecka-Graca, R. Bloniarz, E. Kopytek, K. Muszka, Precipitation -based Strengthening Mechanisms in Ultrafine-grained and Nano-Structures in Microalloyed Steel Wires, 11th International Conference on Advanced Computational Engineering and Experimenting ACE-X, Vienna, Austria, 3-6 July 2017.
# Experimental, analytical, and numerical analysis of the copper wire multi-pass drawing process

DI DONATO Sara¹,ᵃ*, NEGOZIO Marco²,ᵇ, PELACCIA Riccardo³,ᶜ, REGGIANI Barbara ³,⁴,>d and DONATI Lorenzo¹,ᵉ

¹DIN Department of Industrial Engineering – University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

²University of Parma – Department of Engineering and Architecture, Parco Area delle Scienze, 181/A, 43124 Parma, Italy

³DISMI Department of Sciences and Methods for Engineering – University of Modena and Reggio Emilia, Via Amendola 2, 42122, Reggio Emilia, Italy

⁴InterMech - MO.RE – University of Modena and Reggio Emilia, Piazzale Europa 1, Reggio Emilia 42124, Italy

ᵃsara.didonato2@unibo.it, ᵇmarco.negozio@unipr.it, ᶜriccardo.pelaccia@unimore.it,

ᵇbarbara.reggiani@unimore.it, ᵈl.donati@unibo.it

**Keywords:** Multi-Pass Wire Drawing Process, Copper, Numerical Simulation

**Abstract.** In the cold wire drawing process, the stress acting on the wire depends on process parameters, as well as on the material flow stress, including the strain-hardening that occurs step by step. It is essential to ensure that the stress applied to the wire at the exit of the die remains below the material's yield stress, to prevent wire necking and fracture. Industrially, the process is carried out continuously using multi-step-multi-wires machines that deform the material to high strain at elevated strain rate values. The application of analytical models for evaluating the stresses acting on the wire assumes simplified boundary conditions, such as an average distribution of strain and strain rate within the die. Further studies are necessary, considering the entire multi-pass industrial case and involving finite element simulation, which is today the main tool for optimizing industrial processes. In this work, the drawing process applied to ETP Pure Copper (99.9% in weight) is analyzed experimentally, analytically, and numerically. The material was characterized by torsion tests and experimental drawing tests were carried out on four steps of the process. Through the analysis of the different analytical methods, it was shown that a careful evaluation of the friction coefficient values is necessary to reduce errors in estimating the drawing forces. The aim is to provide a reliable numerical model for predicting the stress acting on the wire during the multi-pass drawing process, through an appropriate characterization of the material flow stress and an evaluation of the friction model.

**Introduction**

The wire drawing technology is a cold metal forming process that consists of reducing the cross-section of an initial rod by deforming it within multiple dies with decreasing diameters. The deformation occurs through the application of a pulling force at the die exit, inducing a triaxial stress state on the wire and subsequent material strain hardening due to extensive deformation in the cold working process. The main limitation of the process arises from the necessity to maintain the pulling stress below the material's yield stress, including strain hardening effects, to prevent necking and wire breaks [1]. This condition restricts the wire reduction achievable in a single pass, necessitating multiple passes to attain the desired final dimensions. The stresses acting on the wire during the drawing process depend on different main parameters: the wire material properties, the die geometry, and the process conditions (such as friction at the interface between the die and the wire and drawing speed). Various analytical models have been proposed in the literature for the

<figure><img src="image_710.png" /></figure>
# Experimental, analytical, and numerical analysis of the copper wire multi-pass drawing process

DI DONATO Sara¹,ᵃ*, NEGOZIO Marco²,ᵇ, PELACCIA Riccardo³,ᶜ, REGGIANI Barbara ³,⁴,ᵈ and DONATI Lorenzo¹,ᵉ

¹DIN Department of Industrial Engineering – University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

²University of Parma – Department of Engineering and Architecture, Parco Area delle Scienze, 181/A, 43124 Parma, Italy

³DISMI Department of Sciences and Methods for Engineering – University of Modena and Reggio Emilia, Via Amendola 2, 42122, Reggio Emilia, Italy

⁴InterMech - MO.RE – University of Modena and Reggio Emilia, Piazzale Europa 1, Reggio Emilia 42124, Italy

ªsara.didonato2@unibo.it, ᵇmarco.negozio@unipr.it, ᶜriccardo.pelaccia@unimore.it,

ᵇbarbara.reggiani@unimore.it, ᵈl.donati@unibo.it

**Keywords:** Multi-Pass Wire Drawing Process, Copper, Numerical Simulation

**Abstract.** In the cold wire drawing process, the stress acting on the wire depends on process parameters, as well as on the material flow stress, including the strain-hardening that occurs step by step. It is essential to ensure that the stress applied to the wire at the exit of the die remains below the material's yield stress, to prevent wire necking and fracture. Industrially, the process is carried out continuously using multi-step-multi-wires machines that deform the material to high strain at elevated strain rate values. The application of analytical models for evaluating the stresses acting on the wire assumes simplified boundary conditions, such as an average distribution of strain and strain rate within the die. Further studies are necessary, considering the entire multi-pass industrial case and involving finite element simulation, which is today the main tool for optimizing industrial processes. In this work, the drawing process applied to ETP Pure Copper (99.9% in weight) is analyzed experimentally, analytically, and numerically. The material was characterized by torsion tests and experimental drawing tests were carried out on four steps of the process. Through the analysis of the different analytical methods, it was shown that a careful evaluation of the friction coefficient values is necessary to reduce errors in estimating the drawing forces. The aim is to provide a reliable numerical model for predicting the stress acting on the wire during the multi-pass drawing process, through an appropriate characterization of the material flow stress and an evaluation of the friction model.

**Introduction**

The wire drawing technology is a cold metal forming process that consists of reducing the cross-section of an initial rod by deforming it within multiple dies with decreasing diameters. The deformation occurs through the application of a pulling force at the die exit, inducing a triaxial stress state on the wire and subsequent material strain hardening due to extensive deformation in the cold working process. The main limitation of the process arises from the necessity to maintain the pulling stress below the material's yield stress, including strain hardening effects, to prevent necking and wire breaks [1]. This condition restricts the wire reduction achievable in a single pass, necessitating multiple passes to attain the desired final dimensions. The stresses acting on the wire during the drawing process depend on different main parameters: the wire material properties, the die geometry, and the process conditions (such as friction at the interface between the die and the wire and drawing speed). Various analytical models have been proposed in the literature for the

<figure><img src="image_718.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
calculation of the stress values acting on the material during wire drawing. The SLAB method, introduced by Siebel [2], is based on the definition of an equilibrium of forces applied to a thin section of the wire (slab) and then integrated into the drawing direction. Another analytical method, developed by Wistreich [3] and applied by Wright [4], calculates a 'deformation zone shape parameter' dependent on the reduction ratio and die's semi-angle. Distortion and friction work are evaluated by introducing a redundant work factor derived from semi-empirical approaches and the deformation zone shape parameter. Avitzur [5] proposed another model employing the Upper-Bound approach, dividing the wire into three zones, in each of which the velocity field is assumed to be continuous. The application of analytical methods requires a large calculation effort; furthermore, simplified boundary conditions are assumed in the models such as an average distribution of the strain and the strain rate of the wire within the die. On the other hand, The Finite Element Method (FEM) has emerged as a prominent tool in contemporary times, playing a significant role in advancing the design and optimization of bulk-forming processes [6]. Several efforts have been made by researchers to apply the FEM method to the analysis of the wire drawing process [7-12], however further investigations are necessary to obtain reliable simulations. The main problems in numerical modeling of the drawing process are two: providing adequate material flow stress considering the large strain values involved in the multi-pass drawing process and determining a friction model between wire and die. In this work, the plastic behavior of the ETP Pure Copper (99.9% in weight), was characterized through torsion tests performed at the University of Bologna - Department of Industrial Engineering. In the torsion test, a specimen is twisted by a torque that acts around its axis at a constant speed. This test is useful for obtaining extended strain values for solid specimens, as it overcomes the limitations of both the tensile test, where the necking onset is generally below 0.8 of true strain, and of the compression test, where the flow curves are usually limited to 1 of true strain due to friction and barreling effects [1,13,14]. Subsequently, using an LD5 Lloyd press, drawing tests were carried out on four steps of the process, from diameter 1.83 mm to 1.122 mm, measuring the force applied to the wire by varying two speed values (60, and 1200 mm/min) in lubricated and non-lubricated conditions. The experimental drawing force values were compared with the force values calculated with the different analytical models, finding different values for the Coulomb friction coefficient derived from each model. Finally, a numerical simulation model was applied to the copper wire multi-pass drawing process using the commercial FE code Qform UK®, comparing the predicted drawing force values with the analytical and experimental data. The analysis of deviations among various analytical methods highlights the importance of evaluating friction coefficient values in each model to prevent errors in estimating drawing forces. The aim to provide a reliable numerical model for predicting wire stress during the multi-pass drawing process will be achieved by accurately characterizing material flow stress and assessing the friction model appropriately.

## Experimental procedures

**Wire drawing.** Wire drawing tests were executed in the ICEL s.c.p.a company (Lugo IT), where copper electrical cables are produced starting from an ETP Pure Copper rod (99.9% in weight) with an initial diameter of 8 mm. The rod was subjected to a preliminary multi-pass drawing process from a diameter of 8.00 mm to 1.83 mm (area reduction of 94.77%). Subsequently, using an LD5 Lloyd press, wire drawing tests were carried out on four steps of the process, from diameter 1.83 mm to 1.122 mm, measuring the drawing force applied to the wire by varying speed values in lubricated and non-lubricated conditions. The speed was varied from a minimum value of 60 mm/min to the maximum value of 1200 mm/min achievable with the press equipment. The wire was not subjected to any annealing treatment between one drawing step and the next, as happens in the industrial process [15], and the parameters of the experimental procedure are shown in Table 1.
calculation of the stress values acting on the material during wire drawing. The SLAB method, introduced by Siebel [2], is based on the definition of an equilibrium of forces applied to a thin section of the wire (slab) and then integrated into the drawing direction. Another analytical method, developed by Wistreich [3] and applied by Wright [4], calculates a 'deformation zone shape parameter' dependent on the reduction ratio and die's semi-angle. Distortion and friction work are evaluated by introducing a redundant work factor derived from semi-empirical approaches and the deformation zone shape parameter. Avitzur [5] proposed another model employing the Upper-Bound approach, dividing the wire into three zones, in each of which the velocity field is assumed to be continuous. The application of analytical methods requires a large calculation effort; furthermore, simplified boundary conditions are assumed in the models such as an average distribution of the strain and the strain rate of the wire within the die. On the other hand, The Finite Element Method (FEM) has emerged as a prominent tool in contemporary times, playing a significant role in advancing the design and optimization of bulk-forming processes [6]. Several efforts have been made by researchers to apply the FEM method to the analysis of the wire drawing process [7-12], however further investigations are necessary to obtain reliable simulations. The main problems in numerical modeling of the drawing process are two: providing adequate material flow stress considering the large strain values involved in the multi-pass drawing process and determining a friction model between wire and die. In this work, the plastic behavior of the ETP Pure Copper (99.9% in weight), was characterized through torsion tests performed at the University of Bologna - Department of Industrial Engineering. In the torsion test, a specimen is twisted by a torque that acts around its axis at a constant speed. This test is useful for obtaining extended strain values for solid specimens, as it overcomes the limitations of both the tensile test, where the necking onset is generally below 0.8 of true strain, and of the compression test, where the flow curves are usually limited to 1 of true strain due to friction and barreling effects [1,13,14]. Subsequently, using an LD5 Lloyd press, drawing tests were carried out on four steps of the process, from diameter 1.83 mm to 1.122 mm, measuring the force applied to the wire by varying two speed values (60, and 1200 mm/min) in lubricated and non-lubricated conditions. The experimental drawing force values were compared with the force values calculated with the different analytical models, finding different values for the Coulomb friction coefficient derived from each model. Finally, a numerical simulation model was applied to the copper wire multi-pass drawing process using the commercial FE code Qform UK®, comparing the predicted drawing force values with the analytical and experimental data. The analysis of deviations among various analytical methods highlights the importance of evaluating friction coefficient values in each model to prevent errors in estimating drawing forces. The aim to provide a reliable numerical model for predicting wire stress during the multi-pass drawing process will be achieved by accurately characterizing material flow stress and assessing the friction model appropriately.

## Experimental procedures

**Wire drawing.** Wire drawing tests were executed in the ICEL s.c.p.a company (Lugo IT), where copper electrical cables are produced starting from an ETP Pure Copper rod (99.9% in weight) with an initial diameter of 8 mm. The rod was subjected to a preliminary multi-pass drawing process from a diameter of 8.00 mm to 1.83 mm (area reduction of 94.77%). Subsequently, using an LD5 Lloyd press, wire drawing tests were carried out on four steps of the process, from diameter 1.83 mm to 1.122 mm, measuring the drawing force applied to the wire by varying speed values in lubricated and non-lubricated conditions. The speed was varied from a minimum value of 60 mm/min to the maximum value of 1200 mm/min achievable with the press equipment. The wire was not subjected to any annealing treatment between one drawing step and the next, as happens in the industrial process [15], and the parameters of the experimental procedure are shown in Table 1.
<table><caption>Table 1 Experimental procedure Step schedule.</caption><thead><tr><th>Nº pass</th><th>Initial diameter [mm]</th><th>Final diameter [mm]</th><th>Reduction Ratio [%]</th><th>Equivalent strain</th><th>Total cumulated strain</th></tr></thead><tbody><tr><td>Preliminary</td><td>8.000</td><td>1.830</td><td></td><td></td><td>2.9503</td></tr><tr><td>1</td><td>1.830</td><td>1.592</td><td>24.3</td><td>0.2786</td><td>3.2289</td></tr><tr><td>2</td><td>1.592</td><td>1.417</td><td>20.8</td><td>0.2329</td><td>3.4618</td></tr><tr><td>3</td><td>1.417</td><td>1.261</td><td>20.8</td><td>0.2333</td><td>3.6951</td></tr><tr><td>4</td><td>1.261</td><td>1.122</td><td>20.8</td><td>0.2336</td><td>3.9287</td></tr></tbody></table>

During each drawing pass, the wire was inserted into the die which was placed into a die holder fixed with the lower plate of the press. At the exit of the die, the wire was pulled by the grip fixed at the upper mobile crosshead with constant speed, and the drawing force was measured by the load cell. The dies used in the experimentation were the standards recommended for copper wire drawing, with a core in polycrystalline diamond (PCD), semi-angle of 9°, and bearing length of 25% of the final diameter. In the tests performed in lubricated condition, the die holder was placed inside a container filled with lubricating emulsion composed of industrial oil TADAL CU 1350 N and water, with a Brix concentration of 8%. The test was stopped after a stroke of 600 mm and two trials were conducted for each condition to avoid possible error by a single repetition and a variability of about ±10 N was found, as shown in Fig. 1.

<figure><img src="image_711.png" /><figcaption>Fig. 1 Wire drawing tests in unlubricated condition conducted at passes 1 in red, 2 in green, 3 in blue, and 4 in orange, at the velocity of 60 Two tests are shown for each condition with solid and dashed lines. On the right picture of the set-up for the wire drawing tests using the LD5 Lloyd press.</figcaption></figure>

**Material characterization.** Torsion tests were performed on specimens extracted from the same ETP Copper rod used in the wire drawing experimental trials. There is no standard for the torsion test specimen design, and in this work solid-bar samples with a gauge length and a diameter of 5 mm were used, as shown in Fig. 2. The trials were performed at room temperature (25°C) with different equivalent strain rates of 0.1, 1, and 5 s⁻¹.
<table><caption>Table 1 Experimental procedure Step schedule.</caption><thead><tr><th>N° pass</th><th>Initial diameter [mm]</th><th>Final diameter [mm]</th><th>Reduction Ratio [%]</th><th>Equivalent strain</th><th>Total cumulated strain</th></tr></thead><tbody><tr><td>Preliminary</td><td>8.000</td><td>1.830</td><td></td><td></td><td>2.9503</td></tr><tr><td>1</td><td>1.830</td><td>1.592</td><td>24.3</td><td>0.2786</td><td>3.2289</td></tr><tr><td>2</td><td>1.592</td><td>1.417</td><td>20.8</td><td>0.2329</td><td>3.4618</td></tr><tr><td>3</td><td>1.417</td><td>1.261</td><td>20.8</td><td>0.2333</td><td>3.6951</td></tr><tr><td>4</td><td>1.261</td><td>1.122</td><td>20.8</td><td>0.2336</td><td>3.9287</td></tr></tbody></table>

During each drawing pass, the wire was inserted into the die which was placed into a die holder fixed with the lower plate of the press. At the exit of the die, the wire was pulled by the grip fixed at the upper mobile crosshead with constant speed, and the drawing force was measured by the load cell. The dies used in the experimentation were the standards recommended for copper wire drawing, with a core in polycrystalline diamond (PCD), semi-angle of 9°, and bearing length of 25% of the final diameter. In the tests performed in lubricated condition, the die holder was placed inside a container filled with lubricating emulsion composed of industrial oil TADAL CU 1350 N and water, with a Brix concentration of 8%. The test was stopped after a stroke of 600 mm and two trials were conducted for each condition to avoid possible error by a single repetition and a variability of about ±10 N was found, as shown in Fig. 1.

<figure><img src="image_719.png" /><figcaption>Fig. 1 Wire drawing tests in unlubricated condition conducted at passes 1 in red, 2 in green, 3 in blue, and 4 in orange, at the velocity of 60 Two tests are shown for each condition with solid and dashed lines. On the right picture of the set-up for the wire drawing tests using the LD5 Lloyd press.</figcaption></figure>

**Material characterization.** Torsion tests were performed on specimens extracted from the same ETP Copper rod used in the wire drawing experimental trials. There is no standard for the torsion test specimen design, and in this work solid-bar samples with a gauge length and a diameter of 5 mm were used, as shown in Fig. 2. The trials were performed at room temperature (25°C) with different equivalent strain rates of 0.1, 1, and 5 s⁻¹.
<figure><img src="image_712.png" /><figcaption>Fig. 2 On the left, geometry of the solid-bar specimen, with dimensions expressed in mm. On the right, picture of the ETP copper sample during the test performed with the Torsion testing machine of the University of Bologna-DIN.</figcaption></figure>

During the test, the temperature was measured by a thermocouple inserted inside the specimen, in the 1.5 mm diameter hole, and a jet of water was activated at regular intervals to keep the temperature constant within a range of ±5°C due to the heat generated during deformation, as shown in Fig. 3.

<figure><img src="image_713.png" /><figcaption>Fig. 3 On the left, the temperature trend during the torsion tests. On the right, the stress-strain experimental values. Blue lines indicate the test at a strain rate of 0.1 s⁻¹, the yellow lines at 1 s⁻¹, and the green ones at 5 s⁻¹. Two tests are shown for each condition, indicated respectively with solid and dashed lines.</figcaption></figure>

The strain rate remains constant throughout the test since the torque is applied to the specimen at constant speed, leading to a stable twist rate (θ[rad/s]), additionally the geometry of the gage length (radius and length) remains unchanged until the specimen ultimately break. From the machine outputs, which are torque M [Nmm] and angle of twist θ [rad], to calculate the shear strain γ and the shear stress τ, the Fields-Backofen solution was applied as explained in the references [16-18]. The results are shown in Fig. 4 in terms of equivalent stress σ and strain ε calculated according to Eq. 1 and Eq. 2, where L is the gauge length and R is the radius.

$$ \bar{\sigma} = \frac{M 3 \sqrt{3}}{2 \pi R^3}, \quad \bar{\epsilon} = \frac{R \theta}{L \sqrt{3}}. \qquad (1, 2) $$
<figure><img src="image_720.png" /><figcaption>Fig. 2 On the left, geometry of the solid-bar specimen, with dimensions expressed in mm. On the right, picture of the ETP copper sample during the test performed with the Torsion testing machine of the University of Bologna-DIN.</figcaption></figure>

During the test, the temperature was measured by a thermocouple inserted inside the specimen, in the 1.5 mm diameter hole, and a jet of water was activated at regular intervals to keep the temperature constant within a range of ±5°C due to the heat generated during deformation, as shown in Fig. 3.

<figure><img src="image_721.png" /><figcaption>Fig. 3 On the left, the temperature trend during the torsion tests. On the right, the stress-strain experimental values. Blue lines indicate the test at a strain rate of 0.1 s⁻¹, the yellow lines at 1 s⁻¹, and the green ones at 5 s⁻¹. Two tests are shown for each condition, indicated respectively with solid and dashed lines.</figcaption></figure>

The strain rate remains constant throughout the test since the torque is applied to the specimen at constant speed, leading to a stable twist rate (θ[rad/s]), additionally the geometry of the gage length (radius and length) remains unchanged until the specimen ultimately break. From the machine outputs, which are torque M [Nmm] and angle of twist θ [rad], to calculate the shear strain γ and the shear stress τ, the Fields-Backofen solution was applied as explained in the references [16-18]. The results are shown in Fig. 4 in terms of equivalent stress σ and strain ε calculated according to Eq. 1 and Eq. 2, where L is the gauge length and R is the radius.

$$ \bar{\sigma} = \frac{M 3 \sqrt{3}}{2 \pi R^3}, \quad \bar{\epsilon} = \frac{R \theta}{L \sqrt{3}}. \qquad (1, 2) $$
<figure><img src="image_714.png" /><figcaption>Fig. 4 Comparison between experimental results from torsion test and predicted values. The experimental tests are indicated with solid thick curves and the predicted values with thin curves. The graph on the right shows the predicted stress vs experimental stress and the red dashed line represents a theoretically perfect agreement between the values.</figcaption></figure>

During the industrial wire drawing process, the material is deformed at higher strain rates than those that can be obtained during the torsion test, reaching values between 5000 and 8000 s⁻¹. Therefore, starting from the experimental results, the coefficients C (strength coefficient in MPa), n (strain-hardening coefficient), and m (strain rate sensitivity coefficient) were calculated to regress the material flow stress law in the form of Eq. 3, dependent on the strain and the strain rate.

$$ \bar{\sigma} = C\bar{\varepsilon}^n\dot{\bar{\varepsilon}}^m \quad (3) $$

Fig. 4 shows the comparison between experimental and predicted values. The trials were stopped when the samples' fracture occurred, that was between strain values of 7.5 and 8.5. The predicted values are reported with solid thin curves with different colors as indicated in the legend, and a good agreement was found with an R² of 0.994.

## Analytical model

Three analytical models for the calculation of the stress applied to the wire during the drawing process were analyzed and compared with the experimental data. The first is the SLAB method [2] which is based on the definition of an equilibrium of forces applied to a thin section of the wire (slab) and then integrated in the drawing direction. The method assumes that the deformation field is uniform across the height of the slab, with normal stress uniformly distributed on the plane parallel to it. The solution to the integral equation is derived from these simplified boundary conditions, and it takes the form of Eq. 4.

$$ \sigma_d = \frac{1+B}{B} \bar{\sigma}_m \left[ 1 - \left( \frac{R_f}{R_i} \right)^{2B} \right] + \frac{2\alpha}{3\sqrt{3}} (\bar{\sigma}_{in} + \bar{\sigma}_{out}), \quad B = \mu \cot \alpha \quad (4,5) $$

where α is the semi-angle of the die, μ is the Coulomb's friction coefficient, $\bar{\sigma}_{in}$ and $\bar{\sigma}_{out}$ are the equivalent flow stress of the material at the inlet and the exit of the die, and $\bar{\sigma}_m$ is the mean flow stress calculated as in Eq. 3. The second method considered is that by Wistreich [3] and Wright [4] and reported in Eq. 6. They introduced a redundant work factor φ for the estimation of inhomogeneous deformation due to the distortion of the wire. The value of the redundant work factor was obtained with a semi-empirical approach and depends on the deformation zone shape Δ.

$$ \sigma_d = \bar{\sigma}_m \phi \left[ \ln \left( \frac{1}{1-r} \right) + \frac{4\mu}{\Delta} \right], \quad \Delta = \frac{\alpha}{r} \left( 1 + \sqrt{1-r} \right)^2, \quad \phi = 0.8 + \frac{\Delta}{4.4} \quad (6,7,8) $$
<figure><img src="image_722.png" /><figcaption>Fig. 4 Comparison between experimental results from torsion test and predicted values. The experimental tests are indicated with solid thick curves and the predicted values with thin curves. The graph on the right shows the predicted stress vs experimental stress and the red dashed line represents a theoretically perfect agreement between the values.</figcaption></figure>

During the industrial wire drawing process, the material is deformed at higher strain rates than those that can be obtained during the torsion test, reaching values between 5000 and 8000 s⁻¹. Therefore, starting from the experimental results, the coefficients C (strength coefficient in MPa), n (strain-hardening coefficient), and m (strain rate sensitivity coefficient) were calculated to regress the material flow stress law in the form of Eq. 3, dependent on the strain and the strain rate.

$$ \bar{\sigma} = C\bar{\epsilon}^n\dot{\bar{\epsilon}}^m \quad (3) $$

Fig. 4 shows the comparison between experimental and predicted values. The trials were stopped when the samples' fracture occurred, that was between strain values of 7.5 and 8.5. The predicted values are reported with solid thin curves with different colors as indicated in the legend, and a good agreement was found with an R² of 0.994.

## Analytical model

Three analytical models for the calculation of the stress applied to the wire during the drawing process were analyzed and compared with the experimental data. The first is the SLAB method [2] which is based on the definition of an equilibrium of forces applied to a thin section of the wire (slab) and then integrated in the drawing direction. The method assumes that the deformation field is uniform across the height of the slab, with normal stress uniformly distributed on the plane parallel to it. The solution to the integral equation is derived from these simplified boundary conditions, and it takes the form of Eq. 4.

$$ \sigma_d = \frac{1+B}{B} \bar{\sigma}_m \left[ 1 - \left( \frac{R_f}{R_i} \right)^{2B} \right] + \frac{2\alpha}{3\sqrt{3}} (\bar{\sigma}_{in} + \bar{\sigma}_{out}), \quad B = \mu \cot \alpha \quad (4,5) $$

where α is the semi-angle of the die, μ is the Coulomb's friction coefficient, $\bar{\sigma}_{in}$ and $\bar{\sigma}_{out}$ are the equivalent flow stress of the material at the inlet and the exit of the die, and $\bar{\sigma}_m$ is the mean flow stress calculated as in Eq.3. The second method considered is that by Wistreich [3] and Wright [4] and reported in Eq. 6. They introduced a redundant work factor $\phi$ for the estimation of inhomogeneous deformation due to the distortion of the wire. The value of the redundant work factor was obtained with a semi-empirical approach and depends on the deformation zone shape $\Delta$.

$$ \sigma_d = \bar{\sigma}_m \phi \left[ \ln \left( \frac{1}{1-r} \right) + \frac{4\mu}{\Delta} \right], \quad \Delta = \frac{\alpha}{r} \left( 1 + \sqrt{1-r} \right)^2, \quad \phi = 0.8 + \frac{\Delta}{4.4} \quad (6,7,8) $$
where r is the reduction ratio. The last method considered was proposed by Avitzur [5], applying the Upper-Bound approach to the drawing problem, and the formulation is shown in Eq. 9. It involved the assumption that the wire can be divided into three zones, in each of which the velocity field is assumed to be continuous: the inlet and exit zones of the die are characterized by the lack of deformation in the wire and the axial component of the velocity is uniform, while in the central zone the plastic flow occurs and the velocity field is directed toward the cone's apex with axial cylindrical symmetry.

$$ \sigma_d = \bar{\sigma}_m \frac{2f(\alpha) \ln\left(\frac{R_i}{R_f}\right) + \frac{2}{\sqrt{3}} \left(\frac{\alpha}{\sin^2(\alpha)} - \cot(\alpha)\right) + 2\mu \left[\cot(\alpha)\left(1 - \ln\left(\frac{R_i}{R_f}\right)\right) \ln\left(\frac{R_i}{R_f}\right) + \frac{L}{R_f}\right]}{\left(1 + 2\mu \frac{L}{R_f}\right)} \quad (9) $$

where f(α) is a function of the die's semi-angle calculated as in Eq.7,

$$ f(\alpha) = \frac{1}{\sin^2(\alpha)} \left[ 1 - \cos(\alpha) \sqrt{1 - \frac{11}{12} \sin^2(\alpha)} + \frac{1}{\sqrt{11,12}} \ln \left( \frac{1 + \frac{11}{12}}{\frac{11}{12} \cos(\alpha) + \sqrt{1 - \frac{11}{12} \sin^2(\alpha)}} \right) \right] \quad (10) $$

All three analytical models involve the effects on the wire caused by uniform work due to the reduction ratio, non-uniform or redundant work due to the wire distortion, and friction work modeled according to Coulomb model. However, the SLAB and Wistreich-Wright models don't account for the additional friction work due to the presence of the bearing length downstream of the drawing cone.

## Numerical model

The simulation of the wire drawing process was performed using the commercial FE code Qform UK® and was represented as axisymmetric Lagrangian model. Accordingly, the following boundary conditions were set: symmetry along z axis and a constant velocity applied at the wire's end.

<figure><img src="image_715.png" /><figcaption>Fig. 5 Numerical simulation of the wire drawing process. Example of the simulation on the fourth drawing pass. On the left the distribution of stress on the wire [MPa]. On the right the strain rate field in the drawing die [1/s].</figcaption></figure>

The geometry of the dies was modeled as described in the experimental procedures and a wire length of 600 mm was simulated to be drawn in four consecutive operations corresponding to the four passes of the drawing experimentation. The flow stress of the copper wire was modeled with the equation regressed from the torsion tests, while the other material parameters were listed in Table 2, and the Coulomb model was used for friction.