heating law identifications in the AFP context featuring varying fiber orientations and volume fractions. Furthermore, internal reflections effects will be also considered.

References

[1] A. Levy, J. Tierney, D. Heider, J. Gillespie, P. Lefebure, D. Lang, Modeling of inter-layer thermal contact resistance during thermoplastic tape placement, in: International SAMPE Technical Conference, Baltimore, MA, USA, 2012.

[2] M. J. Donough, N. A. St John, A. W. Philips, B. G. Prusty, et al., Process modelling of in-situ consolidated thermoplastic composite by automated fibre placement-a review, Composites Part A: Applied Science and Manufacturing (2022) 107179. https://doi.org/10.1016/j.compositesa.2022.107179

[3] P. Hörmann, D. Stelzl, R. Lichtinger, S. Van Nieuwenhove, G. M. Carro, K. Drechsler, On the numerical prediction of radiative heat transfer for thermoset automated fiber placement, Composites Part A: Applied Science and Manufacturing 67 (2014) 282-288. https://doi.org/10.1016/j.compositesa.2014.08.019

[4] V. Le Louët, Etude du comportement thermique de bandes composites pré-imprégnées au cours du procédé de fabrication AFP avec chauffage laser, Ph.D. thesis, Université de Nantes (2018).

[5] O. Baho, G. Ausias, Y. Grohens, J. Férec, Simulation of laser heating distribution for a thermoplastic composite: Effects of AFP head parameters, The International Journal of Advanced Manufacturing Technology 110 (2020) 2105-2117. https://doi.org/10.1007/s00170-020-05876-9

[6] A. Le Reun, V. Le Louët, V. Sobotka, S. Le Corre, Numerical simulation at the micro-scale for the heat transfer modeling in the thermoplastic composites laser-assisted AFP process, Composites Part A: Applied Science and Manufacturing 28 (2023) 1897-1906. https://doi.org/10.1016/j.compositesa.2024.108010

[7] V. Le Louët, A. Le Reun, V. Sobotka, S. Le Corre, Experimental measurement of CF/PEKK tapes heating behavior in the laser assisted automated fiber placement process, in: 26th International ESAFORM Conference, 28 (2023) 1897-1906. https://doi.org/10.21741/9781644902479-205

[8] C. Stokes-Griffin, P. Compston, A combined optical-thermal model for near-infrared laser heating of thermoplastic composites in an automated tape placement process, Composites Part A: Applied Science and Manufacturing 75 (2015) 104-115. https://doi.org/10.1016/j.compositesa.2014.08.006

[9] F. Hecht, New development in FreeFEM++, Journal of numerical mathematics 20 (3-4) (2012) 251-266. https://doi.org/10.1515/jnum-2012-0013

[10] M. A. Badri, Efficient finite element strategies for solving the radiative transfer equation, Ph.D. thesis, Université de Nantes (2018). https://doi.org/10.1016/j.jqsrt.2018.03.024

[11] M. Badri, P. Jolivet, B. Rousseau, S. Le Corre, H. Digonnet, Y. Favennec, Vectorial finite elements for solving the radiative transfer equation, Journal of Quantitative Spectroscopy and Radiative Transfer 212 (2018) 59-74. https://doi.org/10.1016/j.jqsrt.2018.03.024

[12] M. Avila, R. Codina, J. Principe, Spatial approximation of the radiation transport equation using a subgrid-scale finite element method, Computer Methods in Applied Mechanics and Engineering 200 (5-8) (2011) 425-438. https://doi.org/10.1016/j.cma.2010.11.003
# Full field Continuous dynamic recrystallization simulations considering precipitates evolutions with DIGIMU®

DE MICHELI Pascal<fnref n="1" />,ᵃ*, ALVARADO Karen<fnref n="1" />, GRAND Victor<fnref n="2" />,ᵇ and BERNACKI Marc<fnref n="3" />,ᶜ*

**Keywords:** Full-Field Simulation, CDRX, Zener Pinning, Precipitates Evolution, RVE

**Abstract.** Full field simulations have proven to be an efficient tool for grain size prediction in industrial processes, with wider ranges of validity and more comprehensive results than other approaches. DIGIMU® is a level-set based solution able to simulate grain growth, Smith-Zener pinning, discontinuous dynamic, post-dynamic, and static recrystallization, and very recently Continuous Dynamic Recrystallization (CDRX). The goal of this work is to explore the capabilities of DIGIMU® to CDRX with evolving second phase particles. First, a new formalism has been implemented to describe particles boundaries with a level-set function. It is then possible to apply growth or dissolution velocities to the boundaries, and to make new particles appear. Secondly, the orientation has been defined in each grain, and the grain boundary energy can then be computed from the intergranular disorientation with Read-Shockley model. A boundary migration solver which considers heterogeneous grain boundary energy is used, which enables the simulation of structure and substructure evolutions. The Gourdet-Montheillet CDRX model is adapted from a mean field to a full field framework, and the corresponding parameters for Zircaloy-4 are identified. Thanks to all those developments, DIGIMU® can simulate full field CDRX in high stacking fault energy materials, coupled with precipitates evolution. Several examples will be presented, some of them compared to experimental results.

**Introduction**

Mastering grain size, solid phase transformations, and dislocations density evolution during industrial processes becomes necessary to fulfill more and more severe microstructure and properties specifications of the processed materials. In multi-pass processes, the competition between boundary migration, recovery and the different types of recrystallizations leads to complex coupled effects, and minor variations of the process parameters may have huge consequences. A correct modelling of the main physical microstructural mechanisms becomes then necessary [1].

Thanks to the explosion of computer capacities, mesoscopic modeling techniques for metallic materials at the solid-state are now available. These lower scale approaches, the so-called full-field models, are based on a full description of the microstructure topology and have demonstrated an exciting potential for an extensive range of microstructure evolutions. First, it is possible to use more physics in the models because of the better description of the microstructure, leading to a wider range of validity [1]. Secondly, observing directly the polycrystal microstructure evolution, in addition to curves and histograms, allows to analyze the results much more comprehensively [1]. Several numerical frameworks are available: Monte Carlo Potts (MC) [1], Cellular Automata

<figure><img src="image_2192.png" /></figure>

<footnote n="1">¹Transvalor, 950 avenue Roumanille, CS40237, Biot, 06904 Sophia Antipolis, France</footnote>

<footnote n="2">²Components Research Center, Framatome, 73403 Ugine, France</footnote>

<footnote n="3">³Mines Paris, PSL University, Centre for material forming (CEMEF), UMR CNRS, 06904 Sophia Antipolis, France</footnote>

<footnote>ᵃPascal.DeMicheli@transvalor.com, ᵇVictor.Grand@framatome.com,</footnote>

<footnote>ᶜMarc.Bernacki@minesparis.psl.eu</footnote>
# Full field Continuous dynamic recrystallization simulations considering precipitates evolutions with DIGIMU®

DE MICHELI Pascal<fnref n="1" />,ᵃ*, ALVARADO Karen<fnref n="1" />, GRAND Victor<fnref n="2" />,ᵇ and BERNACKI Marc<fnref n="3" />,ᶜ*

**Keywords:** Full-Field Simulation, CDRX, Zener Pinning, Precipitates Evolution, RVE

**Abstract.** Full field simulations have proven to be an efficient tool for grain size prediction in industrial processes, with wider ranges of validity and more comprehensive results than other approaches. DIGIMU® is a level-set based solution able to simulate grain growth, Smith-Zener pinning, discontinuous dynamic, post-dynamic, and static recrystallization, and very recently Continuous Dynamic Recrystallization (CDRX). The goal of this work is to explore the capabilities of DIGIMU® to CDRX with evolving second phase particles. First, a new formalism has been implemented to describe particles boundaries with a level-set function. It is then possible to apply growth or dissolution velocities to the boundaries, and to make new particles appear. Secondly, the orientation has been defined in each grain, and the grain boundary energy can then be computed from the intergranular disorientation with Read-Shockley model. A boundary migration solver which considers heterogeneous grain boundary energy is used, which enables the simulation of structure and substructure evolutions. The Gourdet-Montheillet CDRX model is adapted from a mean field to a full field framework, and the corresponding parameters for Zircaloy-4 are identified. Thanks to all those developments, DIGIMU® can simulate full field CDRX in high stacking fault energy materials, coupled with precipitates evolution. Several examples will be presented, some of them compared to experimental results.

**Introduction**

Mastering grain size, solid phase transformations, and dislocations density evolution during industrial processes becomes necessary to fulfill more and more severe microstructure and properties specifications of the processed materials. In multi-pass processes, the competition between boundary migration, recovery and the different types of recrystallizations leads to complex coupled effects, and minor variations of the process parameters may have huge consequences. A correct modelling of the main physical microstructural mechanisms becomes then necessary [1].

Thanks to the explosion of computer capacities, mesoscopic modeling techniques for metallic materials at the solid-state are now available. These lower scale approaches, the so-called full-field models, are based on a full description of the microstructure topology and have demonstrated an exciting potential for an extensive range of microstructure evolutions. First, it is possible to use more physics in the models because of the better description of the microstructure, leading to a wider range of validity [1]. Secondly, observing directly the polycrystal microstructure evolution, in addition to curves and histograms, allows to analyze the results much more comprehensively [1]. Several numerical frameworks are available: Monte Carlo Potts (MC) [1], Cellular Automata

<figure><img src="image_2183.png" /></figure>

<footnote n="1">¹Transvalor, 950 avenue Roumanille, CS40237, Biot, 06904 Sophia Antipolis, France</footnote>

<footnote n="2">²Components Research Center, Framatome, 73403 Ugine, France</footnote>

<footnote n="3">³Mines Paris, PSL University, Centre for material forming (CEMEF), UMR CNRS, 06904 Sophia Antipolis, France</footnote>

<footnote>ᵃPascal.DeMicheli@transvalor.com, ᵇVictor.Grand@framatome.com,</footnote>

<footnote>ᶜMarc.Bernacki@minesparis.psl.eu</footnote>
(CA) [2], Multi-Phase Field (MPF) [3, 4], Vertex [5] or LS models [6,7]. Transvalor distributes for several years the software DIGIMU®, based on a FE-LS framework associated with remeshing algorithms, which is the most adapted framework to simulate large polycrystal deformations. DIGIMU® enables full field simulations of industrial processes and heat treatments with grain growth, discontinuous DRX, PDRX or SRX, with or without second phase particles (SPP) [8,9,10,11,12,13,14]. User-friendliness, CPU time and accuracy have been optimized, and a userroutine module is proposed to modify and adapt default hardening, recovery and nucleation models.

Recently, two new functionalities have been added to broaden the scope of DIGIMU® in terms of thermomechanical conditions and material types. First, an evolutive precipitates formalism is proposed, where a LS function is used to describe the boundaries of an evolutive precipitates population [15,16]. It allows to perform simulations close to solvus, at temperatures where the precipitation state can evolve significantly. Secondly, DIGIMU® can now model the evolution of grain boundaries with heterogeneous energies, and a recent full-field adaptation of the mean field Gourdet-Montheillet model [17, 18] has been implemented to propose a full field CDRX model. It opens the door to the simulation of high stacking fault energy materials, tending to recrystallize continuously [1]. The model was applied to Zircaloy-4 recrystallization and compared with experimental observations [18,19,20].

However, in CDRX, the evolution of the microstructure during the post-dynamic phase is very important, and for numerous materials, this post dynamic phase occurs in presence of precipitates, that may evolve during the heat treatment or the cooling. This is for example the case for several zirconium and aluminum alloys. This article will present more in detail the coupling between precipitates evolution and CDRX models, showing how they can interact to better simulate more advanced CDRX processes with precipitates dissolution, and what are the actual limits. Examples will be provided for Zircaloy-4.

## Simulate coupled SPP and grain evolutions

Grain boundaries pinning by second phase particles (SPP) was first discussed by Smith [21] and then detailed by Zener one year after [22]. Under certain conditions, SPP can pin the grain boundaries, slowing down the grain growth process or even stopping it at a limit grain size. The idea in DIGIMU® is to model the dragging pressure induced by the SPP on grain boundaries naturally, by representing the local curvature modification of the grain boundaries induced by the SPP geometry, without the use of any additional parameter [23,13]. The SPP populations are simply geometrically generated in the polycrystal from a statistical distribution law, or directly imported from microscopy images.

In the evolutive formalism, several distinct SPP populations can be described, each one using its own LS function [15,16]. For each population, one evolution law is provided, describing the normal velocity of the SPP boundary in function of a wide range of variables (temperature, particle radius, mean particle radius, mean particle fraction, particle fraction at equilibrium for this temperature...) and the precipitation rate and radius if precipitation is considered (function of the actual temperature, mean particle fraction, equilibrium particle fraction for this temperature ...). Of course, more advanced phenomena like the dissolution and immediate reprecipitation after the grain boundary passed through a precipitate (the precipitate seems cut by the grain boundary), are not taken into account. However, this simple framework enables to render quite efficiently precipitation, particles growth or dissolution, and Ostwald ripening (see Fig. 1).
(CA) [2], Multi-Phase Field (MPF) [3, 4], Vertex [5] or LS models [6,7]. Transvalor distributes for several years the software DIGIMU®, based on a FE-LS framework associated with remeshing algorithms, which is the most adapted framework to simulate large polycrystal deformations. DIGIMU® enables full field simulations of industrial processes and heat treatments with grain growth, discontinuous DRX, PDRX or SRX, with or without second phase particles (SPP) [8,9,10,11,12,13,14]. User-friendliness, CPU time and accuracy have been optimized, and a userroutine module is proposed to modify and adapt default hardening, recovery and nucleation models.

Recently, two new functionalities have been added to broaden the scope of DIGIMU® in terms of thermomechanical conditions and material types. First, an evolutive precipitates formalism is proposed, where a LS function is used to describe the boundaries of an evolutive precipitates population [15,16]. It allows to perform simulations close to solvus, at temperatures where the precipitation state can evolve significantly. Secondly, DIGIMU® can now model the evolution of grain boundaries with heterogeneous energies, and a recent full-field adaptation of the mean field Gourdet-Montheillet model [17, 18] has been implemented to propose a full field CDRX model. It opens the door to the simulation of high stacking fault energy materials, tending to recrystallize continuously [1]. The model was applied to Zircaloy-4 recrystallization and compared with experimental observations [18,19,20].

However, in CDRX, the evolution of the microstructure during the post-dynamic phase is very important, and for numerous materials, this post dynamic phase occurs in presence of precipitates, that may evolve during the heat treatment or the cooling. This is for example the case for several zirconium and aluminum alloys. This article will present more in detail the coupling between precipitates evolution and CDRX models, showing how they can interact to better simulate more advanced CDRX processes with precipitates dissolution, and what are the actual limits. Examples will be provided for Zircaloy-4.

## Simulate coupled SPP and grain evolutions

Grain boundaries pinning by second phase particles (SPP) was first discussed by Smith [21] and then detailed by Zener one year after [22]. Under certain conditions, SPP can pin the grain boundaries, slowing down the grain growth process or even stopping it at a limit grain size. The idea in DIGIMU® is to model the dragging pressure induced by the SPP on grain boundaries naturally, by representing the local curvature modification of the grain boundaries induced by the SPP geometry, without the use of any additional parameter [23,13]. The SPP populations are simply geometrically generated in the polycrystal from a statistical distribution law, or directly imported from microscopy images.

In the evolutive formalism, several distinct SPP populations can be described, each one using its own LS function [15,16]. For each population, one evolution law is provided, describing the normal velocity of the SPP boundary in function of a wide range of variables (temperature, particle radius, mean particle radius, mean particle fraction, particle fraction at equilibrium for this temperature...) and the precipitation rate and radius if precipitation is considered (function of the actual temperature, mean particle fraction, equilibrium particle fraction for this temperature ...). Of course, more advanced phenomena like the dissolution and immediate reprecipitation after the grain boundary passed through a precipitate (the precipitate seems cut by the grain boundary), are not taken into account. However, this simple framework enables to render quite efficiently precipitation, particles growth or dissolution, and Ostwald ripening (see Fig. 1).
<figure><img src="image_2193.png" /><figcaption>Fig. 1: Simulation of a heat treatment with a wide range of precipitates evolution mechanisms. From a to b) Precipitation and growth of the particles, from b to c) Ostwald ripening, particles with higher radius growth at the expend of the smaller ones, from c to f) increase of the temperature, and lowering of the equilibrium fraction up to the solvus point, followed by the effective particles dissolution, with a progressive release of the grain boundaries. g) Evolution of the simulated particle fraction, number of particles and temperature in the polycrystal during the heat treatment.</figcaption></figure>

## Simulate grain substructure evolution and CDRX

Even if the LAGB's energies are lower than the HAGB's energies, their effect on microstructure evolution can however sometimes be of first order. This is especially the case in high stacking fault energy materials, where the progressive formation of a sub grain structure and its progressive evolution to an unhardened grain structure is the main mechanism of recrystallization, the so called CDRX. In this case, the evolution of LAGB cannot be neglected anymore, and must be considered by using a grain boundary migration solver working with heterogeneous grain boundary energies. Recent developments in DIGIMU® enable to define and store the crystallographic orientation for each grain. The grain boundary energy is computed from the intergranular disorientation using the Read-Shockley model, and the grain boundary migration has been adapted and is now able to deal reliably with high energy gradients [24,25,26]. In order to deal with sub boundaries creation and progressive disorientation, the model has been coupled to an adaptation of the CDRX Gourdet-Montheillet model in a full field context [18,19,26,27]. The dynamically recovered energy provided by the Yoshi-Lasraoui-Jonas hardening law is split into two terms.

* Some dislocations stacks together, leading to the creation of new sub boundaries $dS^+$, converted in the full field model into the creation of new closed sub grains:

$$dS^+ = \frac{\alpha b dE_r}{h\theta_0}.$$

* Some dislocations are stacking into existing boundaries, leading to their disorientation, converted in the model into the rotation of the existing sub grains by an angle $d\theta$:

$$d\theta = \frac{(1-\alpha) b D dE_r}{2h}.$$

Here, $\alpha$ is a coefficient describing the balance between the two phenomena, function of the current grain diameter $D$ and of a reference grain diameter $D_0$:

$$\alpha = 1 - \exp\left(\frac{D}{D_0}\right)^m,$$
<figure><img src="image_2184.png" /><figcaption>Fig. 1: Simulation of a heat treatment with a wide range of precipitates evolution mechanisms. From a to b) Precipitation and growth of the particles, from b to c) Ostwald ripening, particles with higher radius growth at the expend of the smaller ones, from c to f) increase of the temperature, and lowering of the equilibrium fraction up to the solvus point, followed by the effective particles dissolution, with a progressive release of the grain boundaries. g) Evolution of the simulated particle fraction, number of particles and temperature in the polycrystal during the heat treatment.</figcaption></figure>

## Simulate grain substructure evolution and CDRX

Even if the LAGB's energies are lower than the HAGB's energies, their effect on microstructure evolution can however sometimes be of first order. This is especially the case in high stacking fault energy materials, where the progressive formation of a sub grain structure and its progressive evolution to an unhardened grain structure is the main mechanism of recrystallization, the so called CDRX. In this case, the evolution of LAGB cannot be neglected anymore, and must be considered by using a grain boundary migration solver working with heterogeneous grain boundary energies. Recent developments in DIGIMU® enable to define and store the crystallographic orientation for each grain. The grain boundary energy is computed from the intergranular disorientation using the Read-Shockley model, and the grain boundary migration has been adapted and is now able to deal reliably with high energy gradients [24,25,26]. In order to deal with sub boundaries creation and progressive disorientation, the model has been coupled to an adaptation of the CDRX Gourdet-Montheillet model in a full field context [18,19,26,27]. The dynamically recovered energy provided by the Yoshi-Lasraoui-Jonas hardening law is split into two terms.

* Some dislocations stacks together, leading to the creation of new sub boundaries $dS^+$, converted in the full field model into the creation of new closed sub grains:

$$dS^+ = \frac{\alpha b dE_r}{h\theta_0}.$$

* Some dislocations are stacking into existing boundaries, leading to their disorientation, converted in the model into the rotation of the existing sub grains by an angle $d\theta$:

$$d\theta = \frac{(1-\alpha) b D dE_r}{2h}.$$

Here, $\alpha$ is a coefficient describing the balance between the two phenomena, function of the current grain diameter $D$ and of a reference grain diameter $D_0$:

$$\alpha = 1 - \exp\left(\frac{D}{D_0}\right)^m,$$
where *b* is the Burger vector norm, *dE*<sub>r</sub> the computed dynamically recovered energy at this time step, *h* the number or of sets of dislocations, θ<sub>0</sub> the disorientation angle of newly formed sub grains and *m* a fixed coefficient.

Parameters for Zircaloy-4 have been identified by Framatome Components Research Center [18,19]. More details about parameters identification and values can be found in [20]. The simulation of the hot deformation of a sample at 650°C, at a strain rate of 1/s up to a strain value of 1, followed by a heat treatment of 100s has been performed. Substructure formation and evolution (Fig. 2) and dislocation density evolution (Fig. 3) show an interesting agreement with the experimental results. Even if the first inserted circular closed sub grains are not really comparable to the physical open sub boundaries which are created, after a few seconds, the simulated substructure evolves in something much closer to the reality and allows reasonably representative simulations.

<figure><img src="image_2194.png" /><figcaption>Fig. 2: Creation and evolution of sub-grains in Zircaloy-4 during CDRX and PDRX simulations (compression and maintain). Color is related to the grain boundaries energy. HAGB are in red, other ones are LAGB, more or less disoriented.</figcaption></figure>
where *b* is the Burger vector norm, *dE*<sub>r</sub> the computed dynamically recovered energy at this time step, *h* the number or of sets of dislocations, θ<sub>0</sub> the disorientation angle of newly formed sub grains and *m* a fixed coefficient.

Parameters for Zircaloy-4 have been identified by Framatome Components Research Center [18,19]. More details about parameters identification and values can be found in [20]. The simulation of the hot deformation of a sample at 650°C, at a strain rate of 1/s up to a strain value of 1, followed by a heat treatment of 100s has been performed. Substructure formation and evolution (Fig. 2) and dislocation density evolution (Fig. 3) show an interesting agreement with the experimental results. Even if the first inserted circular closed sub grains are not really comparable to the physical open sub boundaries which are created, after a few seconds, the simulated substructure evolves in something much closer to the reality and allows reasonably representative simulations.

<figure><img src="image_2185.png" /><figcaption>Fig. 2: Creation and evolution of sub-grains in Zircaloy-4 during CDRX and PDRX simulations (compression and maintain). Color is related to the grain boundaries energy. HAGB are in red, other ones are LAGB, more or less disoriented.</figcaption></figure>
<figure><img src="image_2195.png" /><figcaption>Fig. 3: Comparison between simulated (up) and experimental (down) dislocation densities evolutions after 25s (left) and 100s (right) of heat treatment.</figcaption></figure>

## Simulate CDRX with an evolutive population of SPP

Zirconium alloys can present some precipitation, with a solvus temperature close to the recrystallization temperatures. Precipitates are present in the microstructure and can evolve during post-dynamic phase. In the previous example, this precipitation has been neglected. The idea here is to see if it could be possible to couple the two previous models in order to present a CDRX-PDRX simulation coupled with particles dissolution, on a process similar to the one presented in the previous chapter.

In experiments, precipitates are present in the material since the beginning of the compression and conserve their equiaxed shape during it. This is one of the limits of our approach, where the SPP are simply deformed with the matrix. Furthermore, the presence of a dense SPP population in the polycrystal may modify the behavior of sub-grain generation, reducing the space available for their nucleation. This should be considered during the CDRX parameters identification, which was not the case here. For those two reasons, the presence of the precipitates during the CDRX simulation step is disturbing. Happily, during the short time of compression, the consequences of Zener pinning are rather small, as sub boundaries mainly appear and disorient, but do not have a lot of time to evolve. For those three reasons, it was decided to insert the particles only at the end of the deformation, with a quick nucleation process of a few tenths of seconds, until the desired fraction of precipitates fraction was reached. SPP are then only present in the PDRX phase, when their effect is the most important. Afterwards, a simple constant dissolution velocity is applied to the boundaries of the precipitates, leading to a complete disappearance in 180s.

Simulation results are presented in Fig. 4. The initial state of precipitation is visible in the first image, just after CDRX. After 40 seconds of PDRX, it is clearly visible that the remaining particles have bent the boundaries and sub boundaries. After 180s, all the particles have disappeared, and the microstructure is already well recrystallized. In this case, it was not clear that the presence of
<figure><img src="image_2186.png" /><figcaption>Fig. 3: Comparison between simulated (up) and experimental (down) dislocation densities evolutions after 25s (left) and 100s (right) of heat treatment.</figcaption></figure>

## Simulate CDRX with an evolutive population of SPP

Zirconium alloys can present some precipitation, with a solvus temperature close to the recrystallization temperatures. Precipitates are present in the microstructure and can evolve during post-dynamic phase. In the previous example, this precipitation has been neglected. The idea here is to see if it could be possible to couple the two previous models in order to present a CDRX-PDRX simulation coupled with particles dissolution, on a process similar to the one presented in the previous chapter.

In experiments, precipitates are present in the material since the beginning of the compression and conserve their equiaxed shape during it. This is one of the limits of our approach, where the SPP are simply deformed with the matrix. Furthermore, the presence of a dense SPP population in the polycrystal may modify the behavior of sub-grain generation, reducing the space available for their nucleation. This should be considered during the CDRX parameters identification, which was not the case here. For those two reasons, the presence of the precipitates during the CDRX simulation step is disturbing. Happily, during the short time of compression, the consequences of Zener pinning are rather small, as sub boundaries mainly appear and disorient, but do not have a lot of time to evolve. For those three reasons, it was decided to insert the particles only at the end of the deformation, with a quick nucleation process of a few tenths of seconds, until the desired fraction of precipitates fraction was reached. SPP are then only present in the PDRX phase, when their effect is the most important. Afterwards, a simple constant dissolution velocity is applied to the boundaries of the precipitates, leading to a complete disappearance in 180s.

Simulation results are presented in Fig. 4. The initial state of precipitation is visible in the first image, just after CDRX. After 40 seconds of PDRX, it is clearly visible that the remaining particles have bent the boundaries and sub boundaries. After 180s, all the particles have disappeared, and the microstructure is already well recrystallized. In this case, it was not clear that the presence of
the SPP has really slowed down the recrystallization process, or even modified the final grain size, which confirms the hypothesis made in the previous paragraph and by Grand et al. [18,19,27]. However, the feasibility of such simulations is proven, and further simulations will be done on different initial material configurations, and different materials.

<figure><img src="image_2196.png" /><figcaption>Fig. 4: Evolution of Grain and sub-grain boundaries and dislocation densities after CDRX for Zircaloy-4, with simultaneous dissolution of precipitates (black dots).</figcaption></figure>

## Conclusion

DIGIMU® has proven these last years to be an efficient and convenient tool to predict and better understand microstructural evolutions in industrial processes and heat treatments, simulating grain growth, Smith-Zener pinning and recrystallization. Very recently, two interesting features have been developed, enabling the simulations of SPP evolution, sub grain evolutions and a first CDRX model.

This article deals with the combination of those two features to simulate CDRX in high stacking fault materials presenting precipitates population whose solvus temperature is close to the recrystallization temperature. Even if the model presents some important approximations, such as the insertion of the sub boundaries as closed circular sub grains, it converges rapidly to realistic microstructures, enabling relevant simulations. It is possible to couple those two models together. In this case, we however preferred to insert the particles after the compression step to avoid their deformation and to not disturb the sub grain nucleation model, identified without the presence of particles. Even if the simulation with precipitates has not been strictly compared to experimental results, the Smith-Zener pinning effect is clearly visible. In this configuration, no global effect on final grain size or recrystallized fraction evolution was predicted, in accordance with Framatome observations.

The actual work will consist in confronting the model to other industrial cases, concerning zirconium alloys first, but also aluminum alloys and ferritic stainless steels, to determine its capabilities and limits in a wider range of material and processes.

## References

[1] Rollett, A., Rohrer, G.S., Humphreys, J., Recrystallization and Related Annealing Phenomena (2017)

[2] K. Janssens, An introductory review of cellular automata modeling of moving grain boundaries in polycrystalline materials, Math. Comput. Simul. 80(7) (2010) pp. 1361–1381.
https://doi.org/10.1016/j.matcom.2009.02.011
the SPP has really slowed down the recrystallization process, or even modified the final grain size, which confirms the hypothesis made in the previous paragraph and by Grand et al. [18,19,27]. However, the feasibility of such simulations is proven, and further simulations will be done on different initial material configurations, and different materials.

<figure><img src="image_2187.png" /><figcaption>Fig. 4: Evolution of Grain and sub-grain boundaries and dislocation densities after CDRX for Zircaloy-4, with simultaneous dissolution of precipitates (black dots).</figcaption></figure>

## Conclusion

DIGIMU® has proven these last years to be an efficient and convenient tool to predict and better understand microstructural evolutions in industrial processes and heat treatments, simulating grain growth, Smith-Zener pinning and recrystallization. Very recently, two interesting features have been developed, enabling the simulations of SPP evolution, sub grain evolutions and a first CDRX model.

This article deals with the combination of those two features to simulate CDRX in high stacking fault materials presenting precipitates population whose solvus temperature is close to the recrystallization temperature. Even if the model presents some important approximations, such as the insertion of the sub boundaries as closed circular sub grains, it converges rapidly to realistic microstructures, enabling relevant simulations. It is possible to couple those two models together. In this case, we however preferred to insert the particles after the compression step to avoid their deformation and to not disturb the sub grain nucleation model, identified without the presence of particles. Even if the simulation with precipitates has not been strictly compared to experimental results, the Smith-Zener pinning effect is clearly visible. In this configuration, no global effect on final grain size or recrystallized fraction evolution was predicted, in accordance with Framatome observations.

The actual work will consist in confronting the model to other industrial cases, concerning zirconium alloys first, but also aluminum alloys and ferritic stainless steels, to determine its capabilities and limits in a wider range of material and processes.

## References

[1] Rollett, A., Rohrer, G.S., Humphreys, J., Recrystallization and Related Annealing Phenomena (2017)

[2] K. Janssens, An introductory review of cellular automata modeling of moving grain boundaries in polycrystalline materials, Math. Comput. Simul. 80(7) (2010) pp. 1361–1381.
https://doi.org/10.1016/j.matcom.2009.02.011
[3] E. Miyoshi, T. Takaki, Multi-phase-field study of the effects of anisotropic grain-boundary properties on polycrystalline grain growth, J. Cryst. Growth 474 (2017) pp. 160–165.
https://doi.org/10.1016/j.jcrysgro.2016.11.097

[4] N. Moelans, B. Blanpain, P. Wollants, Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems, Phys. Rev. B 78(2) (2008) pp. 024113. https://doi.org/10.1103/PhysRevB.78.024113

[5] S. Florez, K. Alvarado, D.P. Muñoz, M. Bernacki, A novel highly efficient lagrangian model for massively multidomain simulation applied to microstructural evolutions, Comput. Methods Appl. Mech. Eng. 367 (2020) p. 113107. https://doi.org/10.1016/j.cma.2020.113107

[6] M. Bernacki, Y. Chastel, T. Coupez, R. Logé, Level set framework for the numerical modelling of primary recrystallization in polycrystalline materials, Scr. Mater. 58(12) (2008) pp. 1129–1132. https://doi.org/10.1016/j.scriptamat.2008.02.016

[7] M. Elsey, S. Esedoglu, P. Smereka, Diffusion generated motion for grain growth in two and three dimensions, J. Comput. Phys. 228(21) (2009) pp. 8015–8033.
https://doi.org/10.1016/j.jcp.2009.07.020

[8] P.O. De Micheli, Maire, L., Cardinaux, D., Moussa, C., N. Bozzolo, M. Bernacki, Digimu® pp. Full field recrystallization simulations for optimization of multi-pass processes, AIP Conf Proc 2113(1) (2019) pp. 040014. https://doi.org/10.1063/1.5112548

[9] B. Scholtes, R. Boulais-Sinou, A. Settefrati, D.P. Muñoz, I. Poitrault, A. Montouchet, N. Bozzolo, M. Bernacki, 3D level set modeling of static recrystallization considering stored energy fields, Comput. Mater. Sci. 122 (2016) pp. 57–71.
https://doi.org/10.1016/j.commatsci.2016.04.045

[10] L. Maire, B. Scholtes, C. Moussa, N. Bozzolo, D.P. Muñoz, A. Settefrati, M. Bernacki, Modeling of dynamic and post-dynamic recrystallization by coupling a full field approach to phenomenological laws, Mater. Des. 133 (2017) pp.498–519.
https://doi.org/10.1016/j.matdes.2017.08.015

[11] A. Agnoli, N. Bozzolo, R. Logé, J.M. Franchet, J. Laigo, M. Bernacki, Development of a level set methodology to simulate grain growth in the presence of real secondary phase particles and stored energy – Application to a nickel-base superalloy, Comput. Mater. Sci. 89 (2014) pp. 233–241.
https://doi.org/10.1016/j.commatsci.2014.03.054

[12] H. Brüggemann, A. Quadfasel, J.A. Nietsch, M. Teller, P. De Micheli, B. Flipon, M. Bernacki (2021), Korngrößenvorhersage beim freiformschmieden von inconel 718 mit DIGIMU®, massiv UMFORMUNG 2021(3) pp. 50–54

[13] F. Villaret, B. Hary, Y. de Carlan, T. Baudin, R. Logé, L. Maire, M. Bernacki, Probabilistic and deterministic full field approaches to simulate recrystallization in ods steels, Comput. Mater. Sci. 179 (2020) pp. 109646. https://doi.org/10.1016/j.commatsci.2020.109646

[14] A. Agnoli, M. Bernacki, R. Logé, J.M. Franchet, J. Laigo, N. Bozzolo, Selective growth of low stored energy grains during δ sub-solvus annealing in the inconel 718 nickel base superalloy, Metall Mater Trans A A 46(9) (2015), pp. 4405–4421. https://doi.org/10.1007/s11661-015-3035-9

[15] K. Alvarado, S. Florez, B. Flipon, N. Bozzolo, M. Bernacki, A level set approach to simulate grain growth with an evolving population of second phase particles, Model Simul Mat Sci Eng 29(3) (2021) p. 035009. https://doi.org/10.1088/1361-651X/abe0a7
[3] E. Miyoshi, T. Takaki, Multi-phase-field study of the effects of anisotropic grain-boundary properties on polycrystalline grain growth, J. Cryst. Growth 474 (2017) pp. 160–165.
https://doi.org/10.1016/j.jcrysgro.2016.11.097

[4] N. Moelans, B. Blanpain, P. Wollants, Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems, Phys. Rev. B 78(2) (2008) pp. 024113. https://doi.org/10.1103/PhysRevB.78.024113

[5] S. Florez, K. Alvarado, D.P. Muñoz, M. Bernacki, A novel highly efficient lagrangian model for massively multidomain simulation applied to microstructural evolutions, Comput. Methods Appl. Mech. Eng. 367 (2020) p. 113107. https://doi.org/10.1016/j.cma.2020.113107

[6] M. Bernacki, Y. Chastel, T. Coupez, R. Logé, Level set framework for the numerical modelling of primary recrystallization in polycrystalline materials, Scr. Mater. 58(12) (2008) pp. 1129–1132. https://doi.org/10.1016/j.scriptamat.2008.02.016

[7] M. Elsey, S. Esedoglu, P. Smereka, Diffusion generated motion for grain growth in two and three dimensions, J. Comput. Phys. 228(21) (2009) pp. 8015–8033.
https://doi.org/10.1016/j.jcp.2009.07.020

[8] P.O. De Micheli, Maire, L., Cardinaux, D., Moussa, C., N. Bozzolo, M. Bernacki, Digimu® pp. Full field recrystallization simulations for optimization of multi-pass processes, AIP Conf Proc 2113(1) (2019) pp. 040014. https://doi.org/10.1063/1.5112548

[9] B. Scholtes, R. Boulais-Sinou, A. Settefrati, D.P. Muñoz, I. Poitrault, A. Montouchet, N. Bozzolo, M. Bernacki, 3D level set modeling of static recrystallization considering stored energy fields, Comput. Mater. Sci. 122 (2016) pp. 57–71.
https://doi.org/10.1016/j.commatsci.2016.04.045

[10] L. Maire, B. Scholtes, C. Moussa, N. Bozzolo, D.P. Muñoz, A. Settefrati, M. Bernacki, Modeling of dynamic and post-dynamic recrystallization by coupling a full field approach to phenomenological laws, Mater. Des. 133 (2017) pp.498–519.
https://doi.org/10.1016/j.matdes.2017.08.015

[11] A. Agnoli, N. Bozzolo, R. Logé, J.M. Franchet, J. Laigo, M. Bernacki, Development of a level set methodology to simulate grain growth in the presence of real secondary phase particles and stored energy – Application to a nickel-base superalloy, Comput. Mater. Sci. 89 (2014) pp. 233–241.
https://doi.org/10.1016/j.commatsci.2014.03.054

[12] H. Brüggemann, A. Quadfasel, J.A. Nietsch, M. Teller, P. De Micheli, B. Flipon, M. Bernacki (2021), Korngrößenvorhersage beim freiformschmieden von inconel 718 mit DIGIMU®, massiv UMFORMUNG 2021(3) pp. 50–54

[13] F. Villaret, B. Hary, Y. de Carlan, T. Baudin, R. Logé, L. Maire, M. Bernacki, Probabilistic and deterministic full field approaches to simulate recrystallization in ods steels, Comput. Mater. Sci. 179 (2020) pp. 109646. https://doi.org/10.1016/j.commatsci.2020.109646

[14] A. Agnoli, M. Bernacki, R. Logé, J.M. Franchet, J. Laigo, N. Bozzolo, Selective growth of low stored energy grains during δ sub-solvus annealing in the inconel 718 nickel base superalloy, Metall Mater Trans A A 46(9) (2015), pp. 4405–4421. https://doi.org/10.1007/s11661-015-3035-9

[15] K. Alvarado, S. Florez, B. Flipon, N. Bozzolo, M. Bernacki, A level set approach to simulate grain growth with an evolving population of second phase particles, Model Simul Mat Sci Eng 29(3) (2021) p. 035009. https://doi.org/10.1088/1361-651X/abe0a7
[16] K. Alvarado, I. Janeiro, S. Florez, B. Flipon, J.M. Franchet, D. Locq, C. Dumont, N. Bozzolo, M. Bernacki, Dissolution of the primary γ' precipitates and grain growth during solution treatment of three nickel base superalloys, *Metals* 11(12) (2021). https://doi.org/10.3390/met11121921

[17] S. Gourdet, F. Montheillet, A model of continuous dynamic recrystallization, *Acta Mater.* 51(9) (2003) pp. 2685-2699. https://doi.org/10.1016/S1359-6454(03)00078-8

[18] V. Grand, B. Flipon, A. Gaillac, M. Bernacki, Simulation of continuous dynamic recrystallization using a level-set method, *Materials* 15(23) (2022). https://doi.org/10.3390/ma15238547

[19] A. Gaillac, V. Grand, A. Arsen, Q. Gaillard, M. Bernacki, Towards multi-scale modeling of zirconium alloys recrystallization and application to thermo-mechanical processes optimization, *Key Eng. Mater.* 926(8) (2022) pp. 443-451. https://doi.org/10.4028/p-54m016

[20] V. Grand, Characterization and modeling of zircaloy-4 recrystallization during hot forming, *PhD manuscript, MINES Paris PSL* (2022)

[21] C.S. Smith, Introduction to grains, phases, and interfaces—an interpretation of microstructure, *Trans. Am. Inst. Min. Metall. Eng.* 175(2) (1948) pp. 15-51

[22] C. Zener, Theory of growth of spherical precipitates from solid solution, *J. Appl. Phys.* 20(10) (1949) pp. 950-953. https://doi.org/10.1063/1.1698258

[23] B. Scholtes, D. Ilin, A. Settefrati, N. Bozzolo, A. Agnoli, M. Bernacki, Full field modeling of the Zener pinning phenomenon in a level set framework - discussion of classical limiting mean grain size equation, *Proceedings of the 13th International Symposium on Superalloys* (2016). https://doi.org/10.1002/9781119075646.ch53

[24] J. Fausty, N. Bozzolo, D.P. Muñoz, M. Bernacki, A novel level-set finite element formulation for grain growth with heterogeneous grain boundary energies, *Mater. Des.* 160 (2018) pp. 578-590. https://doi.org/10.1016/j.matdes.2018.09.050

[25] J. Fausty, N. Bozzolo, M. Bernacki, A 2d level set finite element grain coarsening study with heterogeneous grain boundary energies, *Appl. Math. Model.* 78 (2020) pp. 505-518. https://doi.org/10.1016/j.apm.2019.10.008

[26] B. Murgas, S. Florez, N. Bozzolo, J. Fausty, M. Bernacki, Comparative study and limits of different level-set formulations for the modeling of anisotropic grain growth, *Materials* 14(14) (2021). https://doi.org/10.3390/ma14143883

[27] V. Grand, B. Flipon, A. Gaillac, M. Bernacki, Characterization and modeling of the influence of initial microstructure on recrystallization of zircaloy-4 during hot forming, *J. ASTM Int. In press* (2022). https://doi.org/10.1520/STP164520220010
[16] K. Alvarado, I. Janeiro, S. Florez, B. Flipon, J.M. Franchet, D. Locq, C. Dumont, N. Bozzolo, M. Bernacki, Dissolution of the primary γ' precipitates and grain growth during solution treatment of three nickel base superalloys, *Metals* 11(12) (2021). https://doi.org/10.3390/met11121921

[17] S. Gourdet, F. Montheillet, A model of continuous dynamic recrystallization, *Acta Mater.* 51(9) (2003) pp. 2685-2699. https://doi.org/10.1016/S1359-6454(03)00078-8

[18] V. Grand, B. Flipon, A. Gaillac, M. Bernacki, Simulation of continuous dynamic recrystallization using a level-set method, *Materials* 15(23) (2022). https://doi.org/10.3390/ma15238547

[19] A. Gaillac, V. Grand, A. Arsen, Q. Gaillard, M. Bernacki, Towards multi-scale modeling of zirconium alloys recrystallization and application to thermo-mechanical processes optimization, *Key Eng. Mater.* 926(8) (2022) pp. 443-451. https://doi.org/10.4028/p-54m016

[20] V. Grand, Characterization and modeling of zircaloy-4 recrystallization during hot forming, *PhD manuscript, MINES Paris PSL* (2022)

[21] C.S. Smith, Introduction to grains, phases, and interfaces—an interpretation of microstructure, *Trans. Am. Inst. Min. Metall. Eng.* 175(2) (1948) pp. 15-51

[22] C. Zener, Theory of growth of spherical precipitates from solid solution, *J. Appl. Phys.* 20(10) (1949) pp. 950-953. https://doi.org/10.1063/1.1698258

[23] B. Scholtes, D. Ilin, A. Settefrati, N. Bozzolo, A. Agnoli, M. Bernacki, Full field modeling of the Zener pinning phenomenon in a level set framework - discussion of classical limiting mean grain size equation, *Proceedings of the 13th International Symposium on Superalloys* (2016). https://doi.org/10.1002/9781119075646.ch53

[24] J. Fausty, N. Bozzolo, D.P. Muñoz, M. Bernacki, A novel level-set finite element formulation for grain growth with heterogeneous grain boundary energies, *Mater. Des.* 160 (2018) pp. 578-590. https://doi.org/10.1016/j.matdes.2018.09.050

[25] J. Fausty, N. Bozzolo, M. Bernacki, A 2d level set finite element grain coarsening study with heterogeneous grain boundary energies, *Appl. Math. Model.* 78 (2020) pp. 505-518. https://doi.org/10.1016/j.apm.2019.10.008

[26] B. Murgas, S. Florez, N. Bozzolo, J. Fausty, M. Bernacki, Comparative study and limits of different level-set formulations for the modeling of anisotropic grain growth, *Materials* 14(14) (2021). https://doi.org/10.3390/ma14143883

[27] V. Grand, B. Flipon, A. Gaillac, M. Bernacki, Characterization and modeling of the influence of initial microstructure on recrystallization of zircaloy-4 during hot forming, *J. ASTM Int. In press* (2022). https://doi.org/10.1520/STP164520220010
# Numerical investigation of the effects of process parameters on forming load and failure in hot nosing process

TURSUN Huseyin¹,ᵃ*, OZASLAN Emre¹,ᵇ*, YETGIN Ali¹,ᶜ and ACAR Bulent¹,ᵈ

¹Repkon Machine and Tools Industry and Trade Inc., Istanbul Turkey

ᵃhuseyin.tursun@repkon.com.tr, ᵇemre.ozaslan@repkon.com.tr, ᶜali.yetgin@repkon.com.tr, ᵈbulent.acar@repkon.com.tr

**Keywords:** Hot Nosing Process, Two-Stage Forming, Buckling, Finite Element Modeling

**Abstract.** Nosing process is generally used to reduce the diameter of the open end of the circular thin-walled tubes. Friction coefficient and workpiece temperature are two important process parameters that should be investigated to understand the process in detail. In addition, forming length and nosing ratio need to be considered to carry out a successful process. If these parameters are not determined correctly, workpiece damage like buckling is inevitable. The purpose of this study is to investigate the effects of the process parameters on the forming load and failure of the workpiece in the hot nosing process.

**Introduction**

The nosing process is a commonly favored method among metal-forming manufacturers for acquiring cost-effective final products with the desired microstructure. Furthermore, owing to its effectiveness in achieving the final product thickness and forming surface, it is considered as a futuristic forming technique, as the production steps do not require any subsequent machining stage. In addition, the process is typically employed for reducing the diameter and increasing the thickness of a circular thin-walled shell through the application of press loads. Depending on the specific requirements, the nosing process can be applied to both cold and hot-temperature workpieces [1,2].

Gouveia et al.[3], emphasized the significance of achieving higher nosing ratios and reducing forming loads through modifications of the contour, radius of the tube, initial gap height, and lubrication conditions. Manabe and Nishimura [4], highlighted the importance of designing the nosing die-forming surface in a circular curved manner to minimize nosing load, achieve maximum nosing ratio, and ensure uniform surface pressure. Tang and Kobayashi [5], conducted a study employing a coupled analysis of deformation and heat transfer. They utilized AISI1045 with a rigid-plastic material idealization, adapting the theory of visco-plasticity. The correlation between experimental results and numerical simulation was in good agreement regarding to maximum force during nosing, thickness distribution, deformed shapes and temperature distribution.

Additionally, it is crucial to address the issue of buckling under axial load during nosing. To mitigate this, it is recommended, as suggested by [6], that preheating be implemented at a rapid rate to prevent the propagation of high temperatures into the cold region of the shell.

The friction coefficient and workpiece preheating length are two important process parameters that should be investigated for a comprehensive understanding of the process. Incorrect determination of these parameters can lead to workpiece damage, such as buckling, becoming unavoidable. The aim of this study is to investigate the influence of process parameters, which are friction coefficient and workpiece preheating length, on both the forming load and the occurrence of workpiece failure within the context of the hot nosing process. Moreover, a two-stage nosing forming approach was studied to prevent workpiece buckling failure and concurrently reduce

<figure><img src="image_2197.png" /></figure>
# Numerical investigation of the effects of process parameters on forming load and failure in hot nosing process

TURSUN Huseyin¹,ᵃ*, OZASLAN Emre¹,ᵇ*, YETGIN Ali¹,ᶜ and ACAR Bulent¹,ᵈ

¹Repkon Machine and Tools Industry and Trade Inc., Istanbul Turkey

ᵃhuseyin.tursun@repkon.com.tr, ᵇemre.ozaslan@repkon.com.tr, ᶜali.yetgin@repkon.com.tr,

ᵇbulent.acar@repkon.com.tr

**Keywords:** Hot Nosing Process, Two-Stage Forming, Buckling, Finite Element Modeling

**Abstract.** Nosing process is generally used to reduce the diameter of the open end of the circular thin-walled tubes. Friction coefficient and workpiece temperature are two important process parameters that should be investigated to understand the process in detail. In addition, forming length and nosing ratio need to be considered to carry out a successful process. If these parameters are not determined correctly, workpiece damage like buckling is inevitable. The purpose of this study is to investigate the effects of the process parameters on the forming load and failure of the workpiece in the hot nosing process.

**Introduction**

The nosing process is a commonly favored method among metal-forming manufacturers for acquiring cost-effective final products with the desired microstructure. Furthermore, owing to its effectiveness in achieving the final product thickness and forming surface, it is considered as a futuristic forming technique, as the production steps do not require any subsequent machining stage. In addition, the process is typically employed for reducing the diameter and increasing the thickness of a circular thin-walled shell through the application of press loads. Depending on the specific requirements, the nosing process can be applied to both cold and hot-temperature workpieces [1,2].

Gouveia et al.[3], emphasized the significance of achieving higher nosing ratios and reducing forming loads through modifications of the contour, radius of the tube, initial gap height, and lubrication conditions. Manabe and Nishimura [4], highlighted the importance of designing the nosing die-forming surface in a circular curved manner to minimize nosing load, achieve maximum nosing ratio, and ensure uniform surface pressure. Tang and Kobayashi [5], conducted a study employing a coupled analysis of deformation and heat transfer. They utilized AISI1045 with a rigid-plastic material idealization, adapting the theory of visco-plasticity. The correlation between experimental results and numerical simulation was in good agreement regarding to maximum force during nosing, thickness distribution, deformed shapes and temperature distribution.

Additionally, it is crucial to address the issue of buckling under axial load during nosing. To mitigate this, it is recommended, as suggested by [6], that preheating be implemented at a rapid rate to prevent the propagation of high temperatures into the cold region of the shell.

The friction coefficient and workpiece preheating length are two important process parameters that should be investigated for a comprehensive understanding of the process. Incorrect determination of these parameters can lead to workpiece damage, such as buckling, becoming unavoidable. The aim of this study is to investigate the influence of process parameters, which are friction coefficient and workpiece preheating length, on both the forming load and the occurrence of workpiece failure within the context of the hot nosing process. Moreover, a two-stage nosing forming approach was studied to prevent workpiece buckling failure and concurrently reduce

<figure><img src="image_2188.png" /></figure>
forming load. A coupled displacement-temperature finite element analysis (FEA) was conducted to investigate the parameters explained.

## Methodology

For this study, a workpiece with a diameter of 300 mm, a thickness of 8 mm, and a length of 1300 mm was selected shown as in Figure 1.

<figure><img src="image_2198.png" /><figcaption>Fig. 1. a) Basic dimensions of the die, b) Basic dimensions of the workpiece</figcaption></figure>

Material. In this study, the material was selected as 42CrMo4 (1.7225) for the workpiece, and material properties varying with strain rate (SR) were utilized in the analyses. Strain rate dependent material data, shown in Figure 2, was prepared with JMatPro [7].

<figure><img src="image_2199.png" /><figcaption>Fig. 2. Stress-strain flow curve depending on strain rate of 42CrMo4 (1.7225) material</figcaption></figure>

Finite Element Method. In order to evaluate effects of the friction coefficient between the upper die – the workpiece, as well as the preheating length of the workpiece, on the process, a finite element analysis study was conducted. Finite element model was introduced in this section, followed by a discussion of the obtained analysis findings.

Throughout the study, the models were modeled in an axisymmetric environment and analyzed using a coupled approach involving deformation and heat transfer in ABAQUS/Explicit. The relevant surfaces of both the upper and the lower die, which were selected to be rigid, were included in the finite element model. The workpiece was modeled as deformable. CAX3RT element type was used to mesh workpiece, as the use of axisymmetric reduced-order solid elements provides much less computational time. This element type includes 4-node thermally coupled
forming load. A coupled displacement-temperature finite element analysis (FEA) was conducted to investigate the parameters explained.

## Methodology

For this study, a workpiece with a diameter of 300 mm, a thickness of 8 mm, and a length of 1300 mm was selected shown as in Figure 1.

<figure><img src="image_2189.png" /><figcaption>Fig. 1. a) Basic dimensions of the die, b) Basic dimensions of the workpiece</figcaption></figure>

Material. In this study, the material was selected as 42CrMo4 (1.7225) for the workpiece, and material properties varying with strain rate (SR) were utilized in the analyses. Strain rate dependent material data, shown in Figure 2, was prepared with JMatPro [7].

<figure><img src="image_2190.png" /><figcaption>Fig. 2. Stress-strain flow curve depending on strain rate of 42CrMo4 (1.7225) material</figcaption></figure>

Finite Element Method. In order to evaluate effects of the friction coefficient between the upper die – the workpiece, as well as the preheating length of the workpiece, on the process, a finite element analysis study was conducted. Finite element model was introduced in this section, followed by a discussion of the obtained analysis findings.

Throughout the study, the models were modeled in an axisymmetric environment and analyzed using a coupled approach involving deformation and heat transfer in ABAQUS/Explicit. The relevant surfaces of both the upper and the lower die, which were selected to be rigid, were included in the finite element model. The workpiece was modeled as deformable. CAX3RT element type was used to mesh workpiece, as the use of axisymmetric reduced-order solid elements provides much less computational time. This element type includes 4-node thermally coupled
axisymmetric quadrilateral, bilinear displacement and temperature, reduced integration and hourglass control. After performing a mesh convergence study, the element size was determined as 0.8 mm for each workpiece. The analysis utilized a total of 5456 elements and 6147 nodes.

In the analysis model, the lower die, modeled as analytically rigid, was fixed. The velocity of 100 mm/s was defined for the upper die, which was modeled to move vertically. The temperature of the cold region of the workpiece and rigid dies were set to 20 °C, whereas the hot region of the workpiece was defined with a temperature of 900 °C. The heating process in the preheating region is made with induction heating method. The preheating length of the workpiece, determined to be full (400mm) and half (200mm) of the nosing length, is illustrated in Figure 3.

<figure><img src="image_2200.png" /><figcaption>Fig. 3. Boundary conditions descriptions of hot nosing process, b) Workpiece mesh model, c) Full preheating region, d) Half preheating region.</figcaption></figure>

The contact interaction between die and workpiece was defined using the general contact definition in ABAQUS /Explicit. The analysis duration was set to 3.8 seconds, and the step included variable mass scaling factor to maintain a constant stable time increment of 2e-06. Mass scaling is a method employed to accelerate computations by augmenting the mass of the workpiece. After the solving stage of the analysis, it was checked that mass scale factor (EMSF indicator in ABAQUS) was sufficiently low to prevent any impact on the solution. A thorough examination of artificial strain and kinetic energy parameters was conducted to ascertain that inertial forces did not exert influence on the overall solution.
axisymmetric quadrilateral, bilinear displacement and temperature, reduced integration and hourglass control. After performing a mesh convergence study, the element size was determined as 0.8 mm for each workpiece. The analysis utilized a total of 5456 elements and 6147 nodes.

In the analysis model, the lower die, modeled as analytically rigid, was fixed. The velocity of 100 mm/s was defined for the upper die, which was modeled to move vertically. The temperature of the cold region of the workpiece and rigid dies were set to 20 °C, whereas the hot region of the workpiece was defined with a temperature of 900 °C. The heating process in the preheating region is made with induction heating method. The preheating length of the workpiece, determined to be full (400mm) and half (200mm) of the nosing length, is illustrated in Figure 3.

<figure><img src="image_2191.png" /><figcaption>Fig. 3. Boundary conditions descriptions of hot nosing process, b) Workpiece mesh model, c) Full preheating region, d) Half preheating region.</figcaption></figure>

The contact interaction between die and workpiece was defined using the general contact definition in ABAQUS /Explicit. The analysis duration was set to 3.8 seconds, and the step included variable mass scaling factor to maintain a constant stable time increment of 2e-06. Mass scaling is a method employed to accelerate computations by augmenting the mass of the workpiece. After the solving stage of the analysis, it was checked that mass scale factor (EMSF indicator in ABAQUS) was sufficiently low to prevent any impact on the solution. A thorough examination of artificial strain and kinetic energy parameters was conducted to ascertain that inertial forces did not exert influence on the overall solution.
## Results and Discussion

Effect of the Preheat Length on the Process. Determining the preheat length in the hot nosing process is crucial to prevent premature failures. If the preheat length is longer than necessary, the workpiece may buckle in the hot region that has not been formed yet. The analysis results, in which the length of heating region was excessive, are given in Figure 4. Buckling failure is evident in the hot region that has not yet entered the die, as illustrated in Figure 4. The hot region outside the die was unable to withstand the instantaneous process load, leading to buckling.

<figure><img src="image_2201.png" /><figcaption>Fig. 4. Strain and temperature distribution of the model with preheating along the nosing length</figcaption></figure>

If the preheat length is shortened to half of the original length, the process can be executed successfully. An illustration of a successful process with a short preheat length is presented in Figure 5. In this case, the entire hot region entered the die rapidly due to its shortened length. Consequently, a significant portion of the process load was borne by the cold region. As a result, no premature buckling failure is observed in this case, thanks to the high buckling resistance of the cold section.

<figure><img src="image_2202.png" /><figcaption>Fig. 5. Strain and temperature distribution of the model with preheating along half of the nosing length</figcaption></figure>

The process loads for long and short preheated workpieces are presented in Figure 6. Preheating the long portion of the workpiece resulted in buckling failure at 65 tons, whereas the workpiece with a shorter preheating region can withstand 250 tons of process loads.
## Results and Discussion

Effect of the Preheat Length on the Process. Determining the preheat length in the hot nosing process is crucial to prevent premature failures. If the preheat length is longer than necessary, the workpiece may buckle in the hot region that has not been formed yet. The analysis results, in which the length of heating region was excessive, are given in Figure 4. Buckling failure is evident in the hot region that has not yet entered the die, as illustrated in Figure 4. The hot region outside the die was unable to withstand the instantaneous process load, leading to buckling.

<figure><img src="image_2192.png" /><figcaption>Fig. 4. Strain and temperature distribution of the model with preheating along the nosing length</figcaption></figure>

If the preheat length is shortened to half of the original length, the process can be executed successfully. An illustration of a successful process with a short preheat length is presented in Figure 5. In this case, the entire hot region entered the die rapidly due to its shortened length. Consequently, a significant portion of the process load was borne by the cold region. As a result, no premature buckling failure is observed in this case, thanks to the high buckling resistance of the cold section.

<figure><img src="image_2193.png" /><figcaption>Fig. 5. Strain and temperature distribution of the model with preheating along half of the nosing length</figcaption></figure>

The process loads for long and short preheated workpieces are presented in Figure 6. Preheating the long portion of the workpiece resulted in buckling failure at 65 tons, whereas the workpiece with a shorter preheating region can withstand 250 tons of process loads.
<figure><img src="image_2203.png" /><figcaption>Fig. 6. The effect of preheat length on forming load</figcaption></figure>

Effect of the Friction Coefficient on the Process. In addition to tip thickness, forming load is dependent on the friction between workpiece and die. Therefore, it is important to examine the impact of the friction factor on the process.

As depicted in Figure 7, there were no evident failures in the processes with low friction coefficients (m=0.05 and m=0.1), but a buckling failure occurred at a high friction coefficient (m=0.2). The observation of a buckling failure in the process with high friction coefficient is attributed to the high forming load. When the forming load exceeded the buckling load capacity of the workpiece, the failure occurred in the cold section of the workpiece.

<figure><img src="image_2204.png" /><figcaption>Fig. 7. Equivalent plastic strain according to friction coefficient, a) m: 0.05, b) m: 0.1, c) m: 0.2</figcaption></figure>

The process loads for the different friction coefficients are presented in Figure 8. It can be seen from the figure that increasing the friction coefficient required a higher nosing forming load. Even if the increase in process load has an effect on the cooling of the workpiece during its contact with the nosing die, the cooling amount was accepted as the same for each friction condition and the cooling situation was not examined in detail within the scope of this study. The forming load was 270 tons in the process with high friction coefficient (0.2), while it was 220 tons in the process with low friction coefficient (0.05, 0.1). The high forming load obtained in the process with a high friction coefficient provides an explanation for the buckling failure observed in this process.
<figure><img src="image_2194.png" /><figcaption>Fig. 6. The effect of preheat length on forming load</figcaption></figure>

Effect of the Friction Coefficient on the Process. In addition to tip thickness, forming load is dependent on the friction between workpiece and die. Therefore, it is important to examine the impact of the friction factor on the process.

As depicted in Figure 7, there were no evident failures in the processes with low friction coefficients (m=0.05 and m=0.1), but a buckling failure occurred at a high friction coefficient (m=0.2). The observation of a buckling failure in the process with high friction coefficient is attributed to the high forming load. When the forming load exceeded the buckling load capacity of the workpiece, the failure occurred in the cold section of the workpiece.

<figure><img src="image_2195.png" /><figcaption>Fig. 7. Equivalent plastic strain according to friction coefficient, a) m: 0.05, b) m: 0.1, c) m: 0.2</figcaption></figure>

The process loads for the different friction coefficients are presented in Figure 8. It can be seen from the figure that increasing the friction coefficient required a higher nosing forming load. Even if the increase in process load has an effect on the cooling of the workpiece during its contact with the nosing die, the cooling amount was accepted as the same for each friction condition and the cooling situation was not examined in detail within the scope of this study. The forming load was 270 tons in the process with high friction coefficient (0.2), while it was 220 tons in the process with low friction coefficient (0.05, 0.1). The high forming load obtained in the process with a high friction coefficient provides an explanation for the buckling failure observed in this process.
<figure><img src="image_2205.png" /><figcaption>Fig. 8. The effect of friction coefficient on forming load</figcaption></figure>

In terms of obtaining final product thickness, friction coefficient is a significant parameter in the process. As given in Figure 9, increasing the friction coefficient leads to the wall thickness increases.

<figure><img src="image_2206.png" /><figcaption>Fig. 9. The effect of friction coefficient on workpiece wall thickness increase</figcaption></figure>

Two-Stage Forming Approach. For minimizing the forming load with a high friction coefficient, preferring the two-stage forming method is a productive way to decrease process loading. Investigating the friction coefficient of 0.2 in the scope of this study was handled again to see with two-stage forming approach. In both the first and the second stages of the process, workpiece formed successfully without buckling damage as shown in Figure 10.
<figure><img src="image_2196.png" /><figcaption>Fig. 8. The effect of friction coefficient on forming load</figcaption></figure>

In terms of obtaining final product thickness, friction coefficient is a significant parameter in the process. As given in Figure 9, increasing the friction coefficient leads to the wall thickness increases.

<figure><img src="image_2197.png" /><figcaption>Fig. 9. The effect of friction coefficient on workpiece wall thickness increase</figcaption></figure>

Two-Stage Forming Approach. For minimizing the forming load with a high friction coefficient, preferring the two-stage forming method is a productive way to decrease process loading. Investigating the friction coefficient of 0.2 in the scope of this study was handled again to see with two-stage forming approach. In both the first and the second stages of the process, workpiece formed successfully without buckling damage as shown in Figure 10.
<figure><img src="image_2207.png" /><figcaption>Fig. 10. Equivalent plastic strain of workpiece in the two-stage forming, a) First stage, b) Second stage</figcaption></figure>

The process loads are given in Figure 11 to explain the effect of two-stage forming. Whereas the nosing process load using single-stage forming approach exceeds 275 ton, the load is under 240 ton in the two-stage forming approach. It is demonstrated that two or multi-stage forming method is the best way of minimizing process loads even if the friction coefficient is over 0.2.

<figure><img src="image_2208.png" /><figcaption>Fig. 11. The process loads in the two-stage forming</figcaption></figure>

## Summary

Investigating the influence of process parameters, which are friction coefficient and workpiece preheating length, and single or two-stage pressing on both the forming load and the occurrence of workpiece failure within the context of the hot nosing process were conducted. The obtained results can be summarized as follows:
<figure><img src="image_2198.png" /><figcaption>Fig. 10. Equivalent plastic strain of workpiece in the two-stage forming, a) First stage, b) Second stage</figcaption></figure>

The process loads are given in Figure 11 to explain the effect of two-stage forming. Whereas the nosing process load using single-stage forming approach exceeds 275 ton, the load is under 240 ton in the two-stage forming approach. It is demonstrated that two or multi-stage forming method is the best way of minimizing process loads even if the friction coefficient is over 0.2.

<figure><img src="image_2199.png" /><figcaption>Fig. 11. The process loads in the two-stage forming</figcaption></figure>

## Summary

Investigating the influence of process parameters, which are friction coefficient and workpiece preheating length, and single or two-stage pressing on both the forming load and the occurrence of workpiece failure within the context of the hot nosing process were conducted. The obtained results can be summarized as follows:
* In the circumstance that the length of the heating region was excessive, the hot region outside the die was unable to withstand the instantaneous process load, leading to buckling. If the preheat length is shortened to half of the original length, the process can be executed successfully. As a result, no premature buckling failure is observed in this case, thanks to the high buckling resistance of the cold section.

* There were no evident failures in the processes with low friction coefficients, but a buckling failure occurred at a high friction coefficient. The observation of a buckling failure in the process with high friction coefficient is attributed to the high forming load. When the forming load exceeded the buckling load capacity of the workpiece, the failure occurred in the cold section of the workpiece.

* Increasing the friction coefficient leads to the wall thickness increases.

* The two-stage forming approach demonstrates that the two or multi-stage forming method is the best way of minimizing process loads even if the friction coefficient is constant.

## Acknowledgement

This work was supported by the Repkon Machine and Tool Industry and Trade Inc.

## References

[1] Lu, Y. H. (2005). Study of preform and loading rate in the tube nosing process by spherical die. Computer methods in applied mechanics and engineering, 194(25-26), 2839-2858.
https://doi.org/10.1016/j.cma.2004.07.032

[2] Chang, C. C., Pan, S. K., & Yu, K. T. (2019). Forming limit in the nosing process of micro copper cups. Key Engineering Materials, 794, 121-126.
https://doi.org/10.4028/www.scientific.net/KEM.794.121

[3] Gouveia, B. P. P., Alves, M. L., Rosa, P. A. R., & Martins, P. A. F. (2006). Compression beading and nosing of thin-walled tubes using a die: experimental and theoretical investigation. international Journal of Mechanics and Materials in Design, 3, 7-16.
https://doi.org/10.1007/s10999-006-9009-8

[4] Manabe, K. I., & Nishimura, H. (1984). Nosing of thin-walled tubes by circular curved dies. Journal of mechanical working technology, 10(3), 287-298.
https://doi.org/10.1016/0378-3804(84)90044-5

[5] Tang, M. C., & Kobayashi, S. (1982). An Investigation of the Shell Nosing Process by the Finite-Element Method. Part 2: Nosing at Elevated Temperatures (Hot Nosing).
https://doi.org/10.1115/1.3185835

[6] Carlson, R. K., (1944). "An Experimental Investigation of the Nosing Shells." Forging of Steel Shells, ASME Trans., p.45.

[7] Saunders, N., Guo, U. K. Z., Li, X., Miodownik, A. P., & Schillé, J. P. (2003). Using JMatPro to model materials properties and behavior. Jom, 55(12), 60-65.
https://doi.org/10.1007/s11837-003-0013-2
* In the circumstance that the length of the heating region was excessive, the hot region outside the die was unable to withstand the instantaneous process load, leading to buckling. If the preheat length is shortened to half of the original length, the process can be executed successfully. As a result, no premature buckling failure is observed in this case, thanks to the high buckling resistance of the cold section.

* There were no evident failures in the processes with low friction coefficients, but a buckling failure occurred at a high friction coefficient. The observation of a buckling failure in the process with high friction coefficient is attributed to the high forming load. When the forming load exceeded the buckling load capacity of the workpiece, the failure occurred in the cold section of the workpiece.

* Increasing the friction coefficient leads to the wall thickness increases.

* The two-stage forming approach demonstrates that the two or multi-stage forming method is the best way of minimizing process loads even if the friction coefficient is constant.

## Acknowledgement

This work was supported by the Repkon Machine and Tool Industry and Trade Inc.

## References

[1] Lu, Y. H. (2005). Study of preform and loading rate in the tube nosing process by spherical die. Computer methods in applied mechanics and engineering, 194(25-26), 2839-2858.
https://doi.org/10.1016/j.cma.2004.07.032

[2] Chang, C. C., Pan, S. K., & Yu, K. T. (2019). Forming limit in the nosing process of micro copper cups. Key Engineering Materials, 794, 121-126.
https://doi.org/10.4028/www.scientific.net/KEM.794.121

[3] Gouveia, B. P. P., Alves, M. L., Rosa, P. A. R., & Martins, P. A. F. (2006). Compression beading and nosing of thin-walled tubes using a die: experimental and theoretical investigation. international Journal of Mechanics and Materials in Design, 3, 7-16.
https://doi.org/10.1007/s10999-006-9009-8

[4] Manabe, K. I., & Nishimura, H. (1984). Nosing of thin-walled tubes by circular curved dies. Journal of mechanical working technology, 10(3), 287-298.
https://doi.org/10.1016/0378-3804(84)90044-5

[5] Tang, M. C., & Kobayashi, S. (1982). An Investigation of the Shell Nosing Process by the Finite-Element Method. Part 2: Nosing at Elevated Temperatures (Hot Nosing).
https://doi.org/10.1115/1.3185835

[6] Carlson, R. K., (1944). "An Experimental Investigation of the Nosing Shells." Forging of Steel Shells, ASME Trans., p.45.

[7] Saunders, N., Guo, U. K. Z., Li, X., Miodownik, A. P., & Schillé, J. P. (2003). Using JMatPro to model materials properties and behavior. Jom, 55(12), 60-65.
https://doi.org/10.1007/s11837-003-0013-2
# Eigenstrain method in simulations of laser peen forming of curved surfaces

POLTL Dominik<fnref n="1" />,ᵃ,*, TEJA SALA Siva<fnref n="2" />,ᵇ, KASHAEV Nikolai<fnref n="2" />,ᶜ and KLUSEMANN Benjamin<fnref n="1" />,<fnref n="2" />,ᵈ

**Keywords:** Finite Element Analysis, Laser Peen Forming, Eigenstrain, Bending, Curvature

**Abstract.** The eigenstrain ansatz allows for the efficient simulation of large-scale applications of Laser Peen Forming (LPF) while being subject to geometric constraints. A setup to investigate the viability of the method for non-uniform curvature is proposed. A small-scale laser processing is simulated on cylinder shells of given curvature. Eigenstrains are determined in representative cells and mapped onto a second cylinder shell with different curvature to simulate a large-scale processing operation. The eigenstrains result in changes in local curvature. This is repeated for four curvatures. The resulting data is used to investigate the dependence of the induced curvature change on the origin geometry of the eigenstrains. A determined regression relation provides insight into the feasibility of the eigenstrain ansatz beyond its constraints.

## Introduction

The titanium alloy Ti-6Al-4V offers the aerospace industry a material with a high strength to weight ratio and excellent high temperature stability. This makes it the most used titanium alloy, but poses some challenges due to its difficult formability [1]. Laser Shock Peening (LSP) and its adaption for small forming operations, Laser Peen Forming (LPF) offers a highly controllable processing method: A pulsed high-energy laser hits a specimen perpendicular to its surface in a repeated, structured manner, thus creating a peening pattern that consists of a high number of individual shots based on parameters such as the shot diameter, the overlap of consecutive shots as well as the dimensions of the peened region. Each individual laser pulse, with an energy density in the order of magnitude GW/cm<fnref n="2" /> and a duration in the ns scale, generates a rapidly expanding plasma on specimen's surface. This plasma in a water confined regime induces mechanical shock waves that propagate through the specimen, causing dynamic yielding and resulting in a plastic strain field [2], [3]. The induced local deformation leads to difference between initial and post-processing curvature.

For modelling of LPF, the direct simulation of a large number of peening shots is computationally very costly. An efficient shortcut for simulation of LPF is the use of the eigenstrain method, as presented by Hu and Grandhi [4]. Investigating laser peening of planar samples, Hu and Grandhi [4] used the periodicity of the in-plane plastic strain field to determine eigenstrains $ε_p$ in representative volumes. Eigenstrains are strains that persist in a specimen without external loading, e.g. thermal strains. The eigenstrains are associated with representative positions in a peening patch. Once determined in a small-scale simulation, the eigenstrains can be mapped onto a large-scale geometry whose peening pattern can be much bigger than on the small-scale geometry. On this larger pattern, the eigenstrains $ε_{ij}$ are introduced as anisotropic thermal expansion coefficients $α_{ij}$ via

<figure><img src="image_2209.png" /></figure>

<footnote n="1">¹Institute of Production Technology and Systems, Leuphana University Lüneburg, Germany</footnote>

<footnote n="2">²Institute of Materials Mechanics, Helmholtz-Zentrum Hereon, Geesthacht, Germany</footnote>

<footnote>ᵃdominik-poeltl@leuphana.de, ᵇsiva.sala@hereon.de, ᶜnikolai.kashaev@hereon.de,</footnote>

<footnote>ᵈbenjamin.klusemann@leuphana.de</footnote>
# Eigenstrain method in simulations of laser peen forming of curved surfaces

POLTL Dominik<fnref n="1" />,ᵃ*, TEJA SALA Siva<fnref n="2" />,ᵇ, KASHAEV Nikolai<fnref n="2" />,ᶜ and KLUSEMANN Benjamin<fnref n="1" />,<fnref n="2" />,ᵈ

**Keywords:** Finite Element Analysis, Laser Peen Forming, Eigenstrain, Bending, Curvature

**Abstract.** The eigenstrain ansatz allows for the efficient simulation of large-scale applications of Laser Peen Forming (LPF) while being subject to geometric constraints. A setup to investigate the viability of the method for non-uniform curvature is proposed. A small-scale laser processing is simulated on cylinder shells of given curvature. Eigenstrains are determined in representative cells and mapped onto a second cylinder shell with different curvature to simulate a large-scale processing operation. The eigenstrains result in changes in local curvature. This is repeated for four curvatures. The resulting data is used to investigate the dependence of the induced curvature change on the origin geometry of the eigenstrains. A determined regression relation provides insight into the feasibility of the eigenstrain ansatz beyond its constraints.

## Introduction

The titanium alloy Ti-6Al-4V offers the aerospace industry a material with a high strength to weight ratio and excellent high temperature stability. This makes it the most used titanium alloy, but poses some challenges due to its difficult formability [1]. Laser Shock Peening (LSP) and its adaption for small forming operations, Laser Peen Forming (LPF) offers a highly controllable processing method: A pulsed high-energy laser hits a specimen perpendicular to its surface in a repeated, structured manner, thus creating a peening pattern that consists of a high number of individual shots based on parameters such as the shot diameter, the overlap of consecutive shots as well as the dimensions of the peened region. Each individual laser pulse, with an energy density in the order of magnitude GW/cm<fnref n="2" /> and a duration in the ns scale, generates a rapidly expanding plasma on specimen's surface. This plasma in a water confined regime induces mechanical shock waves that propagate through the specimen, causing dynamic yielding and resulting in a plastic strain field [2], [3]. The induced local deformation leads to difference between initial and post-processing curvature.

For modelling of LPF, the direct simulation of a large number of peening shots is computationally very costly. An efficient shortcut for simulation of LPF is the use of the eigenstrain method, as presented by Hu and Grandhi [4]. Investigating laser peening of planar samples, Hu and Grandhi [4] used the periodicity of the in-plane plastic strain field to determine eigenstrains $ε_p$ in representative volumes. Eigenstrains are strains that persist in a specimen without external loading, e.g. thermal strains. The eigenstrains are associated with representative positions in a peening patch. Once determined in a small-scale simulation, the eigenstrains can be mapped onto a large-scale geometry whose peening pattern can be much bigger than on the small-scale geometry. On this larger pattern, the eigenstrains $ε_{ij}$ are introduced as anisotropic thermal expansion coefficients $α_{ij}$ via

<figure><img src="image_2200.png" /></figure>

<footnote n="1">¹Institute of Production Technology and Systems, Leuphana University Lüneburg, Germany</footnote>

<footnote n="2">²Institute of Materials Mechanics, Helmholtz-Zentrum Hereon, Geesthacht, Germany</footnote>

<footnote>ᵃdominik-poeltl@leuphana.de, ᵇsiva.sala@hereon.de, ᶜnikolai.kashaev@hereon.de,</footnote>

<footnote>ᵈbenjamin.klusemann@leuphana.de</footnote>
$$ \begin{bmatrix} \varepsilon_{rr} & \varepsilon_{rt} & \varepsilon_{ra} \\ \varepsilon_{tt} & \varepsilon_{ta} & \varepsilon_{ta} \\ \varepsilon_{aa} & \varepsilon_{aa} & \varepsilon_{aa} \end{bmatrix} = \Delta T \begin{bmatrix} \alpha_{rr} & \alpha_{rt} & \alpha_{ra} \\ \alpha_{tt} & \alpha_{ta} & \alpha_{ta} \\ \alpha_{aa} & \alpha_{aa} & \alpha_{aa} \end{bmatrix} \qquad (1) $$

in a notation for cylindrical coordinates and $\Delta T = 1K$. The use of the ansatz is not exclusive to LPF, as Faucheux et al. [5] demonstrated it for shot peening. Keller et al. [6] used the eigenstrain ansatz to investigate the influence of LSP on fatigue crack propagation rate. Sala et al. [7] used the method to build a database of arbitrary peening patterns to train a neural network on the prediction of the pattern based on the deformation. However, on the use of eigenstrains on curved geometries, only few studies exist. For instance, Cai and Zhang [8] transformed the determined eigenstrains with respect to the normal to an arbitrarily curved surface.

In this work, the term *source* is coined for the small-scale geometry that is the origin of the eigenstrains while the term *target* is coined for the large-scale geometry that is the destination of the eigenstrains. The main focus of this work will be the investigation of the influence of the difference of local curvature between the source and the target geometry. Other than a plane sheet, which trivially has the uniform curvature $\kappa_{planar} = 0 \text{ mm}^{-1}$, only the cylinder shell features a uniform curvature: For a cylinder shell of outer radius $r$, the curvature is

$$ \kappa(r) = r^{-1}. \qquad (2) $$

## Simulation setup

*Specimen geometry.* In this work, five discrete values of initial curvature are investigated, namely $\kappa \in \{10, 12, 14, 16, 20\} \cdot 10^{-3} \text{mm}^{-1}$. This corresponds to five discrete radii of cylinder shell cuts. For each curvature, the source geometry is a 50 mm × 50 mm × 1 mm sheet bent to the corresponding outer radius. The target geometry is a 80 mm × 20 mm × 1 mm sheet that features the same radius. In order to describe the resulting strain fields via a common cylindrical coordinate system, all geometries are centered around the same origin and oriented along the same axis. As boundary conditions, the source geometry features one slim face with dimensions 50 mm × 1 mm with a prescribed encastre, as shown in Fig. 1. In contrast, the target geometry is fixed in the initial length of 10 mm. Figure 1 shows source and target geometries for two exemplary curvatures, including the corresponding boundary conditions. Using the commercial software package for Finite Element Method (FEM) simulations ABAQUS, continuous 8-node 3D elements with reduced integration and hourglass control (C3D8R) are used to mesh all geometries. For the source geometry, the peened region is meshed with and in-plane average element edge length of 0.1 mm. The target geometry features an in plane average element edge length of 0.2 mm. For both geometries, the thickness is discretized with 20 layers of elements, guaranteeing a fine enough discretization of the in-depth stress and strain fields.

*Peening pattern.* In the center of each source geometry, a peening pattern consisting of 4×4 laser shots is modelled. Each shot has a diameter of 3 mm. In order to model the angle of attack in industrial LSP systems, each circular spot is distorted to an ellipsis. The pattern exhibits an alternating row offset. Due to its structured generation, the pattern exhibits periodicity. This allows for the identification of representative cells. In this work, nine patches, namely one center patch (C) and eight wind rose patches (N, NE, E and so on), make up the set of the representative cells. The dimension and location of each cell is geometrically determined by the spot diameter, the overlap ratio and the alternating offset. Figure 3 a) features 16 shots and the induced origin and dimension of the representative cells. Figure 3 b) indicates the tessellation of the representative
$$ \begin{bmatrix} \varepsilon_{rr} & \varepsilon_{rt} & \varepsilon_{ra} \\ \varepsilon_{tt} & \varepsilon_{ta} & \varepsilon_{ta} \\ \varepsilon_{aa} & \varepsilon_{aa} & \varepsilon_{aa} \end{bmatrix} = \Delta T \begin{bmatrix} \alpha_{rr} & \alpha_{rt} & \alpha_{ra} \\ \alpha_{tt} & \alpha_{ta} & \alpha_{ta} \\ \alpha_{aa} & \alpha_{aa} & \alpha_{aa} \end{bmatrix} \qquad (1) $$

in a notation for cylindrical coordinates and $\Delta T = 1K$. The use of the ansatz is not exclusive to LPF, as Faucheux et al. [5] demonstrated it for shot peening. Keller et al. [6] used the eigenstrain ansatz to investigate the influence of LSP on fatigue crack propagation rate. Sala et al. [7] used the method to build a database of arbitrary peening patterns to train a neural network on the prediction of the pattern based on the deformation. However, on the use of eigenstrains on curved geometries, only few studies exist. For instance, Cai and Zhang [8] transformed the determined eigenstrains with respect to the normal to an arbitrarily curved surface.

In this work, the term *source* is coined for the small-scale geometry that is the origin of the eigenstrains while the term *target* is coined for the large-scale geometry that is the destination of the eigenstrains. The main focus of this work will be the investigation of the influence of the difference of local curvature between the source and the target geometry. Other than a plane sheet, which trivially has the uniform curvature $\kappa_{planar} = 0 \text{ mm}^{-1}$, only the cylinder shell features a uniform curvature: For a cylinder shell of outer radius $r$, the curvature is

$$ \kappa(r) = r^{-1}. \qquad (2) $$

## Simulation setup

*Specimen geometry.* In this work, five discrete values of initial curvature are investigated, namely $\kappa \in \{10, 12, 14, 16, 20\} \cdot 10^{-3} \text{mm}^{-1}$. This corresponds to five discrete radii of cylinder shell cuts. For each curvature, the source geometry is a 50 mm × 50 mm × 1 mm sheet bent to the corresponding outer radius. The target geometry is a 80 mm × 20 mm × 1 mm sheet that features the same radius. In order to describe the resulting strain fields via a common cylindrical coordinate system, all geometries are centered around the same origin and oriented along the same axis. As boundary conditions, the source geometry features one slim face with dimensions 50 mm × 1 mm with a prescribed encastre, as shown in Fig. 1. In contrast, the target geometry is fixed in the initial length of 10 mm. Figure 1 shows source and target geometries for two exemplary curvatures, including the corresponding boundary conditions. Using the commercial software package for Finite Element Method (FEM) simulations ABAQUS, continuous 8-node 3D elements with reduced integration and hourglass control (C3D8R) are used to mesh all geometries. For the source geometry, the peened region is meshed with and in-plane average element edge length of 0.1 mm. The target geometry features an in plane average element edge length of 0.2 mm. For both geometries, the thickness is discretized with 20 layers of elements, guaranteeing a fine enough discretization of the in-depth stress and strain fields.

*Peening pattern.* In the center of each source geometry, a peening pattern consisting of 4×4 laser shots is modelled. Each shot has a diameter of 3 mm. In order to model the angle of attack in industrial LSP systems, each circular spot is distorted to an ellipsis. The pattern exhibits an alternating row offset. Due to its structured generation, the pattern exhibits periodicity. This allows for the identification of representative cells. In this work, nine patches, namely one center patch (C) and eight wind rose patches (N, NE, E and so on), make up the set of the representative cells. The dimension and location of each cell is geometrically determined by the spot diameter, the overlap ratio and the alternating offset. Figure 3 a) features 16 shots and the induced origin and dimension of the representative cells. Figure 3 b) indicates the tessellation of the representative
cells on the surface of the target geometry: While corners stay unchanged in their dimension, edges get extended in one dimension and the center patch gets extended in two dimensions.

<figure><img src="image_2210.png" /><figcaption>Figure 1: Investigated geometries with boundary conditions; a) source geometry of lowest curvature, b) source geometry with highest curvature, c) target geometry with lowest curvature, d) target geometry with highest curvature.</figcaption></figure>

Pulse modelling. For an individual laser pulse, the pressure p, acting on the surface can be expressed as

$$p = p(x, t), \qquad (3)$$

where vector x is a coordinate point on the surface and t is the time. In this work, the spatial distribution of the pressure pulse is assumed uniform, leaving only the temporal evolution. To model the rapid pressure rise caused by the plasma expansion and the following pressure decay, two time parameters t<sub>amp</sub> and t<sub>70</sub> were introduced, yielding

$$ \begin{aligned} p(t_{amp}) &= p_{max} & \square \\ p(t_{70}) &= 0.7 p_{max} & \square \\ p(t) &\propto \exp(-t) & \text{for } t > t_{70}. \end{aligned} \qquad (4) $$
cells on the surface of the target geometry: While corners stay unchanged in their dimension, edges get extended in one dimension and the center patch gets extended in two dimensions.

<figure><img src="image_2201.png" /><figcaption>Figure 1: Investigated geometries with boundary conditions; a) source geometry of lowest curvature, b) source geometry with highest curvature, c) target geometry with lowest curvature, d) target geometry with highest curvature.</figcaption></figure>

*Pulse modelling.* For an individual laser pulse, the pressure *p*, acting on the surface can be expressed as

$$p = p(x, t), \qquad (3)$$

where vector **x** is a coordinate point on the surface and *t* is the time. In this work, the spatial distribution of the pressure pulse is assumed uniform, leaving only the temporal evolution. To model the rapid pressure rise caused by the plasma expansion and the following pressure decay, two time parameters *t*<sub>amp</sub> and *t*<sub>70</sub> were introduced, yielding

$$ \begin{aligned} p(t_{amp}) &= p_{max} & \square \\ p(t_{70}) &= 0.7 p_{max} & \square \\ p(t) &\propto \exp(-t) & \text{for } t > t_{70}. \end{aligned} \qquad (4) $$
<figure><img src="image_2211.png" /><figcaption>Figure 3: a) Representative cells with underlying peening pattern on the source geometry, b) tessellation with representative cells on the target geometry.</figcaption></figure>

<figure><img src="image_2212.png" /><figcaption>Figure 2: Temporal evolution of the pressure pulse.</figcaption></figure>

The maximum pressure $p_{max}$ applied in all simulations is chosen such that dynamic yielding results in deformations of >1 mm at the specimen's free tip. The exponential decay is truncated once 5% of the maximum pressure is reached. The timespan between reaching half maximum in the increasing phase and again in the decreasing phase is called full width half maximum (FWHM) and is well established in the literature. For the applied pulse it is 100 ns. Figure 2 shows the plot of this characteristic.

*Material Model.* The constitutive modelling of Ti-6Al-4V must account for the extremely high strain rate during laser processing [9]. The Johnson-Cook material model with neglected temperature term models the viscoplastic material behavior [10]. The governing equation reads

$$ \sigma_y = [A_{JC} + B_{JC}\epsilon_{BC}^{n_{JC}}] \left[1 + C_{JC} \ln\left(\frac{\dot{\epsilon}_p}{\dot{\epsilon}_{p,0}}\right)\right], \quad (5) $$

where $\epsilon_p$ respective $\dot{\epsilon}_p$ is the equivalent plastic strain and its rate while $A_{JC}$, $B_{JC}$, $C_{JC}$, $n_{JC}$ and $\dot{\epsilon}_{p,0}$ are material parameters, listed for Ti-6Al-4V in Table 1.
<figure><img src="image_2202.png" /><figcaption>Figure 3: a) Representative cells with underlying peening pattern on the source geometry, b) tessellation with representative cells on the target geometry.</figcaption></figure>

<figure><img src="image_2203.png" /><figcaption>Figure 2: Temporal evolution of the pressure pulse.</figcaption></figure>

The maximum pressure $p_{max}$ applied in all simulations is chosen such that dynamic yielding results in deformations of >1 mm at the specimen's free tip. The exponential decay is truncated once 5% of the maximum pressure is reached. The timespan between reaching half maximum in the increasing phase and again in the decreasing phase is called full width half maximum (FWHM) and is well established in the literature. For the applied pulse it is 100 ns. Figure 2 shows the plot of this characteristic.

*Material Model.* The constitutive modelling of Ti-6Al-4V must account for the extremely high strain rate during laser processing [9]. The Johnson-Cook material model with neglected temperature term models the viscoplastic material behavior [10]. The governing equation reads

$$ \sigma_y = [A_{JC} + B_{JC} \varepsilon_{BC}^{n_{JC}}] \left[ 1 + C_{JC} \ln \left( \frac{\dot{\varepsilon}_p}{\dot{\varepsilon}_{p,0}} \right) \right], \quad (5) $$

where $\varepsilon_p$ respective $\dot{\varepsilon}_p$ is the equivalent plastic strain and its rate while $A_{JC}$, $B_{JC}$, $C_{JC}$, $n_{JC}$ and $\dot{\varepsilon}_{p,0}$ are material parameters, listed for Ti-6Al-4V in Table 1.
<table><caption>Table 1: Elastic and Johnson-Cook material parameters for Ti-6Al-4V [11]</caption><tr><td>Density ρ, [g/cm³]</td><td>Young's modulus E, [MPa]</td><td>Poisson's ratio ν, [-]</td><td>Yield strength A<sub>JC</sub>, [MPa]</td></tr><tr><td>4.47</td><td>108500</td><td>0.33</td><td>728.7</td></tr><tr><td>Strain hardening coefficient B<sub>JC</sub>, [MPa]</td><td>Strain hardening exponent n<sub>JC</sub>, [-]</td><td>Strain rate hardening coefficient C<sub>JC</sub>, [-]</td><td>Reference strain rate ε̇<sub>p,0</sub>, [s⁻¹]</td></tr><tr><td>498.4</td><td>0.28</td><td>28·10⁻³</td><td>1·10⁻⁵</td></tr></table>

*Source simulation.* The peening process is simulated using ABAQUS/Explicit. After each pressure pulse, a relaxation phase of 50 µs ensures the decay of transient behavior in the stress and strain fields. In all source simulations, the 16 spots are exposed to the pressure pulse in a spatial zig-zag order, meaning: The four shots of line 1 are irradiated from left to right; afterwards, the four shots of line 2 are irradiated from right to left; and so on. This is looped three times, resulting in three simulated peening sequences. After this explicit simulation, an implicit simulation is run

<figure><img src="image_2213.png" /><figcaption>Figure 4: Tangential plastic strains for a) center and corners of wind rose patches, b) edges of wind rose patches (scale of a) applies) and c) 4 exemplary layers of center patch (scale of a) applies); d) radial and tangential plastic strains through thickness for center patch.</figcaption></figure>
<table><caption>Table 1: Elastic and Johnson-Cook material parameters for Ti-6Al-4V [11]</caption><thead><tr><th></th><th>Young's modulus E, [MPa]</th><th>Poisson's ratio v, [-]</th><th>Yield strength A<sub>JC</sub>, [MPa]</th></tr></thead><tbody><tr><td>Density ρ, [g/cm<sup>3</sup>]</td><td>108500</td><td>0.33</td><td>728.7</td></tr><tr><td>Strain hardening coefficient B<sub>JC</sub>, [MPa]</td><td>Strain hardening exponent n<sub>JC</sub>, [-]</td><td>Strain rate hardening coefficient C<sub>JC</sub>, [-]</td><td>Reference strain rate ε̇<sub>p,0</sub>, [s<sup>-1</sup>]</td></tr><tr><td>498.4</td><td>0.28</td><td>28·10<sup>-3</sup></td><td>1·10<sup>-5</sup></td></tr></tbody></table>

*Source simulation.* The peening process is simulated using ABAQUS/Explicit. After each pressure pulse, a relaxation phase of 50 µs ensures the decay of transient behavior in the stress and strain fields. In all source simulations, the 16 spots are exposed to the pressure pulse in a spatial zig-zag order, meaning: The four shots of line 1 are irradiated from left to right; afterwards, the four shots of line 2 are irradiated from right to left; and so on. This is looped three times, resulting in three simulated peening sequences. After this explicit simulation, an implicit simulation is run

<figure><img src="image_2204.png" /><figcaption>Figure 4: Tangential plastic strains for a) center and corners of wind rose patches, b) edges of wind rose patches (scale of a) applies) and c) 4 exemplary layers of center patch (scale of a) applies); d) radial and tangential plastic strains through thickness for center patch.</figcaption></figure>
to reach equilibrium. The limited number of shots result in no significant deflection and therefore no change in the curvature of the source geometry.

*Eigenstrain mapping.* After the explicit and implicit simulation of the source geometry, the eigenstrains are determined in a mid-processing procedure. Each of the 9 representative cells is split into 20 in-depth slices, resulting a total of 180 slices. Based on the coordinate of the integration point, an element was assigned to one of the 180 slices or not. For all elements assigned to a certain slice, the plastic strain tensors were averaged using an element-volume-weighted averaging scheme. This resulted in one eigenstrain tensor for the whole slice and 180 eigenstrain tensors for the peening pattern. Figure 4 a) and b) show a stacking of the 20 in-depth cells for the 9 windrose patches while c) tries to indicate the stacking of 20 cells by showing 4 exemplary cells. Figure 4 d) shows the profile of the radial as well as tangential plastic strains for the center patch over all 20 slices. On the target geometry, based on the larger-scale peening pattern, the origin and dimensions of the representative cells were calculated. All elements were assigned to a cell according to the coordinate of the integration point as depicted in Figure 3 b). The elements were assigned the averaged eigenstrain as an anisotropic thermal expansion coefficient according to Eq. 1.

*Target simulation.* Once the eigenstrains have been introduced as thermal expansion coefficients, the simulation of a temperature increase of $\Delta T = 1K$ results in the deflection of the geometry as if a large number of individual shots in the large-scale peening pattern have been simulated. A curve $c(s)$ on the surface of the probe in the axial middle is determined based on data from the mesh nodes, where $s \in [0, ..., 80] mm$ is the arc length of the curve starting from the encastre. In a cartesian coordinate system, the coordinates of the curve after deformation are parametrized as $c(s) = (x(s), z(s))$ with the two scalar functions $x(s)$ and $z(s)$.

## Cross-mapping and curvature calculation

The eigenstrain ansatz presumes that the eigenstrains originate from a domain that has identical properties as the domain where they are applied, e.g. material, thickness, orientation, etc. In this work, this presumption is investigated by cross-mapping of eigenstrains from and onto geometries with different curvatures. The eigenstrains from each of the 5 source geometries can be mapped onto each of the 5 corresponding target geometries, resulting in 25 simulations.

Let $\kappa_s$ be the initial curvature of the source geometry and $\kappa_t$ be the initial curvature of the target geometry, then the source curvature difference

$$ \Delta\kappa_s = \kappa_t - \kappa_s \qquad (6) $$

is defined as the curvature difference between the destination and origin of the eigenstrains. Figure 5 a) depicts three examples for $\Delta\kappa_s$ less than, equal to and greater than zero. Once $c(s)$ is extracted from the simulation results, the two functions $x(s)$ and $z(s)$ enable the calculation of the local curvature $\kappa(s)$ as

$$ \kappa(s) = \frac{\dot{x}(s)\ddot{z}(s) - \ddot{x}(s)\dot{z}(s)}{(\dot{x}(s))^2 + (\dot{z}(s))^2} \qquad (7) $$

where discrete derivation is done via a central difference scheme. In terms of the arc length, the origin and dimension of the peening pattern is located at $s \in [17mm, ..., 71mm]$. In order to get one scalar curvature for the whole deflected surface, the curvature after deflection $\bar{\kappa}_t$ is calculated as average over all discrete values for the curvature in this interval. To finally describe a difference of curvature deflection, one last quantity needs to be introduced: Let $\bar{\kappa}_{t,0}$ be the curvature after
to reach equilibrium. The limited number of shots result in no significant deflection and therefore no change in the curvature of the source geometry.

*Eigenstrain mapping.* After the explicit and implicit simulation of the source geometry, the eigenstrains are determined in a mid-processing procedure. Each of the 9 representative cells is split into 20 in-depth slices, resulting a total of 180 slices. Based on the coordinate of the integration point, an element was assigned to one of the 180 slices or not. For all elements assigned to a certain slice, the plastic strain tensors were averaged using an element-volume-weighted averaging scheme. This resulted in one eigenstrain tensor for the whole slice and 180 eigenstrain tensors for the peening pattern. Figure 4 a) and b) show a stacking of the 20 in-depth cells for the 9 windrose patches while c) tries to indicate the stacking of 20 cells by showing 4 exemplary cells. Figure 4 d) shows the profile of the radial as well as tangential plastic strains for the center patch over all 20 slices. On the target geometry, based on the larger-scale peening pattern, the origin and dimensions of the representative cells were calculated. All elements were assigned to a cell according to the coordinate of the integration point as depicted in Figure 3 b). The elements were assigned the averaged eigenstrain as an anisotropic thermal expansion coefficient according to Eq. 1.

*Target simulation.* Once the eigenstrains have been introduced as thermal expansion coefficients, the simulation of a temperature increase of $\Delta T = 1K$ results in the deflection of the geometry as if a large number of individual shots in the large-scale peening pattern have been simulated. A curve $c(s)$ on the surface of the probe in the axial middle is determined based on data from the mesh nodes, where $s \in [0, ..., 80] mm$ is the arc length of the curve starting from the encastre. In a cartesian coordinate system, the coordinates of the curve after deformation are parametrized as $c(s) = (x(s), z(s))$ with the two scalar functions $x(s)$ and $z(s)$.

### Cross-mapping and curvature calculation

The eigenstrain ansatz presumes that the eigenstrains originate from a domain that has identical properties as the domain where they are applied, e.g. material, thickness, orientation, etc. In this work, this presumption is investigated by cross-mapping of eigenstrains from and onto geometries with different curvatures. The eigenstrains from each of the 5 source geometries can be mapped onto each of the 5 corresponding target geometries, resulting in 25 simulations.

Let $\kappa_s$ be the initial curvature of the source geometry and $\kappa_t$ be the initial curvature of the target geometry, then the source curvature difference

$$ \Delta\kappa_s = \kappa_t - \kappa_s \qquad (6) $$

is defined as the curvature difference between the destination and origin of the eigenstrains. Figure 5 a) depicts three examples for $\Delta\kappa_s$ less than, equal to and greater than zero. Once $c(s)$ is extracted from the simulation results, the two functions $x(s)$ and $z(s)$ enable the calculation of the local curvature $\kappa(s)$ as

$$ \kappa(s) = \frac{\dot{x}(s)\ddot{z}(s) - \ddot{x}(s)\dot{z}(s)}{(\dot{x}(s))^2 + (\dot{z}(s))^2} \qquad (7) $$

where discrete derivation is done via a central difference scheme. In terms of the arc length, the origin and dimension of the peening pattern is located at $s \in [17mm, ..., 71mm]$. In order to get one scalar curvature for the whole deflected surface, the curvature after deflection $\bar{\kappa}_t$ is calculated as average over all discrete values for the curvature in this interval. To finally describe a difference of curvature deflection, one last quantity needs to be introduced: Let $\bar{\kappa}_{t,0}$ be the curvature after
deflection for the case where the eigenstrains originated from a source geometry with the same curvature. Then the target curvature difference $\Delta\kappa_t$ as a function of $\Delta\kappa_s$ is defined as

$$ \Delta\kappa_t(\Delta\kappa_s) = \bar{\kappa}_t(\Delta\kappa_s) - \bar{\kappa}_{t,0}. \qquad (8) $$

Figure 5 b) depicts the logic behind this definition.

<figure><img src="image_2214.png" /><figcaption>Figure 5: a) The origins of the source curvature difference $\Delta\kappa_s$ and the target curvature difference $\Delta\kappa_t$.</figcaption></figure>

## Results

In order to find a relation between the mapping curvature difference between and the reached induced curvature difference, a regression of the data was determined. Two constraints for the regression were set: $\Delta\kappa_t(0) = 0$ and $\Delta\kappa_t(\epsilon) \neq \Delta\kappa_t(-\epsilon)$ for $\epsilon > 0$. The linear regression of all data $\Delta\kappa_t(\Delta\kappa_s) = m(\Delta\kappa_s) + b$ with $b = 0$ and a value of $m$ determined by a least squares error ansatz is chosen. This way, when the source curvature difference is zero, the target curvature difference is zero. The regression yields a value of the slope of $0 < m < 1$. Figure 6 plots the target curvature difference over the source curvature difference of the 25 simulations and the forced mean. The significant standard deviation of the data, indicated with the error bars, is due to the lack of preliminary smoothing before the discrete derivation and the following averaging process.

<figure><img src="image_2215.png" /><figcaption>Figure 6: Target curvature difference against source curvature difference with forced mean relation.</figcaption></figure>
deflection for the case where the eigenstrains originated from a source geometry with the same curvature. Then the target curvature difference Δκ_t as a function of Δκ_s is defined as

$$ \Delta\kappa_t(\Delta\kappa_s) = \bar{\kappa}_t(\Delta\kappa_s) - \bar{\kappa}_{t,0}. \qquad (8) $$

Figure 5 b) depicts the logic behind this definition.

<figure><img src="image_2205.png" /><figcaption>Figure 5: a) The origins of the source curvature difference Δκ_s and the target curvature difference Δκ_t.</figcaption></figure>

## Results

In order to find a relation between the mapping curvature difference between and the reached induced curvature difference, a regression of the data was determined. Two constraints for the regression were set: Δκ_t(0) = 0 and Δκ_t(ε) ≠ Δκ_t(-ε) for ε > 0. The linear regression of all data Δκ_t(Δκ_s) = m(Δκ_s) + b with b = 0 and a value of m determined by a least squares error ansatz is chosen. This way, when the source curvature difference is zero, the target curvature difference is zero. The regression yields a value of the slope of 0 < m < 1. Figure 6 plots the target curvature difference over the source curvature difference of the 25 simulations and the forced mean. The significant standard deviation of the data, indicated with the error bars, is due to the lack of preliminary smoothing before the discrete derivation and the following averaging process.

<figure><img src="image_2206.png" /><figcaption>Figure 6: Target curvature difference against source curvature difference with forced mean relation.</figcaption></figure>
The positive slope of the forced mean indicates the geometry dependence of the eigenstrains: For positive Δκs, the curvature change of the target geometry will be overstated; for negative Δκs, the curvature change of the target geometry will be understated. The value of m < 1 indicates a tolerance of the eigenstrain ansatz for deviations in the eigenstrain. While the data points stem from discrete values of cylinder shells with uniform curvature, the continuous relation is able to improve LPF simulations of geometries with a non-uniform, continuous curvature profile. In the following, two ways, how knowledge of this relation gives an advantage when simulating small forming operations, are outlined: Assume, a small forming operation on target geometry with a non-uniform curvature distribution $\tilde{\kappa}_t(s)$ should be simulated. If one eigenstrain tensor on a geometry with uniform curvature $\tilde{\kappa}_s$ is known, then the relation gives an estimate on the maximal error caused by using the one known eigenstrain tensor, explicitly

$$|\max(\Delta\kappa_t)| = m \max_s(|\tilde{\kappa}_t(s) - \tilde{\kappa}_s|) \qquad (9)$$

Also, for a given target geometry with a non-uniform curvature distribution $\tilde{\kappa}_t(s)$, if the error in overall shape prediction needs to be limited to a given value, then the relation dictates the number of eigenstrain tensors $n_\epsilon$ needed to stay below a corresponding target curvature difference $\tilde{\epsilon}$, namely

$$n_{\epsilon} = \left[ \frac{m \left( \max_{s} \tilde{\kappa}_{t}(s) - \min_{s} \tilde{\kappa}_{t}(s) \right)}{\tilde{\epsilon}} \right] \qquad (10)$$

The results are derived from geometries with a mono-directional curvature. No statement on geometries with multi-dimensional, non-trivial curvature can be made.

## Conclusions

In this work, the simulation workflow using the eigenstrain ansatz was set up to model LSP processing of thin-walled Ti-6Al-4V specimen. For a smaller scale geometry, a small number of laser shots were simulated and the resulting eigenstrains were determined based on representative cells defined by the peening pattern. The eigenstrains were mapped onto a larger scale geometry featuring a large scale pattern. The introduced eigenstrains result in a global deflection curve, for which local curvatures and a global quantity as its proxy were calculated. The dependence of the induced curvature on the curvature of the origin of the eigenstrains was investigated.

The results yield a linear relation between the source curvature difference and the target curvature difference. This hints at a geometry sensitivity of the eigenstrains contrary to current literature. A possible explanation is the area moment of inertia, which in this case features an additional term due to Steiner's theorem. The higher the curvature of the geometry, the higher the axial area moment of inertia. This also yields: For identical radial load, a geometry with higher curvature will exhibit higher plastic strains than a geometry with lower curvature. For positive Δκs, mapped plastic strains that are lower than physically correct result in a deflection that is larger than physically correct, which corresponds to positive Δκt.

The results can be applied to the simulation of small forming operations of complex shaped parts: A known eigenstrain tensor can be used in a maximally large volume or the number of eigenstrain tensors needed can be estimated. For both cases, the gain in efficiency by use of the eigenstrain ansatz, compared to direct simulation, can be maintained.

Using the determined relation, future work of the authors will aim to investigate the simulation error that arises from the discretization of a target geometry with a continuous curvature in a given closed interval. A generalization to multidimensional curvature will also be tackled in upcoming work.
The positive slope of the forced mean indicates the geometry dependence of the eigenstrains: For positive Δκs, the curvature change of the target geometry will be overstated; for negative Δκs, the curvature change of the target geometry will be understated. The value of m < 1 indicates a tolerance of the eigenstrain ansatz for deviations in the eigenstrain. While the data points stem from discrete values of cylinder shells with uniform curvature, the continuous relation is able to improve LPF simulations of geometries with a non-uniform, continuous curvature profile. In the following, two ways, how knowledge of this relation gives an advantage when simulating small forming operations, are outlined: Assume, a small forming operation on target geometry with a non-uniform curvature distribution $\tilde{\kappa}_t(s)$ should be simulated. If one eigenstrain tensor on a geometry with uniform curvature $\tilde{\kappa}_s$ is known, then the relation gives an estimate on the maximal error caused by using the one known eigenstrain tensor, explicitly

$$|\max(\Delta\kappa_t)| = m \max_s(|\tilde{\kappa}_t(s) - \tilde{\kappa}_s|) \qquad (9)$$

Also, for a given target geometry with a non-uniform curvature distribution $\tilde{\kappa}_t(s)$, if the error in overall shape prediction needs to be limited to a given value, then the relation dictates the number of eigenstrain tensors $n_ε$ needed to stay below a corresponding target curvature difference $\tilde{e}$, namely

$$n_{\epsilon} = \left[ \frac{m \left( \max_{s} \tilde{\kappa}_{t}(s) - \min_{s} \tilde{\kappa}_{t}(s) \right)}{\tilde{e}} \right] \qquad (10)$$

The results are derived from geometries with a mono-directional curvature. No statement on geometries with multi-dimensional, non-trivial curvature can be made.

## Conclusions

In this work, the simulation workflow using the eigenstrain ansatz was set up to model LSP processing of thin-walled Ti-6Al-4V specimen. For a smaller scale geometry, a small number of laser shots were simulated and the resulting eigenstrains were determined based on representative cells defined by the peening pattern. The eigenstrains were mapped onto a larger scale geometry featuring a large scale pattern. The introduced eigenstrains result in a global deflection curve, for which local curvatures and a global quantity as its proxy were calculated. The dependence of the induced curvature on the curvature of the origin of the eigenstrains was investigated.

The results yield a linear relation between the source curvature difference and the target curvature difference. This hints at a geometry sensitivity of the eigenstrains contrary to current literature. A possible explanation is the area moment of inertia, which in this case features an additional term due to Steiner's theorem. The higher the curvature of the geometry, the higher the axial area moment of inertia. This also yields: For identical radial load, a geometry with higher curvature will exhibit higher plastic strains than a geometry with lower curvature. For positive Δκs, mapped plastic strains that are lower than physically correct result in a deflection that is larger than physically correct, which corresponds to positive Δκt.

The results can be applied to the simulation of small forming operations of complex shaped parts: A known eigenstrain tensor can be used in a maximally large volume or the number of eigenstrain tensors needed can be estimated. For both cases, the gain in efficiency by use of the eigenstrain ansatz, compared to direct simulation, can be maintained.

Using the determined relation, future work of the authors will aim to investigate the simulation error that arises from the discretization of a target geometry with a continuous curvature in a given closed interval. A generalization to multidimensional curvature will also be tackled in upcoming work.
## Acknowledgements

The work was carried out within the PEENCOR project funded by the German Federal Ministry of Economic Affairs and Climate Action (BMWK) under the LuFo VI-1 program (project numbers: 20Q1920C, 20Q1920D), which is gratefully acknowledged.

## References

[1] R. R. Boyer, "An overview on the use of titanium in the aerospace industry," *Materials Science and Engineering: A*, vol. 213, no. 1, pp. 103-114, Aug. 1996.
https://doi.org/10.1016/0921-5093(96)10233-1

[2] L. Berthe, R. Fabbro, P. Peyre, L. Tollier, and E. Bartnicki, "Shock waves from a water-confined laser-generated plasma," *Journal of Applied Physics*, vol. 82, no. 6, pp. 2826-2832, Sep. 1997. https://doi.org/10.1063/1.366113

[3] P. Peyre and R. Fabbro, "Laser shock processing: a review of the physics and applications," *Optical and Quantum Electronics*, vol. 27, no. 12, pp. 1213-1229, Dec. 1995.
https://doi.org/10.1007/BF00326477

[4] Y. Hu and R. V. Grandhi, "Efficient numerical prediction of residual stress and deformation for large-scale laser shock processing using the eigenstrain methodology," *Surface and Coatings Technology*, vol. 206, no. 15, pp. 3374-3385, Mar. 2012.
https://doi.org/10.1016/j.surfcoat.2012.01.050

[5] P. A. Faucheux, F. P. Gosselin, and M. Lévesque, "Simulating shot peen forming with eigenstrains," *Journal of Materials Processing Technology*, vol. 254, pp. 135-144, Apr. 2018.
https://doi.org/10.1016/j.jmatprotec.2017.11.036

[6] S. Keller, M. Horstmann, N. Kashaev, and B. Klusemann, "Experimentally validated multi-step simulation strategy to predict the fatigue crack propagation rate in residual stress fields after laser shock peening," *International Journal of Fatigue*, vol. 124, pp. 265-276, Jul. 2019. https://doi.org/10.1016/j.ijfatigue.2018.12.014

[7] S. T. Sala, R. Körner, N. Huber, and N. Kashaev, "On the use of machine learning and genetic algorithm to predict the region processed by laser peen forming," *Manufacturing Letters*, vol. 38, pp. 60-64, Nov. 2023. https://doi.org/10.1016/j.mfglet.2023.09.006

[8] S. Cai and Y. Zhang, "A novel approach to reconstruct residual stress fields induced by surface treatments in arbitrary 3D geometries using the eigenstrain method," *International Journal of Solids and Structures*, vol. 236-237, p. 111372, Feb. 2022.
https://doi.org/10.1016/j.ijsolstr.2021.111372

[9] Z. Zhou et al., "Thermal relaxation of residual stress in laser shock peened Ti-6Al-4V alloy," *Surface and Coatings Technology*, vol. 206, no. 22, pp. 4619-4627, Jun. 2012.
https://doi.org/10.1016/j.surfcoat.2012.05.022

[10] G. R. Johnson and W. H. Cook, "A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates, and High Temperatures. Proceedings of the 7th International Symposium on Ballistic," The Hague, Apr. 1983, pp. 541-547.

[11] W.-S. Lee and C.-F. Lin, "High-temperature deformation behaviour of Ti6Al4V alloy evaluated by high strain-rate compression tests," *Journal of Materials Processing Technology*, vol. 75, no. 1, pp. 127-136, Mar. 1998. https://doi.org/10.1016/S0924-0136(97)00302-6
## Acknowledgements

The work was carried out within the PEENCOR project funded by the German Federal Ministry of Economic Affairs and Climate Action (BMWK) under the LuFo VI-1 program (project numbers: 20Q1920C, 20Q1920D), which is gratefully acknowledged.

## References

[1] R. R. Boyer, "An overview on the use of titanium in the aerospace industry," *Materials Science and Engineering: A*, vol. 213, no. 1, pp. 103-114, Aug. 1996.
https://doi.org/10.1016/0921-5093(96)10233-1

[2] L. Berthe, R. Fabbro, P. Peyre, L. Tollier, and E. Bartnicki, "Shock waves from a water-confined laser-generated plasma," *Journal of Applied Physics*, vol. 82, no. 6, pp. 2826-2832, Sep. 1997. https://doi.org/10.1063/1.366113

[3] P. Peyre and R. Fabbro, "Laser shock processing: a review of the physics and applications," *Optical and Quantum Electronics*, vol. 27, no. 12, pp. 1213-1229, Dec. 1995.
https://doi.org/10.1007/BF00326477

[4] Y. Hu and R. V. Grandhi, "Efficient numerical prediction of residual stress and deformation for large-scale laser shock processing using the eigenstrain methodology," *Surface and Coatings Technology*, vol. 206, no. 15, pp. 3374-3385, Mar. 2012.
https://doi.org/10.1016/j.surfcoat.2012.01.050

[5] P. A. Faucheux, F. P. Gosselin, and M. Lévesque, "Simulating shot peen forming with eigenstrains," *Journal of Materials Processing Technology*, vol. 254, pp. 135-144, Apr. 2018.
https://doi.org/10.1016/j.jmatprotec.2017.11.036

[6] S. Keller, M. Horstmann, N. Kashaev, and B. Klusemann, "Experimentally validated multi-step simulation strategy to predict the fatigue crack propagation rate in residual stress fields after laser shock peening," *International Journal of Fatigue*, vol. 124, pp. 265-276, Jul. 2019. https://doi.org/10.1016/j.ijfatigue.2018.12.014

[7] S. T. Sala, R. Körner, N. Huber, and N. Kashaev, "On the use of machine learning and genetic algorithm to predict the region processed by laser peen forming," *Manufacturing Letters*, vol. 38, pp. 60-64, Nov. 2023. https://doi.org/10.1016/j.mfglet.2023.09.006

[8] S. Cai and Y. Zhang, "A novel approach to reconstruct residual stress fields induced by surface treatments in arbitrary 3D geometries using the eigenstrain method," *International Journal of Solids and Structures*, vol. 236-237, p. 111372, Feb. 2022.
https://doi.org/10.1016/j.ijsolstr.2021.111372

[9] Z. Zhou et al., "Thermal relaxation of residual stress in laser shock peened Ti-6Al-4V alloy," *Surface and Coatings Technology*, vol. 206, no. 22, pp. 4619-4627, Jun. 2012.
https://doi.org/10.1016/j.surfcoat.2012.05.022

[10] G. R. Johnson and W. H. Cook, "A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates, and High Temperatures. Proceedings of the 7th International Symposium on Ballistic," The Hague, Apr. 1983, pp. 541-547.

[11] W.-S. Lee and C.-F. Lin, "High-temperature deformation behaviour of Ti6Al4V alloy evaluated by high strain-rate compression tests," *Journal of Materials Processing Technology*, vol. 75, no. 1, pp. 127-136, Mar. 1998. https://doi.org/10.1016/S0924-0136(97)00302-6
Non-conventional processes

<figure><img src="image_2216.png" /></figure>
Non-conventional processes

<figure><img src="image_2207.png" /></figure>
# Experimental derivation of process input parameters for electrochemical precision machining of a powder metallurgical tool steel

PETERMANN Richard<sup>1,a*</sup>, CLAUß Pascal<sup>1,b</sup>, DAMM Philipp<sup>1,c</sup>, MEICHSNER Gunnar<sup>1,d</sup> and HACKERT-OSCHÄTZCHEN Matthias<sup>1,e</sup>

<sup>1</sup>Chair of Manufacturing Technology with Focus Machining, Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany

<sup>a</sup>richard.petermann@ovgu.de, <sup>b</sup>pascal.clauss@ovgu.de, <sup>c</sup>philipp.damm@ovgu.de,

<sup>d</sup>gunnar.meichsner@ovgu.de, <sup>e</sup>matthias.hackert-oschaetzchen@ovgu.de

**Keywords:** Anodic Dissolution, Electrochemical Precision Machining (PECM), Removal Rate, DIN SPEC 91399

**Abstract.** Powder metallurgical steels are used for tools because of their properties, such as toughness, wear resistance and hardness. During the machining of the tool, the material is hardened. Consequently, the subsequent finishing of the tool is a challenge, as conventional manufacturing processes reach their limit of machinability due to the hardness. An alternative manufacturing technology is electrochemical precision machining (PECM). The PECM process, which is based on anodic metal dissolution, can be applied to machine metallic materials without force, cracks or burrs, regardless of their hardness [1]. Due to the process principle, the mechanical properties of the workpiece are not changed. As a result, the PECM process is suitable for finish machining of tools and finishing of tools. A significant challenge for PECM is the material-specific removal functions [2]. In this paper, the material removal functions of the hardened powder metallurgical steel S390 [3] are analyzed using experimental tests and the achievable surface quality is shown. Subsequently, process input parameters for electrochemical precision machining of S390 are derived.

**Introduction**

The working principle of electrochemical machining (ECM) is based on the principle of anodic metal dissolution. It allows the machining of metallic materials regardless of their mechanical properties [1,2,4,5,6]. The removal properties of a material depend on the electrochemical interactions between the workpiece material and the electrolyte.

Currently, the design of the cathode geometry and the derivation of process parameters is challenging due to the material-dependent removal properties. Usually, the removal properties are determined by the user with the help of an empirical removal experiment series [2].

This paper presents the determination of the removal parameters of the hardened tool steel S390 [3]. The determination is done by applying the PECM variant of the ECM process. The PECM process differs from the ECM process in that an oscillating cathode and a pulsed direct current are applied.

**Material**

The tool steel S390 was selected as the anodic material. S390 is a powder metallurgical steel and was examined in a hardened heat treatment condition. The hardness was measured at 890 ±30 HV<sub>30</sub>. Round samples with a diameter of 12 mm were selected for the experiments. The composition of the material is given by the manufacturer as follows: 67.46 % Fe, 1.64 % C, 0.60 % Si, 0.30 % Mn, 4.80 % Cr, 2.00 % Mo, 4.80 % V, 10.40 % W and 8.00 % Co [3]. The

<figure><img src="image_2217.png" /></figure>
# Experimental derivation of process input parameters for electrochemical precision machining of a powder metallurgical tool steel

PETERMANN Richard¹,ª*, CLAUß Pascal¹,ᵇ, DAMM Philipp¹,ᶜ, MEICHSNER Gunnar¹,ᵈ and HACKERT-OSCHÄTZCHEN Matthias¹,ᵉ

¹Chair of Manufacturing Technology with Focus Machining, Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany

ªrichard.petermann@ovgu.de, ᵇpascal.clauss@ovgu.de, ᶜphilipp.damm@ovgu.de,

ᵈgunnar.meichsner@ovgu.de, ᶜmatthias.hackert-oschaetzchen@ovgu.de

**Keywords:** Anodic Dissolution, Electrochemical Precision Machining (PECM), Removal Rate, DIN SPEC 91399

**Abstract.** Powder metallurgical steels are used for tools because of their properties, such as toughness, wear resistance and hardness. During the machining of the tool, the material is hardened. Consequently, the subsequent finishing of the tool is a challenge, as conventional manufacturing processes reach their limit of machinability due to the hardness. An alternative manufacturing technology is electrochemical precision machining (PECM). The PECM process, which is based on anodic metal dissolution, can be applied to machine metallic materials without force, cracks or burrs, regardless of their hardness [1]. Due to the process principle, the mechanical properties of the workpiece are not changed. As a result, the PECM process is suitable for finish machining of tools and finishing of tools. A significant challenge for PECM is the material-specific removal functions [2]. In this paper, the material removal functions of the hardened powder metallurgical steel S390 [3] are analyzed using experimental tests and the achievable surface quality is shown. Subsequently, process input parameters for electrochemical precision machining of S390 are derived.

**Introduction**

The working principle of electrochemical machining (ECM) is based on the principle of anodic metal dissolution. It allows the machining of metallic materials regardless of their mechanical properties [1,2,4,5,6]. The removal properties of a material depend on the electrochemical interactions between the workpiece material and the electrolyte.

Currently, the design of the cathode geometry and the derivation of process parameters is challenging due to the material-dependent removal properties. Usually, the removal properties are determined by the user with the help of an empirical removal experiment series [2].

This paper presents the determination of the removal parameters of the hardened tool steel S390 [3]. The determination is done by applying the PECM variant of the ECM process. The PECM process differs from the ECM process in that an oscillating cathode and a pulsed direct current are applied.

**Material**

The tool steel S390 was selected as the anodic material. S390 is a powder metallurgical steel and was examined in a hardened heat treatment condition. The hardness was measured at 890 ±30 HV₃₀. Round samples with a diameter of 12 mm were selected for the experiments. The composition of the material is given by the manufacturer as follows: 67.46 % Fe, 1.64 % C, 0.60 % Si, 0.30 % Mn, 4.80 % Cr, 2.00 % Mo, 4.80 % V, 10.40 % W and 8.00 % Co [3]. The

<figure><img src="image_2208.png" /></figure>
initial surface was examined using a Hirox SEM SH-5500 scanning electron microscope (SEM) for SE and BSE images. These images are shown in Fig. 1.

<figure><img src="image_2218.png" /><figcaption>Figure 1: SEM image (left) and BSE image (right) of the base material</figcaption></figure>

Several particles can be observed on the surface of the sample. The size of these particles varies between 0.5 µm and 2 µm. The main elements were determined by applying energy dispersive X-ray analysis (EDX) as follows: 46 % Fe, 29 % W and 11 % C. According to [7], these particles are comprised of Fe₃W₃C and VC carbides.

## Experimental Setup

The experiments were conducted on a commercially available PEM 800S from PEMTec Snc. The design of the PECM device is based on DIN SPEC 91399 [4] and is shown in Fig. 2. A flushing chamber around the working gap encloses the cathode and anode. The flushing chamber channels the electrolyte, thus ensuring an exchange of the electrolyte and the removal of dissolved material. According to [4], the diameters of the cathode and anode are 12 mm, resulting in an effective removal area $A_{sample}$ of 1.13 cm². The anode is made from 1.4301 (X5CrNi18-10).

<figure><img src="image_2219.png" /><figcaption>Figure 2: Schematic of the experimental setup (left) and schematic representation of removal and required current characteristics during the PECM process (right) according to [5]</figcaption></figure>

In PECM, the cathode is oscillated, and therefore, the working distance is varied. In addition, in PECM, a pulsed direct current is applied. As shown in Fig. 2, the removal-effective current pulse is centered around the bottom dead center of the sine oscillation of the cathode. As a result, the working distance during the removal process is minimal. This ensures a high localization of the material removal. By opening the working gap and pausing the current, the working gap is more effectively flushed with fresh electrolyte, increasing the removal process's stability and reducing the working gap to up to 10 µm at the bottom dead center. [6]
initial surface was examined using a Hirox SEM SH-5500 scanning electron microscope (SEM) for SE and BSE images. These images are shown in Fig. 1.

<figure><img src="image_2209.png" /><figcaption>Figure 1: SEM image (left) and BSE image (right) of the base material</figcaption></figure>

Several particles can be observed on the surface of the sample. The size of these particles varies between 0.5 µm and 2 µm. The main elements were determined by applying energy dispersive X-ray analysis (EDX) as follows: 46 % Fe, 29 % W and 11 % C. According to [7], these particles are comprised of Fe₃W₃C and VC carbides.

## Experimental Setup

The experiments were conducted on a commercially available PEM 800S from PEMTec Snc. The design of the PECM device is based on DIN SPEC 91399 [4] and is shown in Fig. 2. A flushing chamber around the working gap encloses the cathode and anode. The flushing chamber channels the electrolyte, thus ensuring an exchange of the electrolyte and the removal of dissolved material. According to [4], the diameters of the cathode and anode are 12 mm, resulting in an effective removal area $A_{sample}$ of 1.13 cm². The anode is made from 1.4301 (X5CrNi18-10).

<figure><img src="image_2210.png" /><figcaption>Figure 2: Schematic of the experimental setup (left) and schematic representation of removal and required current characteristics during the PECM process (right) according to [5]</figcaption></figure>

In PECM, the cathode is oscillated, and therefore, the working distance is varied. In addition, in PECM, a pulsed direct current is applied. As shown in Fig. 2, the removal-effective current pulse is centered around the bottom dead center of the sine oscillation of the cathode. As a result, the working distance during the removal process is minimal. This ensures a high localization of the material removal. By opening the working gap and pausing the current, the working gap is more effectively flushed with fresh electrolyte, increasing the removal process's stability and reducing the working gap to up to 10 µm at the bottom dead center. [6]
## Design of Experiments

The process parameters presented in Table 1 were kept constant over the various series of experiments. A NaNO₃ solution with an 8 % mass fraction was applied as an electrolyte. The specific removal volume Vsp was calculated as 2.495 e⁻⁵ cm³/C based on the chemical composition. The oscillation frequency and the pulse frequency were set to 50 Hz. The pulse width was determined to be 4 ms.

<table><caption>Table 1: Constant parameters</caption><thead><tr><th>Symbol</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td></td><td>Electrolyte NaNO<sub>3</sub></td><td>8 [% mass fraction]</td></tr><tr><td>σ</td><td>Electric conductivity</td><td>67 [mS/cm]</td></tr><tr><td>p<sub>el</sub></td><td>Electrolyte inlet pressure</td><td>350 [kPa]</td></tr><tr><td>f</td><td>Oscillation frequency</td><td>50 [Hz]</td></tr><tr><td>Z</td><td>Oscillation amplitude</td><td>200 [µm]</td></tr><tr><td>f<sub>p</sub></td><td>Pulse frequency</td><td>50 [Hz]</td></tr><tr><td>t<sub>p</sub></td><td>Pulse width</td><td>4 [ms]</td></tr></tbody></table>

Current densities J in the range from 2 A/cm² to 100 A/cm² were realized in the experiments. The aimed working distances were between 20 µm and 90 µm. For this purpose, the process input variables shown in Table 2 were varied in the specified ranges.

<table><caption>Table 2: Experimental parameters</caption><thead><tr><th>Symbol</th><th>Parameter</th><th>Value Range</th></tr></thead><tbody><tr><td>U<sub>q</sub></td><td>Voltage</td><td>5 – 18 [V]</td></tr><tr><td>v<sub>f</sub></td><td>Feed rate</td><td>0.002 – 0.321 [mm/min]</td></tr></tbody></table>

## Experimental Results

After completion of the removal experiments, the optical appearance of the machined surfaces was analyzed. Fig. 3 shows the macroscopic surface appearance for different current densities. The working distance af was kept constant at 50 µm for each experiment. It can be observed that the surface appearance changes depending on the current density. The color change can be divided into three areas. A dark surface is observed in the range between 20 A/cm² and 40 A/cm². The samples ranging from 50 A/cm² to 67 A/cm² appear to have a reddish-gray surface. A slight metallic shine can be seen in some areas of the surface. Beginning with a current density of 80 A/cm², a shiny metallic surface is generated.

Another influencing factor is the flow direction of the electrolyte. The fresh electrolyte is fed into the working gap from the left and transported out on the right side. For samples with a current density above 50 A/cm², it can be seen that the surfaces in the area of the inlet show a metallic shine, which decreases in the direction of the electrolyte flow. This can be seen particularly clearly on the sample with a current density of 50 A/cm². The reason for this color change is the change in the electrolyte composition. The electrolyte is contaminated by the dissolution already taking place in the inlet area, which changes the removal conditions in the flow direction. This change in the electrolyte is reflected in the optical surface appearance.
## Design of Experiments

The process parameters presented in Table 1 were kept constant over the various series of experiments. A NaNO₃ solution with an 8 % mass fraction was applied as an electrolyte. The specific removal volume Vsp was calculated as 2.495 e⁻⁵ cm³/C based on the chemical composition. The oscillation frequency and the pulse frequency were set to 50 Hz. The pulse width was determined to be 4 ms.

<table><caption>Table 1: Constant parameters</caption><thead><tr><th>Symbol</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td></td><td>Electrolyte NaNO<sub>3</sub></td><td>8 [% mass fraction]</td></tr><tr><td>σ</td><td>Electric conductivity</td><td>67 [mS/cm]</td></tr><tr><td>p<sub>el</sub></td><td>Electrolyte inlet pressure</td><td>350 [kPa]</td></tr><tr><td>f</td><td>Oscillation frequency</td><td>50 [Hz]</td></tr><tr><td>Z</td><td>Oscillation amplitude</td><td>200 [µm]</td></tr><tr><td>f<sub>p</sub></td><td>Pulse frequency</td><td>50 [Hz]</td></tr><tr><td>t<sub>p</sub></td><td>Pulse width</td><td>4 [ms]</td></tr></tbody></table>

Current densities J in the range from 2 A/cm² to 100 A/cm² were realized in the experiments. The aimed working distances were between 20 µm and 90 µm. For this purpose, the process input variables shown in Table 2 were varied in the specified ranges.

<table><caption>Table 2: Experimental parameters</caption><thead><tr><th>Symbol</th><th>Parameter</th><th>Value Range</th></tr></thead><tbody><tr><td>U<sub>q</sub></td><td>Voltage</td><td>5 – 18 [V]</td></tr><tr><td>v<sub>f</sub></td><td>Feed rate</td><td>0.002 – 0.321 [mm/min]</td></tr></tbody></table>

## Experimental Results

After completion of the removal experiments, the optical appearance of the machined surfaces was analyzed. Fig. 3 shows the macroscopic surface appearance for different current densities. The working distance af was kept constant at 50 µm for each experiment. It can be observed that the surface appearance changes depending on the current density. The color change can be divided into three areas. A dark surface is observed in the range between 20 A/cm² and 40 A/cm². The samples ranging from 50 A/cm² to 67 A/cm² appear to have a reddish-gray surface. A slight metallic shine can be seen in some areas of the surface. Beginning with a current density of 80 A/cm², a shiny metallic surface is generated.

Another influencing factor is the flow direction of the electrolyte. The fresh electrolyte is fed into the working gap from the left and transported out on the right side. For samples with a current density above 50 A/cm², it can be seen that the surfaces in the area of the inlet show a metallic shine, which decreases in the direction of the electrolyte flow. This can be seen particularly clearly on the sample with a current density of 50 A/cm². The reason for this color change is the change in the electrolyte composition. The electrolyte is contaminated by the dissolution already taking place in the inlet area, which changes the removal conditions in the flow direction. This change in the electrolyte is reflected in the optical surface appearance.
<figure><img src="image_2220.png" /><figcaption>Figure 3: Photos of machined surfaces at different current densities J, constant working distance a_f of 50 µm</figcaption></figure>

Fig. 4 shows the feed rate v_f of the cathode as a function of the current density J. The graph can be divided into three areas. Only minimal dissolution occurs in the range of low current densities below 30 A/cm². The feed rate is near zero in this area (area 1). In the range from 33 A/cm² to 36 A/cm², a steep increase in the feed rate can be seen. For a current density higher than 38 A/cm², a linear increase with a lower gradient can be observed.

<figure><img src="image_2221.png" /><figcaption>Figure 4: Feed rate v_f as a function of the current density for PECM with a frequency f of 50 Hz and a pulse width t_p of 4 ms</figcaption></figure>

The functional relationship is described using linear functions in equation 1:

$$v_f = v_{a-PC} = \frac{\Delta v_{a-PC}}{\Delta J} \cdot J + v_0. \qquad (1)$$

The parameters of these functions are listed in Table 3.
<figure><img src="image_2211.png" /><figcaption>Figure 3: Photos of machined surfaces at different current densities J, constant working distance a_f of 50 µm</figcaption></figure>

Fig. 4 shows the feed rate v_f of the cathode as a function of the current density J. The graph can be divided into three areas. Only minimal dissolution occurs in the range of low current densities below 30 A/cm². The feed rate is near zero in this area (area 1). In the range from 33 A/cm² to 36 A/cm², a steep increase in the feed rate can be seen. For a current density higher than 38 A/cm², a linear increase with a lower gradient can be observed.

<figure><img src="image_2212.png" /><figcaption>Figure 4: Feed rate v_f as a function of the current density for PECM with a frequency f of 50 Hz and a pulse width t_p of 4 ms</figcaption></figure>

The functional relationship is described using linear functions in equation 1:

$$v_f = v_{a-PC} = \frac{\Delta v_{a-PC}}{\Delta J} \cdot J + v_0. \quad (1)$$

The parameters of these functions are listed in Table 3.
<table><caption>Table 3: Parameters for the different areas of the function $v_f(J)$</caption><thead><tr><th>Area</th><th>J<br>[A/cm²]</th><th>Δv<sub>a-PC</sub>/ΔJ<br>[mm/min / A/cm²]</th><th>v<sub>0</sub><br>[mm/min]</th></tr></thead><tbody><tr><td>1</td><td>12 – 30</td><td>0.0007</td><td>-0.0066</td></tr><tr><td>2</td><td>33 – 36</td><td>0.0162</td><td>-0.5116</td></tr><tr><td>3</td><td>38 – 117</td><td>0.0033</td><td>-0.0093</td></tr></tbody></table>

As only minimal dissolution occurs up to a current density of 30 A/cm², the material dissolution in NaNO₃ solution is classified as trans-passive according to [6]. The surfaces of the samples after PECM were examined more closely by applying a scanning electron microscope (SEM). Fig. 5 shows the surfaces of 4 samples at a magnification of 3000. The data points are additionally labelled (a) to (d) in Fig. 4.

Surface (a) represents a surface after machining in the passive dissolution range. Surface (b) corresponds with machining in the transition area between passive and trans-passive dissolution. Surface (c) and surface (d) represent the trans-passive dissolution.

<figure><img src="image_2222.png" /><figcaption>Figure 5: SEM images of selected surfaces (a) $J = 20 A/cm^2$, (b) $J = 39 A/cm^2$, (c) $J = 49 A/cm^2$, (d) $J = 117 A/cm^2$</figcaption></figure>

When analyzing the SEM images, it is noticeable that the appearance of the surfaces changes significantly between samples which were machined with different current densities. The surface of the passive area (see Fig. 5(a)) has a fractured structure. Several of the particles already present on the initial surface can be recognized. These particles lie on a surface which appears porous. Surface (b) also shows the presence of numerous particles, but the surface seems less porous. Surface (c) represents the initial area of trans-passive dissolution. In contrast to (a) and (b), the
<table><caption>Table 3: Parameters for the different areas of the function $v_f(J)$</caption><thead><tr><th>Area</th><th>J<br>[A/cm²]</th><th>Δv<sub>a-PC</sub>/ΔJ<br>[mm/min / A/cm²]</th><th>v<sub>0</sub><br>[mm/min]</th></tr></thead><tbody><tr><td>1</td><td>12 – 30</td><td>0.0007</td><td>-0.0066</td></tr><tr><td>2</td><td>33 – 36</td><td>0.0162</td><td>-0.5116</td></tr><tr><td>3</td><td>38 – 117</td><td>0.0033</td><td>-0.0093</td></tr></tbody></table>

As only minimal dissolution occurs up to a current density of 30 A/cm², the material dissolution in NaNO₃ solution is classified as trans-passive according to [6]. The surfaces of the samples after PECM were examined more closely by applying a scanning electron microscope (SEM). Fig. 5 shows the surfaces of 4 samples at a magnification of 3000. The data points are additionally labelled (a) to (d) in Fig. 4.

Surface (a) represents a surface after machining in the passive dissolution range. Surface (b) corresponds with machining in the transition area between passive and trans-passive dissolution. Surface (c) and surface (d) represent the trans-passive dissolution.

<figure><img src="image_2213.png" /><figcaption>Figure 5: SEM images of selected surfaces (a) $J = 20 A/cm^2$, (b) $J = 39 A/cm^2$, (c) $J = 49 A/cm^2$, (d) $J = 117 A/cm^2$</figcaption></figure>

When analyzing the SEM images, it is noticeable that the appearance of the surfaces changes significantly between samples which were machined with different current densities. The surface of the passive area (see Fig. 5(a)) has a fractured structure. Several of the particles already present on the initial surface can be recognized. These particles lie on a surface which appears porous. Surface (b) also shows the presence of numerous particles, but the surface seems less porous. Surface (c) represents the initial area of trans-passive dissolution. In contrast to (a) and (b), the
surface appears noticeably smoother and has only a few pores. As with the previous samples, the particles can also be seen on this sample surface. Sample (d) represents a surface machined with a high current density (J = 117 A/cm²). The surface appears similar to the surface of sample (c).

All sample surfaces are covered with brightly highlighted particles. The particles are also present on the initial surface. The particles consist mainly of Fe (46 %), W (29 %) and C (11 %) and do not appear to be dissolved by anodic dissolution. Therefore, it can be assumed that these are covalent compounds of Fe₃W₃C and VC [7]. The pores, which can be seen in surfaces (c) and (d), are caused by particles that have already been loosened. It should be noted that although the particles themselves cannot be dissolved, they can be released from the material by dissolving the matrix material around them [8]. The particles are then transported by the flow of the electrolyte.

Fig. 6 shows the roughness Sz and the roughness Sa of the sample surface as a function of the current density J. The roughness values were determined by applying a Nanofocus µsurf expert. The measured data is processed by applying the factory settings with a 0.25 mm Gaussian filter. For each sample, three measurement squares were measured in the area of the inlet and the outlet. The roughness values are shown for the inlet and the outlet area. Furthermore, the arithmetic mean value formed from all six values is shown as the average roughness.

<figure><img src="image_2223.png" /><figcaption>Figure 6: Surface roughness Sz and Sa as a function of the current density J</figcaption></figure>

When looking at Sz, it is noticeable that the roughness decreases from Sz 9 µm at 20 A/cm² to Sz 6.1 µm and then rises slightly to a plateau of approx. Sz 7.5 µm at current densities between 80 A/cm² and 100 A/cm². Finally, the roughness falls again to a value of Sz 6 µm at 117 A/cm². The roughness Sa also decreases from 0.38 µm at 20 A/cm² to 0.21 µm at 50 A/cm². This trend is interrupted by the point of 0.42 µm at 38 A/cm². Sa then remains constant at around 0.21 µm. In the range around 50 A/cm² to 70 A/cm², a deviation of Sa between inlet and outlet of approx. 0.05 µm is noticeable. The deviations of Sz are larger than the deviations of Sa and scatter over the entire course between 0.3 µm at 25 A/cm² and 2.6 µm at 48 A/cm².
surface appears noticeably smoother and has only a few pores. As with the previous samples, the particles can also be seen on this sample surface. Sample (d) represents a surface machined with a high current density (J = 117 A/cm²). The surface appears similar to the surface of sample (c).

All sample surfaces are covered with brightly highlighted particles. The particles are also present on the initial surface. The particles consist mainly of Fe (46 %), W (29 %) and C (11 %) and do not appear to be dissolved by anodic dissolution. Therefore, it can be assumed that these are covalent compounds of Fe₃W₃C and VC [7]. The pores, which can be seen in surfaces (c) and (d), are caused by particles that have already been loosened. It should be noted that although the particles themselves cannot be dissolved, they can be released from the material by dissolving the matrix material around them [8]. The particles are then transported by the flow of the electrolyte.

Fig. 6 shows the roughness Sz and the roughness Sa of the sample surface as a function of the current density J. The roughness values were determined by applying a Nanofocus µsurf expert. The measured data is processed by applying the factory settings with a 0.25 mm Gaussian filter. For each sample, three measurement squares were measured in the area of the inlet and the outlet. The roughness values are shown for the inlet and the outlet area. Furthermore, the arithmetic mean value formed from all six values is shown as the average roughness.

<figure><img src="image_2214.png" /><figcaption>Figure 6: Surface roughness Sz and Sa as a function of the current density J</figcaption></figure>

When looking at Sz, it is noticeable that the roughness decreases from Sz 9 µm at 20 A/cm² to Sz 6.1 µm and then rises slightly to a plateau of approx. Sz 7.5 µm at current densities between 80 A/cm² and 100 A/cm². Finally, the roughness falls again to a value of Sz 6 µm at 117 A/cm². The roughness Sa also decreases from 0.38 µm at 20 A/cm² to 0.21 µm at 50 A/cm². This trend is interrupted by the point of 0.42 µm at 38 A/cm². Sa then remains constant at around 0.21 µm. In the range around 50 A/cm² to 70 A/cm², a deviation of Sa between inlet and outlet of approx. 0.05 µm is noticeable. The deviations of Sz are larger than the deviations of Sa and scatter over the entire course between 0.3 µm at 25 A/cm² and 2.6 µm at 48 A/cm².
Fig. 7 shows the roughness values *Rz* and *Ra* in a similar format to Fig. 6. A line profile was evaluated for each of the previously analyzed measurement squares. Accordingly, 3 values are averaged in the inlet and outlet area. The *Rz* values initially fall from a roughness of 2.8 µm at 20 A/cm² to 1.4 µm at 66 A/cm². This trend is interrupted by the point at 40 A/cm² with 2.7 µm. In the range between 60 A/cm² and 117 A/cm², *Rz* is almost constant at 1.4 µm. The course of the roughness *Ra* is also decreasing and starts at 20 A/cm² with 0.3 µm. *Ra* falls on average to 0.19 µm at 50 A/cm². Like *Rz*, the graph of *Ra* also has a higher value at the point at 40 A/cm² with 0.31 µm. In the range between 50 A/cm² and 117 A/cm² *Ra* falls slightly from 0.19 µm to 0.13 µm. The differences between inlet and outlet are maximum for both *Rz* and *Ra* at 66 A/cm² with Δ*Rz* approx. 1.2 µm and Δ*Ra* approx. 0.13 µm. These differences decrease towards higher and lower current densities.

<figure><img src="image_2224.png" /><figcaption>Figure 7: Roughness *Rz* and *Ra* as a function of the current density *J*</figcaption></figure>

A comparison of the graphs from Fig. 6 and Fig. 7 shows that all graphs display a globally decreasing trend of the roughness with increasing current densities. This is consistent with the macroscopic (see Fig. 3) and microscopic (see Fig. 5) appearances. It is also noticeable that all graphs except *Sz* have a local maximum of 40 A/cm². Looking at the removal characteristics shown in Fig. 5, it is noticeable that these maxima match the transition area between the passive and the trans-passive removal areas. The microscopic surface structure (see Fig. 5(b)) also appears clearly fractured at this current density. This fractured structure particularly influences the roughness *Sa* and *Ra*, as these are based on the average profile height. The roughness *Sz* and *Rz*, which only consider the local distance between hill and valley, on the other hand, are less or not noticeably impacted by the fractured structure. It should also be noted that *Rz*, in particular, is subject to greater statistical fluctuation due to the presence of particles, as shown in Fig. 5. In general, the deviation of *Rz* and *Ra* values between inlet and outlet is maximum in the range of 50 A/cm² to 80 A/cm² and decreases towards higher and lower current densities.
Fig. 7 shows the roughness values *Rz* and *Ra* in a similar format to Fig. 6. A line profile was evaluated for each of the previously analyzed measurement squares. Accordingly, 3 values are averaged in the inlet and outlet area. The *Rz* values initially fall from a roughness of 2.8 µm at 20 A/cm² to 1.4 µm at 66 A/cm². This trend is interrupted by the point at 40 A/cm² with 2.7 µm. In the range between 60 A/cm² and 117 A/cm², *Rz* is almost constant at 1.4 µm. The course of the roughness *Ra* is also decreasing and starts at 20 A/cm² with 0.3 µm. *Ra* falls on average to 0.19 µm at 50 A/cm². Like *Rz*, the graph of *Ra* also has a higher value at the point at 40 A/cm² with 0.31 µm. In the range between 50 A/cm² and 117 A/cm² *Ra* falls slightly from 0.19 µm to 0.13 µm. The differences between inlet and outlet are maximum for both *Rz* and *Ra* at 66 A/cm² with Δ*Rz* approx. 1.2 µm and Δ*Ra* approx. 0.13 µm. These differences decrease towards higher and lower current densities.

<figure><img src="image_2215.png" /><figcaption>Figure 7: Roughness *Rz* and *Ra* as a function of the current density *J*</figcaption></figure>

A comparison of the graphs from Fig. 6 and Fig. 7 shows that all graphs display a globally decreasing trend of the roughness with increasing current densities. This is consistent with the macroscopic (see Fig. 3) and microscopic (see Fig. 5) appearances. It is also noticeable that all graphs except *Sz* have a local maximum of 40 A/cm². Looking at the removal characteristics shown in Fig. 5, it is noticeable that these maxima match the transition area between the passive and the trans-passive removal areas. The microscopic surface structure (see Fig. 5(b)) also appears clearly fractured at this current density. This fractured structure particularly influences the roughness *Sa* and *Ra*, as these are based on the average profile height. The roughness *Sz* and *Rz*, which only consider the local distance between hill and valley, on the other hand, are less or not noticeably impacted by the fractured structure. It should also be noted that *Rz*, in particular, is subject to greater statistical fluctuation due to the presence of particles, as shown in Fig. 5. In general, the deviation of *Rz* and *Ra* values between inlet and outlet is maximum in the range of 50 A/cm² to 80 A/cm² and decreases towards higher and lower current densities.
Fig. 8 displays the current efficiency $\eta_{v-a}$ as a function of the current density $J$. The current efficiency $\eta_{v-a}$ is defined as the quotient of the effective removal volume $V_{eff}$ and the specific removal volume $V_{sp}$ according to equation 2 [6]. The effective removal volume $V_{eff}$ is calculated from the feed or removal rate $v_f$ and the removal area $A_{sample}$.

$$ \eta_{v-a} = \frac{V_{eff}}{V_{sp}} = \frac{v_f \cdot A_{sample}}{V_{sp}} \qquad (2) $$

<figure><img src="image_2225.png" /><figcaption>Figure 8: Current efficiency $\eta_{v-a}$ as a function of the current density $J$</figcaption></figure>

It can be seen that the current efficiency in the range from 20 A/cm² to 40 A/cm² is below 20 % and increases slightly. Around 40 A/cm², the current efficiency jumps to a value between 100 % and 110 % and stays constant for rising current densities. The jump in the current efficiency is overlapped by the jump in the feed rate, as seen in Fig. 4. Usually, a current efficiency of 70 % to 80 % is to be expected for a material and electrolyte combination with trans-passive dissolution characteristics [8]. In the case of the analyzed material, however, it should be noted that the material is reinforced with covalent bonded particles. As mentioned, these particles are not dissolved anodically but instead drop out of the material when the matrix material around the particle is removed sufficiently. However, the volume of the particles is included in the removal volume of the current efficiency, which significantly increases the current efficiency.

As a result of this analysis, different process input parameters can be derived for the subsequent design of a PECM application for the machining of S390. It is to be noted that the following parameters and expected results only apply to parts and surfaces machined in the frontal working gap and application with sufficient flushing of electrolyte. The creation of current densities of 50 A/cm² up to 117 A/cm² results in a possible feed rate $v_f$ of 0.15 mm/min up to 0.47 mm/min. The machined surfaces can be expected to have a roughness of $Rz$ around 2 µm and below and $Ra$ at 0.22 µm and below. A metallic shin of the machined surfaces can be expected. To reduce the machining time, a high current density and, therefore, a high feed rate is advised.

The creation of rougher and porous surfaces with $Rz$ of 2.9 µm and $Ra$ of 0.3 µm can be achieved using lower current densities below 35 A/cm². The machining with current densities below 35 A/cm² also creates a dark greyish surface appearance. Lower current densities, however, result in a low feed rate of 0.02 mm/min and lower, which increases the machining time. As these parameters only apply to machining in the frontal working gap, the usage of low current densities
Fig. 8 displays the current efficiency $\eta_{v-a}$ as a function of the current density $J$. The current efficiency $\eta_{v-a}$ is defined as the quotient of the effective removal volume $V_{eff}$ and the specific removal volume $V_{sp}$ according to equation 2 [6]. The effective removal volume $V_{eff}$ is calculated from the feed or removal rate $v_f$ and the removal area $A_{sample}$.

$$ \eta_{v-a} = \frac{V_{eff}}{V_{sp}} = \frac{v_f \cdot A_{sample}}{V_{sp}} \qquad (2) $$

<figure><img src="image_2216.png" /><figcaption>Figure 8: Current efficiency $\eta_{v-a}$ as a function of the current density $J$</figcaption></figure>

It can be seen that the current efficiency in the range from 20 A/cm² to 40 A/cm² is below 20 % and increases slightly. Around 40 A/cm², the current efficiency jumps to a value between 100 % and 110 % and stays constant for rising current densities. The jump in the current efficiency is overlapped by the jump in the feed rate, as seen in Fig. 4. Usually, a current efficiency of 70 % to 80 % is to be expected for a material and electrolyte combination with trans-passive dissolution characteristics [8]. In the case of the analyzed material, however, it should be noted that the material is reinforced with covalent bonded particles. As mentioned, these particles are not dissolved anodically but instead drop out of the material when the matrix material around the particle is removed sufficiently. However, the volume of the particles is included in the removal volume of the current efficiency, which significantly increases the current efficiency.

As a result of this analysis, different process input parameters can be derived for the subsequent design of a PECM application for the machining of S390. It is to be noted that the following parameters and expected results only apply to parts and surfaces machined in the frontal working gap and application with sufficient flushing of electrolyte. The creation of current densities of 50 A/cm² up to 117 A/cm² results in a possible feed rate $v_f$ of 0.15 mm/min up to 0.47 mm/min. The machined surfaces can be expected to have a roughness of $Rz$ around 2 µm and below and $Ra$ at 0.22 µm and below. A metallic shin of the machined surfaces can be expected. To reduce the machining time, a high current density and, therefore, a high feed rate is advised.

The creation of rougher and porous surfaces with $Rz$ of 2.9 µm and $Ra$ of 0.3 µm can be achieved using lower current densities below 35 A/cm². The machining with current densities below 35 A/cm² also creates a dark greyish surface appearance. Lower current densities, however, result in a low feed rate of 0.02 mm/min and lower, which increases the machining time. As these parameters only apply to machining in the frontal working gap, the usage of low current densities
for the creation of specific surfaces is advised only for surface finishing to avoid high machining time.

**Summary**

In this study, the removal characteristics of the hardened tool steel S390 under PECM conditions were investigated according to DIN SPEC 91399. The material dissolution was found to be transpassive starting at a current density *J* of approx. 33 A/cm². During this investigation, the material removal was characterized by a pulse width of 4 ms. It could be shown that three linear functions describe the feed rate *vf* as a function of the current density *J*. Furthermore, the machined surfaces were examined macroscopically and microscopically. In addition, the roughness of the individual samples was examined for different current densities. A minimal roughness of *Rz* 1.36 µm and *Ra* 0.109 µm is achieved. Finally, the current efficiency was calculated.

**Acknowledgements**

The project work was funded by the Forschungsvereinigung Antriebstechnik e. V. with project number 986-I. The machine tool PEM 800S was funded by the German Research Foundation (DFG) with project number 467011871.

**References**

[1] J.F. Wilson: Practice and Theory of Electrochemical Machining. New York: Wiley-Interscience, 1971, ISBN 9780471949701

[2] G. Meichsner, M. Hackert-Oschätzchen, M. Krönert, J. Edelmann, A. Schubert, M. Putz, Fast Determination of the Material Removal Characteristics in Pulsed Electrochemical Machining, Procedia CIRP 46 (2016) 123-126. https://doi.org/10.1016/j.procir.2016.03.175

[3] N.N., SCHNELLARBEITSSTÄHLE- BÖHLER S390 MICROCLEA, Datasheet, URL: https://www.bohler.de/app/uploads/sites/92/2023/08/productdb/api/s390-microclean_de.pdf, last checked 20.12.2023

[4] DIN SPEC 91399 - Method for determining process input parameters for precision electrochemical machining – requirements, criteria, definitions. Berlin: Beuth Verlag, 2018. https://dx.doi.org/10.31030/3007935

[5] G. Meichsner, M. Hackert-Oschätzchen, T. Petzold, M. Krönert, J. Edelmann, A. Schubert, M. Putz, Removal Characteristics of Stainless Steel in Pulsed Electrochemical Machining, in 12th International Symposium on Electrochemical Machining Technology, INSECT 2016 Proceedings, Brussels, 2016, 73-83. ISBN 978-9-4619746-1-7

[6] F. Klocke, W. Koenig.: Fertigungsverfahren Band 3. Springer Verlag, 2007. https://doi.org/10.1007/978-3-540-48954-2

[7] H. Peng, L. Hu, L. Li, L. Zhang, X. Zhang, Evolution of the microstructure and mechanical properties of powder metallurgical high-speed steel S390 after heat treatment, Journal of Alloys and Compounds 740 (2018) 766-773. https://doi.org/10.1016/j.jallcom.2017.12.264

[8] T. Haisch, E. Mittemeijer, J.W. Schultze, Electrochemical machining of the steel 100Cr6 in aqueous NaCl and NaNO3 solutions in the microstructure of surface films formed by carbides, Electrochimica Acta 47 (2001) 235-241. https://doi.org/10.1016/S0013-4686(01)00561-8
for the creation of specific surfaces is advised only for surface finishing to avoid high machining time.

## Summary

In this study, the removal characteristics of the hardened tool steel S390 under PECM conditions were investigated according to DIN SPEC 91399. The material dissolution was found to be transpassive starting at a current density *J* of approx. 33 A/cm². During this investigation, the material removal was characterized by a pulse width of 4 ms. It could be shown that three linear functions describe the feed rate *vf* as a function of the current density *J*. Furthermore, the machined surfaces were examined macroscopically and microscopically. In addition, the roughness of the individual samples was examined for different current densities. A minimal roughness of *Rz* 1.36 µm and *Ra* 0.109 µm is achieved. Finally, the current efficiency was calculated.

## Acknowledgements

The project work was funded by the Forschungsvereinigung Antriebstechnik e. V. with project number 986-I. The machine tool PEM 800S was funded by the German Research Foundation (DFG) with project number 467011871.

## References

[1] J.F. Wilson: Practice and Theory of Electrochemical Machining. New York: Wiley-Interscience, 1971, ISBN 9780471949701

[2] G. Meichsner, M. Hackert-Oschätzchen, M. Krönert, J. Edelmann, A. Schubert, M. Putz, Fast Determination of the Material Removal Characteristics in Pulsed Electrochemical Machining, Procedia CIRP 46 (2016) 123-126. https://doi.org/10.1016/j.procir.2016.03.175

[3] N.N., SCHNELLARBEITSSTÄHLE- BÖHLER S390 MICROCLEA, Datasheet, URL: https://www.bohler.de/app/uploads/sites/92/2023/08/productdb/api/s390-microclean_de.pdf, last checked 20.12.2023

[4] DIN SPEC 91399 - Method for determining process input parameters for precision electrochemical machining – requirements, criteria, definitions. Berlin: Beuth Verlag, 2018. https://dx.doi.org/10.31030/3007935

[5] G. Meichsner, M. Hackert-Oschätzchen, T. Petzold, M. Krönert, J. Edelmann, A. Schubert, M. Putz, Removal Characteristics of Stainless Steel in Pulsed Electrochemical Machining, in 12th International Symposium on Electrochemical Machining Technology, INSECT 2016 Proceedings, Brussels, 2016, 73-83. ISBN 978-9-4619746-1-7

[6] F. Klocke, W. Koenig.: Fertigungsverfahren Band 3. Springer Verlag, 2007. https://doi.org/10.1007/978-3-540-48954-2

[7] H. Peng, L. Hu, L. Li, L. Zhang, X. Zhang, Evolution of the microstructure and mechanical properties of powder metallurgical high-speed steel S390 after heat treatment, Journal of Alloys and Compounds 740 (2018) 766-773. https://doi.org/10.1016/j.jallcom.2017.12.264

[8] T. Haisch, E. Mittemeijer, J.W. Schultze, Electrochemical machining of the steel 100Cr6 in aqueous NaCl and NaNO3 solutions in the microstructure of surface films formed by carbides, Electrochimica Acta 47 (2001) 235-241. https://doi.org/10.1016/S0013-4686(01)00561-8
# Novel approach for reducing thinning during forming of foil metal

MEERKAMP Max<fnref n="1" />,ª*, UHLMANN Lars<fnref n="1" />,ᵇ, MÜLLER Martina<fnref n="1" />,ᶜ, HERRIG Tim<fnref n="1" />,ᵈ and BERGS Thomas<fnref n="1" />,<fnref n="2" />,ᵉ

**Keywords:** Bipolar Plate, Foil Metal, Novel Forming Approach, Stress State, Thinning

**Abstract.** The mobility sector is one of the largest contributors to greenhouse gas emissions in the European Union. One approach to reduce these emissions and thereby reach the climate goals set by the European Union is the use of alternative drive concepts, such as the hydrogen fuel cell. The economic feasibility of the hydrogen fuel cell depends on its performance which is mostly determined by the volumetric flows of reactants conveyed within the cell. In this context, the key component is the bipolar plate, as its channel depth defines the volumetric flows of reactants. The channel depth is, however, limited by the formability of the material being used and the occurring sheet thinning during forming. This paper presents a novel approach of segmental forming, meaning the forming of individual channels in a defined sequence. The aim is to reduce local stress concentration in the component and to homogenize the spatial thinning within the component in order to avoid a pronounced local sheet thinning. Therefore, five channels are formed by conventional forming in a single step and by segmental forming. The results are then analyzed in terms of the thinning of the sheet thickness that occurs and conclusions are drawn in terms of the stress states that occur.

**Introduction**

One of the largest contributors to the greenhouse gas emissions in the European Union is the mobility sector [1]. To reach the climate goals of the European Union to become a zero-greenhouse gas economy by 2050, it is necessary to reduce the greenhouse gas emissions in the mobility sector [2]. The aviation sector as part of the mobility sector accounts for 13.9% of the total greenhouse gas emissions of the mobility sector [3]. One approach to reduce these emissions is the use of alternative drive concepts in aviation, such as the hydrogen fuel cell [4]. For a long-term change in drive concepts, however, the chosen alternative drive concepts must be economically feasible [5]. In case of the hydrogen fuel cell, its feasibility largely depends on its performance which is mostly determined by the volumetric flows of reactants conveyed within the cell [6]. An increased flow of volumetric reactants leads to more chemical reactions, therefore to a higher electrical output and finally to greater performance of the fuel cell [7]. In this context, a key component for the performance is the bipolar plate, as the geometry of its channels defines the volumetric flows of reactants [8]. One method to increase the volumetric flows of reactants is to use deeper channels [9]. The channel depth is, however, limited by the formability of the material being used [10]. The limited formability during forming of bipolar plates is mostly due to the high tensile stress states within the component which lead to pronounced sheet thinning and ultimately to failure [11]. In this paper segmental forming is introduced as a novel forming process to produce channel geometries made of foil metal. Segmental forming is derived from a traditional deep drawing process currently used for the production of bipolar plates and is defined as the forming

<figure><img src="image_2226.png" /></figure>

<footnote n="1">¹Manufacturing Technology Institute MTI of RWTH Aachen University, Campus-Boulevard 30, 52074 Aachen, Germany</footnote>

<footnote n="2">²Fraunhofer Institute for Production Technology IPT, Steinbachstr. 17, 52074 Aachen, Germany</footnote>

<footnote>ªm.meerkamp@mti.rwth-aachen.de, bl.uhlmann@mti.rwth-aachen.de,</footnote>

<footnote>ᶜm.mueller@mti.rwth-aachen.de, dt.herrig@mti.rwth-aachen.de, et.bergs@mti.rwth-aachen.de</footnote>
# Novel approach for reducing thinning during forming of foil metal

MEERKAMP Max¹,ª*, UHLMANN Lars¹,ᵇ, MÜLLER Martina¹,ᶜ, HERRIG Tim¹,ᵈ and BERGS Thomas¹,²,ᵉ

¹Manufacturing Technology Institute MTI of RWTH Aachen University, Campus-Boulevard 30, 52074 Aachen, Germany

²Fraunhofer Institute for Production Technology IPT, Steinbachstr. 17, 52074 Aachen, Germany

ªm.meerkamp@mti.rwth-aachen.de, bl.uhlmann@mti.rwth-aachen.de,

ᶜm.mueller@mti.rwth-aachen.de, ᵈt.herrig@mti.rwth-aachen.de, ᶜt.bergs@mti.rwth-aachen.de

**Keywords:** Bipolar Plate, Foil Metal, Novel Forming Approach, Stress State, Thinning

**Abstract.** The mobility sector is one of the largest contributors to greenhouse gas emissions in the European Union. One approach to reduce these emissions and thereby reach the climate goals set by the European Union is the use of alternative drive concepts, such as the hydrogen fuel cell. The economic feasibility of the hydrogen fuel cell depends on its performance which is mostly determined by the volumetric flows of reactants conveyed within the cell. In this context, the key component is the bipolar plate, as its channel depth defines the volumetric flows of reactants. The channel depth is, however, limited by the formability of the material being used and the occurring sheet thinning during forming. This paper presents a novel approach of segmental forming, meaning the forming of individual channels in a defined sequence. The aim is to reduce local stress concentration in the component and to homogenize the spatial thinning within the component in order to avoid a pronounced local sheet thinning. Therefore, five channels are formed by conventional forming in a single step and by segmental forming. The results are then analyzed in terms of the thinning of the sheet thickness that occurs and conclusions are drawn in terms of the stress states that occur.

**Introduction**

One of the largest contributors to the greenhouse gas emissions in the European Union is the mobility sector [1]. To reach the climate goals of the European Union to become a zero-greenhouse gas economy by 2050, it is necessary to reduce the greenhouse gas emissions in the mobility sector [2]. The aviation sector as part of the mobility sector accounts for 13.9% of the total greenhouse gas emissions of the mobility sector [3]. One approach to reduce these emissions is the use of alternative drive concepts in aviation, such as the hydrogen fuel cell [4]. For a long-term change in drive concepts, however, the chosen alternative drive concepts must be economically feasible [5]. In case of the hydrogen fuel cell, its feasibility largely depends on its performance which is mostly determined by the volumetric flows of reactants conveyed within the cell [6]. An increased flow of volumetric reactants leads to more chemical reactions, therefore to a higher electrical output and finally to greater performance of the fuel cell [7]. In this context, a key component for the performance is the bipolar plate, as the geometry of its channels defines the volumetric flows of reactants [8]. One method to increase the volumetric flows of reactants is to use deeper channels [9]. The channel depth is, however, limited by the formability of the material being used [10]. The limited formability during forming of bipolar plates is mostly due to the high tensile stress states within the component which lead to pronounced sheet thinning and ultimately to failure [11]. In this paper segmental forming is introduced as a novel forming process to produce channel geometries made of foil metal. Segmental forming is derived from a traditional deep drawing process currently used for the production of bipolar plates and is defined as the forming

<figure><img src="image_2217.png" /></figure>
of individual channels in a chronological sequence rather than simultaneous forming of all channels. The idea is to reduce the area between the foil and the tool that is in contact simultaneously, resulting in lower tensile stress states within the component. Thereby the aim is to reduce local stress concentration in the component and to homogenize the spatial thinning within the component in order to avoid a pronounced local sheet thinning. As segmental forming is a novel approach, the possibilities offered by segmental forming to reduce local stress concentration, homogenize spatial thinning and avoid pronounced local sheet thinning are currently unknown.

## Materials and Methods

Forming process. In this work, the forming of five channels with the same channel geometry is analyzed. The blank extends beyond the width of the die. This creates additional material that can be drawn into the tool during the forming process and thereby a potentially improved material flow due to the change in process routes can be depicted. The material used is foil metal made of stainless steel 1.4404 with an initial sheet thickness $t_0$ of $t_0 = 0.1$ mm. The channel geometry, the material used, and the initial sheet thickness $t_0$ are derived from a state-of-the-art bipolar plate. The punch movement is position controlled. The punch travel $h$ is set to achieve an after forming distance between the die and punch equal to the initial sheet thickness $t_0$. The punch velocity $v$ is set to $v = 1$ mm/s. The forming of the five channels is performed using three different process routes. As a reference process route, a classical deep drawing process is performed (REF). In the process route REF, all five channels are formed in one step of forming (Fig. 1a)). The other process routes compared in this work are derived from a segmental forming approach. In the first segmental forming process route (S123) the channel in the middle is formed first (segment 1). Afterwards the channels next to the channel in the middle (segment 2) and lastly the outer channels (segment 3) are formed (Fig. 1b)).

<figure><img src="image_2227.png" /><figcaption>Figure 1: Schematic illustration of process routes a) REF and b) S123</figcaption></figure>

For the second process route using segmental forming (S321) the process route S123 is reversed, resulting in firstly forming the most outer channels (segment 3), then forming the channels next to the most outer channels (segment 2) and lastly forming the channel in the middle (segment 1).

Process model. The process routes considered in this work are modelled using Abaqus/Explicit 2021. To save computational time, the geometry is modelled in two-dimensional space and reduced to half using symmetry. The material is modelled using a HENSEL-SPITTEL material model as it was successfully used in preliminary work [12]. For contact modelling a COULOMB friction
of individual channels in a chronological sequence rather than simultaneous forming of all channels. The idea is to reduce the area between the foil and the tool that is in contact simultaneously, resulting in lower tensile stress states within the component. Thereby the aim is to reduce local stress concentration in the component and to homogenize the spatial thinning within the component in order to avoid a pronounced local sheet thinning. As segmental forming is a novel approach, the possibilities offered by segmental forming to reduce local stress concentration, homogenize spatial thinning and avoid pronounced local sheet thinning are currently unknown.

## Materials and Methods

Forming process. In this work, the forming of five channels with the same channel geometry is analyzed. The blank extends beyond the width of the die. This creates additional material that can be drawn into the tool during the forming process and thereby a potentially improved material flow due to the change in process routes can be depicted. The material used is foil metal made of stainless steel 1.4404 with an initial sheet thickness $t_0$ of $t_0 = 0.1$ mm. The channel geometry, the material used, and the initial sheet thickness $t_0$ are derived from a state-of-the-art bipolar plate. The punch movement is position controlled. The punch travel $h$ is set to achieve an after forming distance between the die and punch equal to the initial sheet thickness $t_0$. The punch velocity $v$ is set to $v = 1$ mm/s. The forming of the five channels is performed using three different process routes. As a reference process route, a classical deep drawing process is performed (REF). In the process route REF, all five channels are formed in one step of forming (Fig. 1a)). The other process routes compared in this work are derived from a segmental forming approach. In the first segmental forming process route (S123) the channel in the middle is formed first (segment 1). Afterwards the channels next to the channel in the middle (segment 2) and lastly the outer channels (segment 3) are formed (Fig. 1b)).

<figure><img src="image_2218.png" /><figcaption>Figure 1: Schematic illustration of process routes a) REF and b) S123</figcaption></figure>

For the second process route using segmental forming (S321) the process route S123 is reversed, resulting in firstly forming the most outer channels (segment 3), then forming the channels next to the most outer channels (segment 2) and lastly forming the channel in the middle (segment 1).

Process model. The process routes considered in this work are modelled using Abaqus/Explicit 2021. To save computational time, the geometry is modelled in two-dimensional space and reduced to half using symmetry. The material is modelled using a HENSEL-SPITTEL material model as it was successfully used in preliminary work [12]. For contact modelling a COULOMB friction
model with a uniform friction coefficient $\mu$ of $\mu = 0.05$ is used. The die and the punch are modelled using analytical rigid bodies to reduce computational time further. Therefore, neither the die nor the punches require meshing. The blank is modelled as a deformable solid body, using a structured mesh with quadrilateral shaped elements (C2D4R).

Reference process, methods of evaluation and area under investigation. A traditional deep drawing process is used as the reference process (REF) and is compared to the process routes S123 and S321 obtained by segmental forming. First, the thinning $\Delta t$ of the workpieces is analysed. The thinning $\Delta t$ is calculated as the distance between the nodes of the mesh on the lower and upper edge of the sheet edge, which is perpendicular to the sheet thickness. To systematically analyse the thinning $\Delta t$, the segments 1, 2 and 3 are divided into halves by the corresponding symmetry line, resulting in segment halve 1.1 (SH 1.1) and 1.2 (SH 1.2), segment halve 2.1 (SH 2.1) and 2.2 (SH 2.2), segment halve 3.1 (SH 3.1) and the area of additional material (AM). Each segment half is further divided into five characteristic areas. The five characteristic areas are defined as the upper and lower bridges (UB, LB), upper and lower radii (UR, LR) and the wall (W). The geometry after forming divided into segments and segment halves can be seen in Fig. 2a), while the characteristic areas of an exemplary segment halve are shown in Fig. 2b).

<figure><img src="image_2228.png" /><figcaption>Figure 2: Schematic illustration of a) channel geometry divided into segments (segment 1,2 and 3) and segment halves (SH 1.1, SH 1.2, SH 2.1, SH 2.2, SH 3.1 and AM) and b) characteristic areas of a segment halve (UB, UR, W, LR and LB)</figcaption></figure>

After analyzing the thinning $\Delta t$, the stress states in the most sensitive characteristic areas of the segment halves regarding thinning are analyzed using triaxiality $\eta$, lode angle parameter $\bar{\theta}$ and the VON MISES stress $\sigma_{vM}$. Stress states are generally defined by the CAUCHY stress tensor. However, the CAUCHY stress tensor does not provide an immediate indication of whether the component is under tensile stress or under compressive stress. In order to determine whether the component is under tensile or compressive stress, triaxiality $\eta$ (Eq. 1), lode angle parameter $\bar{\theta}$ (Eq. 2) and the VON MISES stress $\sigma_{vM}$ (Eq. 3) can be derived from the CAUCHY stress tensor. Whereby $\sigma_3$ is the minor, $\sigma_2$ is the mid and $\sigma_1$ is the major principal stress of the CAUCHY stress tensor.

$$ \eta = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3\sqrt{0.5 [(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}} \quad (1) $$

$$ \bar{\theta} = 1 - \frac{6}{\pi} \arccot \left( \frac{2}{\sqrt{3}} \left( \frac{\sigma_1 - \sigma_3}{\sigma_2 - \sigma_3} - \frac{1}{2} \right) \right). \quad (2) $$

$$ \sigma_{vM} = \sqrt{0.5 [(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}. \quad (3) $$
model with a uniform friction coefficient $\mu$ of $\mu = 0.05$ is used. The die and the punch are modelled using analytical rigid bodies to reduce computational time further. Therefore, neither the die nor the punches require meshing. The blank is modelled as a deformable solid body, using a structured mesh with quadrilateral shaped elements (C2D4R).

Reference process, methods of evaluation and area under investigation. A traditional deep drawing process is used as the reference process (REF) and is compared to the process routes S123 and S321 obtained by segmental forming. First, the thinning $\Delta t$ of the workpieces is analysed. The thinning $\Delta t$ is calculated as the distance between the nodes of the mesh on the lower and upper edge of the sheet edge, which is perpendicular to the sheet thickness. To systematically analyse the thinning $\Delta t$, the segments 1, 2 and 3 are divided into halves by the corresponding symmetry line, resulting in segment halve 1.1 (SH 1.1) and 1.2 (SH 1.2), segment halve 2.1 (SH 2.1) and 2.2 (SH 2.2), segment halve 3.1 (SH 3.1) and the area of additional material (AM). Each segment half is further divided into five characteristic areas. The five characteristic areas are defined as the upper and lower bridges (UB, LB), upper and lower radii (UR, LR) and the wall (W). The geometry after forming divided into segments and segment halves can be seen in Fig. 2a), while the characteristic areas of an exemplary segment halve are shown in Fig. 2b).

<figure><img src="image_2219.png" /><figcaption>Figure 2: Schematic illustration of a) channel geometry divided into segments (segment 1,2 and 3) and segment halves (SH 1.1, SH 1.2, SH 2.1, SH 2.2, SH 3.1 and AM) and b) characteristic areas of a segment halve (UB, UR, W, LR and LB)</figcaption></figure>

After analyzing the thinning $\Delta t$, the stress states in the most sensitive characteristic areas of the segment halves regarding thinning are analyzed using triaxiality $\eta$, lode angle parameter $\bar{\theta}$ and the VON MISES stress $\sigma_{vM}$. Stress states are generally defined by the CAUCHY stress tensor. However, the CAUCHY stress tensor does not provide an immediate indication of whether the component is under tensile stress or under compressive stress. In order to determine whether the component is under tensile or compressive stress, triaxiality $\eta$ (Eq. 1), lode angle parameter $\bar{\theta}$ (Eq. 2) and the VON MISES stress $\sigma_{vM}$ (Eq. 3) can be derived from the CAUCHY stress tensor. Whereby $\sigma_3$ is the minor, $\sigma_2$ is the mid and $\sigma_1$ is the major principal stress of the CAUCHY stress tensor.

$$ \eta = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3\sqrt{0.5 [(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}} \quad (1) $$

$$ \bar{\theta} = 1 - \frac{6}{\pi} \arccot \left( \frac{2}{\sqrt{3}} \left( \frac{\sigma_1 - \sigma_3}{\sigma_2 - \sigma_3} - \frac{1}{2} \right) \right). \quad (2) $$

$$ \sigma_{vM} = \sqrt{0.5 [(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}. \quad (3) $$
## Results

Influence of process route on thinning. The workpiece after forming divided into segment halves and characteristic areas is visualized in Fig. 3a), while the thinning $\Delta t$ in dependence of the distance to the symmetry line of the workpiece $x$ is shown in Fig. 3b).

<figure><img src="image_2229.png" /><figcaption>Figure 3: a) Workpiece divided into segment halves (SH 1.1, SH 1.2, SH 2.1, SH 2.2, SH 3.1 and AM) and characteristic areas (UB, UR, W, LR and LB) and b) thinning $\Delta t$ in dependence of distance to symmetry line $x$</figcaption></figure>

Overall, the thinning $\Delta t$ varies greatly for all process routes within the workpiece. However, the thinning $\Delta t$ of S321 is in almost all characteristic areas of the segment halves significantly higher than the thinning $\Delta t$ of REF. The thinning $\Delta t$ of REF in turn is slightly higher than the thinning $\Delta t$ of S123. Only in the characteristic areas UR of SH 2.1, LB of SH 2.2 and UB and LR of SH 3.1 the thinning $\Delta t$ of S123 briefly exceeds the thinning $\Delta t$ of REF. In addition, the thinning $\Delta t$ of S321 in the characteristic area UB of SH 2.2 and in the characteristic areas UR and LR of SH 3.1 is below the thinning $\Delta t$ of REF. In AM, almost no thinning $\Delta t$ occurs. It is noticeable that the thinning $\Delta t$ increases significantly in the characteristic area LR of each segment halve for all process routes. For each segment halve the local maximum of thinning $\Delta t_{max,local}$ is reached in the characteristic area LR (Table 1).

<table><caption>Table 1: Local maxima of thinning $\Delta t_{max,local}$ in characteristic area LR of each segment halve</caption><thead><tr><th rowspan="2">Segment halve</th><th colspan="3">Local maxima of thinning Δt<sub>max,local</sub>/%</th></tr><tr><th>REF</th><th>S123</th><th>S321</th></tr></thead><tbody><tr><td>SH 1.1</td><td>3.23</td><td>2.10</td><td>10.93</td></tr><tr><td>SH 1.2</td><td>3.16</td><td>1.84</td><td>9.86</td></tr><tr><td>SH 2.1</td><td>2.68</td><td>2.06</td><td>5.65</td></tr><tr><td>SH 2.2</td><td>2.63</td><td>1.86</td><td>5.71</td></tr><tr><td>SH 3.1</td><td>1.60</td><td>1.95</td><td>4.26</td></tr></tbody></table>

In the characteristic area LR of each segment the highest local maxima of thinning $\Delta t_{max,local}$ occur in S321, while the lowest local maxima of thinning $\Delta t_{max,local}$ are reached in S123. For all
## Results

Influence of process route on thinning. The workpiece after forming divided into segment halves and characteristic areas is visualized in Fig. 3a), while the thinning $\Delta t$ in dependence of the distance to the symmetry line of the workpiece $x$ is shown in Fig. 3b).

<figure><img src="image_2220.png" /><figcaption>Figure 3: a) Workpiece divided into segment halves (SH 1.1, SH 1.2, SH 2.1, SH 2.2, SH 3.1 and AM) and characteristic areas (UB, UR, W, LR and LB) and b) thinning $\Delta t$ in dependence of distance to symmetry line $x$</figcaption></figure>

Overall, the thinning $\Delta t$ varies greatly for all process routes within the workpiece. However, the thinning $\Delta t$ of S321 is in almost all characteristic areas of the segment halves significantly higher than the thinning $\Delta t$ of REF. The thinning $\Delta t$ of REF in turn is slightly higher than the thinning $\Delta t$ of S123. Only in the characteristic areas UR of SH 2.1, LB of SH 2.2 and UB and LR of SH 3.1 the thinning $\Delta t$ of S123 briefly exceeds the thinning $\Delta t$ of REF. In addition, the thinning $\Delta t$ of S321 in the characteristic area UB of SH 2.2 and in the characteristic areas UR and LR of SH 3.1 is below the thinning $\Delta t$ of REF. In AM, almost no thinning $\Delta t$ occurs. It is noticeable that the thinning $\Delta t$ increases significantly in the characteristic area LR of each segment halve for all process routes. For each segment halve the local maximum of thinning $\Delta t_{max,local}$ is reached in the characteristic area LR (Table 1).

<table><caption>Table 1: Local maxima of thinning $\Delta t_{max,local}$ in characteristic area LR of each segment halve</caption><thead><tr><th rowspan="2">Segment halve</th><th colspan="3">Local maxima of thinning Δt<sub>max,local</sub>/%</th></tr><tr><th>REF</th><th>S123</th><th>S321</th></tr></thead><tbody><tr><td>SH 1.1</td><td>3.23</td><td>2.10</td><td>10.93</td></tr><tr><td>SH 1.2</td><td>3.16</td><td>1.84</td><td>9.86</td></tr><tr><td>SH 2.1</td><td>2.68</td><td>2.06</td><td>5.65</td></tr><tr><td>SH 2.2</td><td>2.63</td><td>1.86</td><td>5.71</td></tr><tr><td>SH 3.1</td><td>1.60</td><td>1.95</td><td>4.26</td></tr></tbody></table>

In the characteristic area LR of each segment the highest local maxima of thinning $\Delta t_{max,local}$ occur in S321, while the lowest local maxima of thinning $\Delta t_{max,local}$ are reached in S123. For all