# Numerical and experimental studies of molten pool phenomena
influence on dissimilar materials coatings by L-PBF

JACQUOT Amaury<fnref n="1" />,ᵃ*, SOVEJA Adriana<fnref n="2" />, LEMAOULT Yannick<fnref n="1" />,
BOHER Christine<fnref n="1" />, MARCOUX Manuel<fnref n="3" /> and ESCAPE Christophe<fnref n="4" />

**Keywords:** L-PBF Process, Heat Transfer, Multiphysics Model, Characterization

**Abstract.** Over the past few years, **Laser Powder Bed Fusion (L-PBF)** has become increasingly popular, as this method enables energy and material savings during the manufacturing process. 3D L-PBF parts, based on computer-designed geometries, are generated layer by layer using laser energy. Since the beginning of these studies several decades ago, significant progress has been made in the understanding of additive manufacturing, particularly with regard to properties, structure and in situ monitoring. However, the scientific manufacturing community has yet to achieve optimal operational reliability. Presently, numerous defects remain a major problem for parts produced by L-PBF. These defects are mainly caused by the movement of the molten material and its rate of solidification within the melt, which is also influenced by the thermal phenomena of the process. In the L-PBF manufacturing process, the main challenge is to control the complex interdependence of these phenomena [1]. The aim of this work is therefore to study the various physical aspects during a Laser Powder Bed Fusion (L-PBF) process. For this purpose, it is necessary to provide a numerical model with specific works on experimental characterizations of materials at high temperature (up to 1000 °C) as a model's input parameters, such as the thermal or optical properties of cobalt based powder (CoCrMo) used in our study. In addition to material characterization, theoretical model studies have been used to determine the thermal properties of L-PBF powder [2], in particular thermal diffusivity. This model is also validated by specific experimental characterizations, involving the dilution of iron substrates in a CoCrMo alloy deposit and temperature measurements during the manufacturing process. Furthermore, the amount of iron transferred from the substrate to the coating can be used as an indicator of the molten metal movement in the melt and, ultimately, of the operating parameters used to apply the coating. Too much iron on the surface impairs the mechanical strength of the substrate-coating assembly, thus indicates poor control of the parameters used for this purpose [1].

**Introduction**

Metallic additive manufacturing allows structure fabrication layer-by-layer deposition process and is revolutionizing the manufacturing industry due to its ability to obtain near-net shape products, in a short time span, with almost no material waste. The L-PBF is one of the most used among the additive manufacturing (AM) process [2] and plays an important role in the industry by offering new possibilities of parts shaping, especially for the complex-shape parts. Its principle is to use a heat source to agglomerate a dense material based on pre-deposited layer of powder [3].

However, the AM suffers from certain weakness. Some of AM-made parts have some default like bad homogenization. Depending on the range of the machine parameters like laser scanning

<figure><img src="image_17.png" /></figure>

<footnote n="1">¹Institut Clément Ader (ICA), Campus Jarlard, 81013 Albi, Fance</footnote>

<footnote n="2">²Institut Clément Ader (ICA), 3 rue Caroline Aigle, 31400 Toulouse, France</footnote>

<footnote n="3">³Institut de Mécanique des Fluides de Toulouse, Allée Prof. Camille Soula, 31400 Toulouse,
France</footnote>

<footnote n="4">⁴PROMES -UPR CNRS 8521, France</footnote>

<footnote>ªamaury.jacquot@mines-albi.fr</footnote>
# Numerical and experimental studies of molten pool phenomena
influence on dissimilar materials coatings by L-PBF

JACQUOT Amaury¹,ª*, SOVEJA Adriana², LEMAOULT Yannick¹,
BOHER Christine¹, MARCOUX Manuel³ and ESCAPE Christophe⁴

¹Institut Clément Ader (ICA), Campus Jarlard, 81013 Albi, Fance

²Institut Clément Ader (ICA), 3 rue Caroline Aigle, 31400 Toulouse, France

³Institut de Mécanique des Fluides de Toulouse, Allée Prof. Camille Soula, 31400 Toulouse,
France

⁴PROMES -UPR CNRS 8521, France

ªamaury.jacquot@mines-albi.fr

**Keywords:** L-PBF Process, Heat Transfer, Multiphysics Model, Characterization

**Abstract.** Over the past few years, **Laser Powder Bed Fusion (L-PBF)** has become increasingly popular, as this method enables energy and material savings during the manufacturing process. 3D L-PBF parts, based on computer-designed geometries, are generated layer by layer using laser energy. Since the beginning of these studies several decades ago, significant progress has been made in the understanding of additive manufacturing, particularly with regard to properties, structure and in situ monitoring. However, the scientific manufacturing community has yet to achieve optimal operational reliability. Presently, numerous defects remain a major problem for parts produced by L-PBF. These defects are mainly caused by the movement of the molten material and its rate of solidification within the melt, which is also influenced by the thermal phenomena of the process. In the L-PBF manufacturing process, the main challenge is to control the complex interdependence of these phenomena [1]. The aim of this work is therefore to study the various physical aspects during a Laser Powder Bed Fusion (L-PBF) process. For this purpose, it is necessary to provide a numerical model with specific works on experimental characterizations of materials at high temperature (up to 1000 °C) as a model's input parameters, such as the thermal or optical properties of cobalt based powder (CoCrMo) used in our study. In addition to material characterization, theoretical model studies have been used to determine the thermal properties of L-PBF powder [2], in particular thermal diffusivity. This model is also validated by specific experimental characterizations, involving the dilution of iron substrates in a CoCrMo alloy deposit and temperature measurements during the manufacturing process. Furthermore, the amount of iron transferred from the substrate to the coating can be used as an indicator of the molten metal movement in the melt and, ultimately, of the operating parameters used to apply the coating. Too much iron on the surface impairs the mechanical strength of the substrate-coating assembly, thus indicates poor control of the parameters used for this purpose [1].

**Introduction**

Metallic additive manufacturing allows structure fabrication layer-by-layer deposition process and is revolutionizing the manufacturing industry due to its ability to obtain near-net shape products, in a short time span, with almost no material waste. The L-PBF is one of the most used among the additive manufacturing (AM) process [2] and plays an important role in the industry by offering new possibilities of parts shaping, especially for the complex-shape parts. Its principle is to use a heat source to agglomerate a dense material based on pre-deposited layer of powder [3].

However, the AM suffers from certain weakness. Some of AM-made parts have some default like bad homogenization. Depending on the range of the machine parameters like laser scanning

<figure><img src="image_18.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
speed, laser power, inner temperature, thermal relaxation time, the shape and the properties of the final part might be different.

To investigate the origins of these issues, we have used the L-PBF process as coating tools. A substrate made of C35 steel has been coated with a thin layer of CoCrMo alloy. In a previous study [4] in the same context, it has been found that iron migration rises into the coating layer. In addition, these dilutions serve as a marker to explain the various physical phenomena present during this AM process and the operating parameters influences on the final results. This work aims to understand the application of L-PBF on these dissimilar materials through an accurate model.

## Materials

For both experimentation and numerical modeling, a set of two types of materials have been used: C35 steel alloy as substrates and a CoCrMo alloy, named stellite 21, as coating material. The chemical compositions of both materials are detailed in Tables 1 and 2 respectively.

<table><caption>Table 1 Chemical composition of the C35 steel alloy</caption><thead><tr><th>Element</th><th>Fe</th><th>C</th><th>Mn</th><th>Si</th><th>P</th><th>S</th><th>Cr</th><th>Ni</th><th>Mo</th><th>Cu</th><th>Al</th></tr></thead><tbody><tr><th>Wt%</th><td>Base</td><td>0.35</td><td>0.68</td><td>0.21</td><td>0.016</td><td>0.024</td><td>0.19</td><td>0.14</td><td>0.02</td><td>0.17</td><td>0.023</td></tr></tbody></table>

<table><caption>Table 2 Chemical composition of the CoCrMo alloy</caption><thead><tr><th>Element</th><th>Co</th><th>Cr</th><th>Mo</th><th>Mn</th><th>Fe</th><th>Si</th><th>P</th><th>C</th></tr></thead><tbody><tr><th>Wt%</th><td>63.45</td><td>28.9</td><td>6.4</td><td>0.3</td><td>0.2</td><td>0.6</td><td>0.04</td><td>0.11</td></tr></tbody></table>

The particles of CoCrMo alloy powder generally have a spherical shape (Fig.1), with a particle size distribution between 30 and 50 µm.

<figure><img src="image_18.png" /><figcaption>Figure 1 SEM observation of the coating powder presenting predominantly a spherical form of particles and some particles having an asteroidal shape (a) and particle size distribution histogram (b)</figcaption></figure>

The properties required to set up the model are well known for the steel composing the substrate. However, little was known about the thermal properties of CoCrMo powder. The next chapter will therefore focus on characterizing these properties.

## Diffusivity Measurement

Numerical models have been widely used to understand complex physical phenomena in the printing process, and to study the effect of causal variables on heat transfer, microstructure or residual stresses in printed parts. Heat transfer analysis is a fundamental step in modeling the process. The accuracy of heat transfer analysis relies on accurate characterizations of the material properties, such as density, specific heat, and thermal conductivity. We have carried out the
speed, laser power, inner temperature, thermal relaxation time, the shape and the properties of the final part might be different.

To investigate the origins of these issues, we have used the L-PBF process as coating tools. A substrate made of C35 steel has been coated with a thin layer of CoCrMo alloy. In a previous study [4] in the same context, it has been found that iron migration rises into the coating layer. In addition, these dilutions serve as a marker to explain the various physical phenomena present during this AM process and the operating parameters influences on the final results. This work aims to understand the application of L-PBF on these dissimilar materials through an accurate model.

## Materials

For both experimentation and numerical modeling, a set of two types of materials have been used: C35 steel alloy as substrates and a CoCrMo alloy, named stellite 21, as coating material. The chemical compositions of both materials are detailed in Tables 1 and 2 respectively.

<table><caption>Table 1 Chemical composition of the C35 steel alloy</caption><thead><tr><th>Element</th><th>Fe</th><th>C</th><th>Mn</th><th>Si</th><th>P</th><th>S</th><th>Cr</th><th>Ni</th><th>Mo</th><th>Cu</th><th>Al</th></tr></thead><tbody><tr><th>Wt%</th><td>Base</td><td>0.35</td><td>0.68</td><td>0.21</td><td>0.016</td><td>0.024</td><td>0.19</td><td>0.14</td><td>0.02</td><td>0.17</td><td>0.023</td></tr></tbody></table>

<table><caption>Table 2 Chemical composition of the CoCrMo alloy</caption><thead><tr><th>Element</th><th>Co</th><th>Cr</th><th>Mo</th><th>Mn</th><th>Fe</th><th>Si</th><th>P</th><th>C</th></tr></thead><tbody><tr><th>Wt%</th><td>63.45</td><td>28.9</td><td>6.4</td><td>0.3</td><td>0.2</td><td>0.6</td><td>0.04</td><td>0.11</td></tr></tbody></table>

The particles of CoCrMo alloy powder generally have a spherical shape (Fig.1), with a particle size distribution between 30 and 50 µm.

<figure><img src="image_19.png" /><figcaption>Figure 1 SEM observation of the coating powder presenting predominantly a spherical form of particles and some particles having an asteroidal shape (a) and particle size distribution histogram (b)</figcaption></figure>

The properties required to set up the model are well known for the steel composing the substrate. However, little was known about the thermal properties of CoCrMo powder. The next chapter will therefore focus on characterizing these properties.

## Diffusivity Measurement

Numerical models have been widely used to understand complex physical phenomena in the printing process, and to study the effect of causal variables on heat transfer, microstructure or residual stresses in printed parts. Heat transfer analysis is a fundamental step in modeling the process. The accuracy of heat transfer analysis relies on accurate characterizations of the material properties, such as density, specific heat, and thermal conductivity. We have carried out the
measurement of thermal diffusivity on this powder, for which there is a lack of data. The thermal diffusivity describes a material's ability to defuse heat through the sample. It is related to transient conduction through the sample and is defined by “$α_{th}$” as the thermal diffusivity, “$k$” as thermal conductivity, “$ρ$” as material density and “$C_p$” is the heat capacity:

$$α_{th} = \frac{k}{ρC_p} \qquad (1)$$

To evaluate the thermal diffusivity of our powder, we use the Netzsch® LFA 457 micro-flash.This system is composed of a furnace which allows elevating the ambient temperature around the specimen from -125°C to 1100°C. A laser is used to elevate slightly the temperature sample for measurement purpose. To calculate the thermal diffusivity, this system evaluates the time needed for the edge of the sample to reach half the maximum temperature after the pulse heating from the laser. In order to recreate the same measurement conditions as during additive manufacturing, the gas surrounding the sample is argon at atmospheric pressure.

The thermal diffusivity has been measured between the room temperature (30°C) and 1000°C in order to have a good knowledge of this property over the temperature range corresponding to the manufacturing conditions. Three “shots” (punctual measurement) has been made in order to quantify thermal diffusivity uncertainties of these measurements as shown in figure 2 and 3.

<figure><img src="image_19.png" /><figcaption>Figure 2 Thermal diffusivity's measurement points</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 3 Theoretical and Experimental Thermal Diffusivities</figcaption></figure>

In addition to experimental measurement, some theoretical models have been established to estimate the thermal conductivity, heat capacity and the thermal diffusivity. This article compares thermal diffusivities obtained from theoretical models with those measured. The apparent density was measured and its value is listed in Table 3. As for the heat capacity, a series of measurements was made between room temperature and 300°C [5]. Finally, the thermal conductivity model is based on that of Sih and Barlow. Sih and Barlow model's [6] integrate the Damköhler term (referred to the contribution of thermal radiation) into the Zehner-Schlünder's model to estimate the Equivalent Thermal Conductivity (ETC).

Figure 3 shows a comparison between model-derived and measured diffusivity. Both trends and orders of magnitude agree. However, the experimental thermal diffusivity shows a much more pronounced curvature than the theoretical one. The origin of these discrepancies may lie in the Sih and Barlow model [6], which may neglect some aspects of heat exchange such as the evolution of contact resistance between particles, in particular towards the warmer temperatures. The differences between the two curves may also stem from heat capacity measurements, where we lack certainty as to values and their evolution above 300°C.
measurement of thermal diffusivity on this powder, for which there is a lack of data. The thermal diffusivity describes a material's ability to defuse heat through the sample. It is related to transient conduction through the sample and is defined by “$α_{th}$” as the thermal diffusivity, “$k$” as thermal conductivity, “$ρ$” as material density and “$C_p$” is the heat capacity:

$$α_{th} = \frac{k}{ρC_p} \qquad (1)$$

To evaluate the thermal diffusivity of our powder, we use the Netzsch® LFA 457 micro-flash.This system is composed of a furnace which allows elevating the ambient temperature around the specimen from -125°C to 1100°C. A laser is used to elevate slightly the temperature sample for measurement purpose. To calculate the thermal diffusivity, this system evaluates the time needed for the edge of the sample to reach half the maximum temperature after the pulse heating from the laser. In order to recreate the same measurement conditions as during additive manufacturing, the gas surrounding the sample is argon at atmospheric pressure.

The thermal diffusivity has been measured between the room temperature (30°C) and 1000°C in order to have a good knowledge of this property over the temperature range corresponding to the manufacturing conditions. Three “shots” (punctual measurement) has been made in order to quantify thermal diffusivity uncertainties of these measurements as shown in figure 2 and 3.

<figure><img src="image_20.png" /><figcaption>Figure 2 Thermal diffusivity's measurement points</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 3 Theoretical and Experimental Thermal Diffusivities</figcaption></figure>

In addition to experimental measurement, some theoretical models have been established to estimate the thermal conductivity, heat capacity and the thermal diffusivity. This article compares thermal diffusivities obtained from theoretical models with those measured. The apparent density was measured and its value is listed in Table 3. As for the heat capacity, a series of measurements was made between room temperature and 300°C [5]. Finally, the thermal conductivity model is based on that of Sih and Barlow. Sih and Barlow model's [6] integrate the Damköhler term (referred to the contribution of thermal radiation) into the Zehner-Schlünder's model to estimate the Equivalent Thermal Conductivity (ETC).

Figure 3 shows a comparison between model-derived and measured diffusivity. Both trends and orders of magnitude agree. However, the experimental thermal diffusivity shows a much more pronounced curvature than the theoretical one. The origin of these discrepancies may lie in the Sih and Barlow model [6], which may neglect some aspects of heat exchange such as the evolution of contact resistance between particles, in particular towards the warmer temperatures. The differences between the two curves may also stem from heat capacity measurements, where we lack certainty as to values and their evolution above 300°C.
## Numerical Model

In this work, a two-dimensional heat and mass transfer numerical model is developed to study the CoCrMo-iron interface composition evolution during additive laser manufacturing of the coating. The numerical model is created using a commercial finite element package COMSOL MultiPhysics®. To represent the L-PBF process, we used a 2D model representing the C35 steel cylinder with a thin layer of CoCrMo alloy powder on the top.

Heat dissipation takes place by convection on the vertical and upper walls, and radiation exchanges are only considered on the cylinder's upper edge. The L-PBF support is maintained at 100°C during the coating process. To model the interaction between the substrate and the support, we considered the temperature at the bottom edge at this imposed temperature.

<figure><img src="image_21.png" /><figcaption>Figure 4 Computation domain and associated mesh</figcaption></figure>

The geometry is meshed with several tetrahedral elements (Fig.4). The mesh generated uses the finest elements in the cobalt powder layer area since this is in this area where the gradient of temperatures and velocities are the most important. With this method, an adaptive mesh, consisting of 15473 tetrahedral elements, has been set up. With the exception of heat capacity and thermal diffusivity, which were measured experimentally for the solid phase.

## Heat Transfer Modeling

For the laser heat source model, Gusarov et al [8] proposed to couple the heat transfers and radiation on a homogeneous absorbing scattering medium. Boley et al [9] and Khairallah et al [10], [11] used the ray-tracing method to simulate the input heat flow in a realistic L-PBF process (both on the size of the particles and their array) process. Most generally, authors used a Gaussian shape to model the heat source distribution. Some authors [10] have used a Gaussian surface distribution and others have used a volumetric distribution of the heat source to model the laser-powder interaction. According to Boley et al [9], the laser powder layer interaction takes place mostly one of the first layer of the particle, especially on agglomerates of tiny particles under the laser beam.

In this study the heat source term is used as a volume source with a penetration corresponding to the multi-reflection under the powder layer:

$$Q = \frac{2 \times A \times P_L}{\pi \cdot R_L^2 \cdot \eta} \exp\left(\frac{2 \times (x - V_{Laser} \times t)^2}{R_L^2}\right) \exp\left(\frac{[z - z_0]}{\eta}\right) \quad (2)$$

Here "A" is the optical absorptivity of the powder, "$P_L$" is the nominal laser power" $R_L$" is the projected laser spot, "$V_{Laser}$" is the scanning speed and "$\eta$" being the laser penetration depth. A Gaussian standard deviation corresponding to D4$\sigma$ method according to Li et al [12] with a radius
## Numerical Model

In this work, a two-dimensional heat and mass transfer numerical model is developed to study the CoCrMo-iron interface composition evolution during additive laser manufacturing of the coating. The numerical model is created using a commercial finite element package COMSOL MultiPhysics®. To represent the L-PBF process, we used a 2D model representing the C35 steel cylinder with a thin layer of CoCrMo alloy powder on the top.

Heat dissipation takes place by convection on the vertical and upper walls, and radiation exchanges are only considered on the cylinder's upper edge. The L-PBF support is maintained at 100°C during the coating process. To model the interaction between the substrate and the support, we considered the temperature at the bottom edge at this imposed temperature.

<figure><img src="image_22.png" /><figcaption>Figure 4 Computation domain and associated mesh</figcaption></figure>

The geometry is meshed with several tetrahedral elements (Fig.4). The mesh generated uses the finest elements in the cobalt powder layer area since this is in this area where the gradient of temperatures and velocities are the most important. With this method, an adaptive mesh, consisting of 15473 tetrahedral elements, has been set up. With the exception of heat capacity and thermal diffusivity, which were measured experimentally for the solid phase.

## Heat Transfer Modeling

For the laser heat source model, Gusarov et al [8] proposed to couple the heat transfers and radiation on a homogeneous absorbing scattering medium. Boley et al [9] and Khairallah et al [10], [11] used the ray-tracing method to simulate the input heat flow in a realistic L-PBF process (both on the size of the particles and their array) process. Most generally, authors used a Gaussian shape to model the heat source distribution. Some authors [10] have used a Gaussian surface distribution and others have used a volumetric distribution of the heat source to model the laser-powder interaction. According to Boley et al [9], the laser powder layer interaction takes place mostly one of the first layer of the particle, especially on agglomerates of tiny particles under the laser beam.

In this study the heat source term is used as a volume source with a penetration corresponding to the multi-reflection under the powder layer:

$$Q = \frac{2 \times A \times P_L}{\pi \cdot R_L^2 \cdot \eta} \exp\left(\frac{2 \times (x - V_{Laser} \times t)^2}{R_L^2}\right) \exp\left(\frac{[z - z_0]}{\eta}\right) \quad (2)$$

Here "A" is the optical absorptivity of the powder, "$P_L$" is the nominal laser power" $R_L$" is the projected laser spot, "$V_{Laser}$" is the scanning speed and "$\eta$" being the laser penetration depth. A Gaussian standard deviation corresponding to D4$\sigma$ method according to Li et al [12] with a radius
corresponding to $1/e^2$ of maximal power has been chosen. According to Boley et al [9], it is assumed that the penetration's depth corresponds to the average of the particle size. With a majority of approximately 30µm particle size (Fig.2), it can be supposed that penetration of a non-negligible part of the heat source can be at this depth. With our assumption of that homogeneous medium for the powder, the simplest approach is to consider the particles are disposed on an ideal stacking with one size of particle i.e. the average particle size.

With these assumptions for the heat input, the heat transfers equation for the melted pool can be defined as:

$$ \rho C_p \frac{\partial T}{\partial t} - \rho C_p \vec{\nabla}(\vec{u}T) + k \vec{\nabla}^2 T = q \quad (3) $$

where " $\rho$ " is the density of the material, " $C_p$ " is the heat capacity, " $u$ " is the velocity vector, " $k$ " is the thermal conductivity and " $q$ " is the term combining the various heat sources and losses. Here, we considered the solid part as a liquid with a high viscosity value to neglect any velocity unless the liquidus temperature is reached.

On the edges of the plots, the heat transfers loss is given by:

$$ -\vec{n} \cdot (-k\vec{\nabla}T) = h_{\infty}(T - T_{\infty}) + \epsilon\sigma(T^4 - T_{\infty}^4) \quad (4) $$

In this equation, " $h$ " is the convective exchange coefficient, " $\epsilon$ " is the thermal emissivity and " $\sigma$ " is the Stefan Boltzmann constant. During the melting of the powder layer, the values of $\rho$, $C_p$, and $k$ to be used are determined by averaging in powder and liquid phase " $f_L$ ". These phases are determined by the condition below:

$$ f_L = \begin{cases} 0 & T < T_s \\ \frac{T - T_s}{T_L - T_s} & T_s \le T < T_L \\ 1 & T > T_L \end{cases} \quad (5) $$

where " $T_s$ " and " $T_L$ " are the solidus and liquid temperature respectively. With this condition, an approximative value of the mixture properties like the heat capacity can be calculated:

$$ C_p = \frac{1}{\rho} \left\{ (1 - f_L) \rho_s C_{ps} + f_L \rho_L C_{pL} \right\} + L_f \frac{\partial \alpha_m}{\partial T} \quad (6) $$

Particularly, only the " $C_{ps}$ " comes from a laboratory in-situ measurement.

For the thermal condition, we calculate its evolution through the temperature between the temperature room (293K) and the Liquidus of each material.

## Melt flow Modeling

For the mass flow inside the melt pool, two approaches have been used for the thermomechanical process. For their dissimilar material welding, Métais et al [13] used a RANS k-ω method to represent the multi-vortex phenomena in this context. For the L-PBF model, the main model found is the laminar model [10], [12], [13] with a Newtonian fluid to representing the melted material.

To model the melted metal flow, the conservation of mass is considered:

$$ \frac{\partial \rho}{\partial t} + \rho \vec{\nabla} \cdot \vec{u} = 0 \quad (7) $$

and the momentum conservation equation is given as:
corresponding to $1/e^2$ of maximal power has been chosen. According to Boley et al [9], it is assumed that the penetration's depth corresponds to the average of the particle size. With a majority of approximately 30µm particle size (Fig.2), it can be supposed that penetration of a non-negligible part of the heat source can be at this depth. With our assumption of that homogeneous medium for the powder, the simplest approach is to consider the particles are disposed on an ideal stacking with one size of particle i.e. the average particle size.

With these assumptions for the heat input, the heat transfers equation for the melted pool can be defined as:

$$ \rho C_p \frac{\partial T}{\partial t} - \rho C_p \vec{\nabla}(\vec{u}T) + k \vec{\nabla}^2 T = q \quad (3) $$

where " $\rho$ " is the density of the material, " $C_p$ " is the heat capacity, " $u$ " is the velocity vector, " $k$ " is the thermal conductivity and " $q$ " is the term combining the various heat sources and losses. Here, we considered the solid part as a liquid with a high viscosity value to neglect any velocity unless the liquidus temperature is reached.

On the edges of the plots, the heat transfers loss is given by:

$$ -\vec{n} \cdot (-k \vec{\nabla} T) = h_{\infty}(T - T_{\infty}) + \epsilon \sigma (T^4 - T_{\infty}^4) \quad (4) $$

In this equation, " $h$ " is the convective exchange coefficient, " $\epsilon$ " is the thermal emissivity and " $\sigma$ " is the Stefan Boltzmann constant. During the melting of the powder layer, the values of $\rho$, $C_p$, and $k$ to be used are determined by averaging in powder and liquid phase " $f_L$ ". These phases are determined by the condition below:

$$ f_L = \begin{cases} 0 & T < T_s \\ \frac{T - T_s}{T_L - T_s} & T_s \le T < T_L \\ 1 & T > T_L \end{cases} \quad (5) $$

where " $T_s$ " and " $T_L$ " are the solidus and liquid temperature respectively. With this condition, an approximative value of the mixture properties like the heat capacity can be calculated:

$$ C_p = \frac{1}{\rho} \left\{ (1 - f_L) \rho_s C_{ps} + f_L \rho_L C_{pL} \right\} + L_f \frac{\partial \alpha_m}{\partial T} \quad (6) $$

Particularly, only the " $C_{ps}$ " comes from a laboratory in-situ measurement.

For the thermal condition, we calculate its evolution through the temperature between the temperature room (293K) and the Liquidus of each material.

## Melt flow Modeling

For the mass flow inside the melt pool, two approaches have been used for the thermomechanical process. For their dissimilar material welding, Métais et al [13] used a RANS k-ω method to represent the multi-vortex phenomena in this context. For the L-PBF model, the main model found is the laminar model [10], [12], [13] with a Newtonian fluid to representing the melted material.

To model the melted metal flow, the conservation of mass is considered:

$$ \frac{\partial \rho}{\partial t} + \rho \vec{\nabla} \cdot \vec{u} = 0 \quad (7) $$

and the momentum conservation equation is given as:
$$ \rho \left( \frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot (\nabla \cdot \vec{u}) \right) = \nabla \cdot \left[ -pI + \mu \left( (\nabla \vec{u}) + (\nabla \vec{u})^T \right) \right] + F_A + K \vec{u} \quad (8) $$

Here “I” is the diagonal hydrostatic pressure matrix “$F_A$” is the buoyancy force, “K” is the Darcy damping force and “μ” is the local viscosity. Here we consider the buoyancy forces on the liquid metal induced by the difference of temperature between the center and the edge of the melting pool. With the Boussinesq approximation, we estimate the buoyancy contribution by:

$$ F_A = \rho_L \vec{g} \beta_T (T - T_f) \quad (9) $$

where “$T_f$” is the melting temperature of both Cobalt alloy and steel, “$\beta_T$” is the thermal expansion coefficient. The transition from the solid to liquid state is treated with a Darcy condition and the equivalent viscosity methods combined. The first term modifies the viscosity of the medium to give it a deliberately high value, and weakens as soon as the local temperature reaches the melting point. It is applied in this form:

$$ \mu = \mu_{\text{solid}} + (\mu - \mu_{\text{solid}}) * flc2hs(T, T_{\text{fusion}}) \quad (10) $$

and the last term is a force to damp the movement outside of the melted area:

$$ K = \frac{-C(1 - f_l)^2}{f_l^3 + b} \quad (11) $$

Where “C” is Darcy's law coefficient deliberately set to a high value to attenuate velocities, “b” is a coefficient to avoid the division by 0 and “$f_l$” is the local liquid proportion seen in equation (5).

## Species transport Modeling

Initial studies of modelling of species transport was carried out for the mixing of species during the welding process of two different materials. These studies were then extended to the context of additive manufacturing [17]. In this work, we consider the movement of the different species inside the melting pool. Being in an area assumed to be liquid, we considered the mix of species by the diluted equation species transports:

$$ \frac{\partial \vec{c}_i}{\partial t} - D_i \vec{\nabla}^2 \vec{c}_i + \vec{u} \times \vec{\nabla} \vec{c}_i = R_i \quad (12) $$

Where “c” is the local material concentration and “$D_i$” represent the diffusion coefficient, we considered dependent on the temperature by using the Stokes-Einstein formula

$$ D_i = \frac{\kappa \times T}{6 \cdot \pi \cdot \mu \cdot r_i} (T > T_m) \quad (13) $$

## Simulation Parameters

Finally, the properties of each material are summarized in the table 3 [7]. Note the physical properties come for the literature excepted the thermal conductivity. For this one, it comes from the characterization of the thermal diffusivity. As explained in the "diffusivity measurement" section, this comes from a calculation based on the definition of this diffusivity and the measurements of each parameter associated with it. In view of the wide temperature variations associated with this type of process, it should be noted that the values are thermally dependent. This is because the behavior of matter is no longer the same between low and "high" temperatures (near melting point).
$$ \rho \left( \frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot (\nabla \cdot \vec{u}) \right) = \vec{\nabla} \cdot \left[ -pI + \mu \left( (\nabla \vec{u}) + (\nabla \vec{u})^T \right) \right] + F_A + K \vec{u} \quad (8) $$

Here “I” is the diagonal hydrostatic pressure matrix “$F_A$” is the buoyancy force, “K” is the Darcy damping force and “μ” is the local viscosity. Here we consider the buoyancy forces on the liquid metal induced by the difference of temperature between the center and the edge of the melting pool. With the Boussinesq approximation, we estimate the buoyancy contribution by:

$$ F_A = \rho_L \vec{g} \beta_T (T - T_f) \quad (9) $$

where “$T_f$” is the melting temperature of both Cobalt alloy and steel, “$\beta_T$” is the thermal expansion coefficient. The transition from the solid to liquid state is treated with a Darcy condition and the equivalent viscosity methods combined. The first term modifies the viscosity of the medium to give it a deliberately high value, and weakens as soon as the local temperature reaches the melting point. It is applied in this form:

$$ \mu = \mu_{\text{solid}} + (\mu - \mu_{\text{solid}}) * flc2hs(T, T_{\text{fusion}}) \quad (10) $$

and the last term is a force to damp the movement outside of the melted area:

$$ K = \frac{-C(1 - f_l)^2}{f_l^3 + b} \quad (11) $$

Where “C” is Darcy's law coefficient deliberately set to a high value to attenuate velocities, “b” is a coefficient to avoid the division by 0 and “$f_l$” is the local liquid proportion seen in equation (5).

## Species transport Modeling

Initial studies of modelling of species transport was carried out for the mixing of species during the welding process of two different materials. These studies were then extended to the context of additive manufacturing [17]. In this work, we consider the movement of the different species inside the melting pool. Being in an area assumed to be liquid, we considered the mix of species by the diluted equation species transports:

$$ \frac{\partial \vec{c}_i}{\partial t} - D_i \vec{\nabla}^2 \vec{c}_i + \vec{u} \times \vec{\nabla} \vec{c}_i = R_i \quad (12) $$

Where “c” is the local material concentration and “$D_i$” represent the diffusion coefficient, we considered dependent on the temperature by using the Stokes-Einstein formula

$$ D_i = \frac{\kappa \times T}{6 \cdot \pi \cdot \mu \cdot r_i} (T > T_m) \quad (13) $$

## Simulation Parameters

Finally, the properties of each material are summarized in the table 3 [7]. Note the physical properties come for the literature excepted the thermal conductivity. For this one, it comes from the characterization of the thermal diffusivity. As explained in the "diffusivity measurement" section, this comes from a calculation based on the definition of this diffusivity and the measurements of each parameter associated with it. In view of the wide temperature variations associated with this type of process, it should be noted that the values are thermally dependent. This is because the behavior of matter is no longer the same between low and "high" temperatures (near melting point).
<figure><img src="image_22.png" /><figcaption>Figure 5 Comparison of experimental and simulated iron dilution near the substrate/coating interface</figcaption></figure>

## Results and Discussions

To validate the numerical modeling, the comparison focuses on the proportion of migrated iron near the interface between the substrate and the coating layer with the experimental results, in line with the experimental results obtained by Younsi et al [4]. The dilution is investigated from 110 µm inside the substrate to the upper of cobalt layer (50µm upside) (Figure 5).

This may be due to an underestimation of upward velocities in the molten pool. To find out the reasons for these differences, the heat and mass transfer modes were analyzed from a dimensional point of view.

To take our analysis further, an evaluation of the Reynolds and Rayleigh numbers over time has been made (figure 6), in order to assess their variance before, during and after the passage of the laser source. The Reynolds numbers evaluated show that the material flow is maintained under laminar conditions, validating our basic hypothesis. As for the Rayleigh number, it is used to determine whether buoyancy forces dominate material movement or whether viscous dissipation predominates. The peaks correspond, for both Reynolds and Rayleigh numbers, to instants when the fluidic mechanisms are at their most effective. This makes it possible to evaluate the peak values to check whether the assumptions are valid at all times for Reynolds numbers. But also, to determine the contribution of buoyancy motions before, during and after local melting. These values confirm that buoyancy forces struggle to set the molten metal in motion. Finally, we evaluate the main type of mixing for the present different species occurring in the melt pool. For this purpose, we estimate the ratio between the convective momentum and the diffusion coefficient of the species using the Peclet number. With the obtained result shown by the Fig. 7, the calculated values are well above the critical value of 1, demonstrating that iron migration is mainly due to fluid motion rather than diffusion.
<figure><img src="image_23.png" /><figcaption>Figure 5 Comparison of experimental and simulated iron dilution near the substrate/coating interface</figcaption></figure>

## Results and Discussions

To validate the numerical modeling, the comparison focuses on the proportion of migrated iron near the interface between the substrate and the coating layer with the experimental results, in line with the experimental results obtained by Younsi et al [4]. The dilution is investigated from 110 µm inside the substrate to the upper of cobalt layer (50µm upside) (Figure 5).

This may be due to an underestimation of upward velocities in the molten pool. To find out the reasons for these differences, the heat and mass transfer modes were analyzed from a dimensional point of view.

To take our analysis further, an evaluation of the Reynolds and Rayleigh numbers over time has been made (figure 6), in order to assess their variance before, during and after the passage of the laser source. The Reynolds numbers evaluated show that the material flow is maintained under laminar conditions, validating our basic hypothesis. As for the Rayleigh number, it is used to determine whether buoyancy forces dominate material movement or whether viscous dissipation predominates. The peaks correspond, for both Reynolds and Rayleigh numbers, to instants when the fluidic mechanisms are at their most effective. This makes it possible to evaluate the peak values to check whether the assumptions are valid at all times for Reynolds numbers. But also, to determine the contribution of buoyancy motions before, during and after local melting. These values confirm that buoyancy forces struggle to set the molten metal in motion. Finally, we evaluate the main type of mixing for the present different species occurring in the melt pool. For this purpose, we estimate the ratio between the convective momentum and the diffusion coefficient of the species using the Peclet number. With the obtained result shown by the Fig. 7, the calculated values are well above the critical value of 1, demonstrating that iron migration is mainly due to fluid motion rather than diffusion.
<table><caption>Table 1: Thermo-physical properties used by the simulation.</caption><thead><tr><th>Property</th><th>CoCrMo</th><th>C35 Steel</th></tr></thead><tbody><tr><td>T Solidus [K]</td><td>1598</td><td>1609</td></tr><tr><td>T Liquidus [K]</td><td>1754</td><td>1768</td></tr><tr><td>T Evaporation [K]</td><td>3200</td><td>/</td></tr><tr><td>Density as solid [kg/m3]</td><td>5100</td><td>7850</td></tr><tr><td>Density as Liquid [kg/m3]</td><td>6870</td><td>6980</td></tr><tr><td>Heat Solid Capacity [J/kg/K]</td><td>441.75 + 0.1871×T</td><td>604</td></tr><tr><td>Heat Liquid Capacity [J/kg/K]</td><td>13</td><td>758</td></tr><tr><td>Thermal conductivity as solid [W/m/K]</td><td>Zehner-Schlünder's model [6]</td><td>51</td></tr><tr><td>Thermal conductivity as Liquid [W/m/K]</td><td>13</td><td>22</td></tr><tr><td>Emissivity</td><td>0.63</td><td>/</td></tr><tr><td>Latent heat of fusion [kJ/kg]</td><td>310</td><td>250</td></tr><tr><td>Latent heat of evaporation [kJ/kg]</td><td>4961</td><td>/</td></tr><tr><td>Thermal expansion as solid [1/K]</td><td>7.62×10<sup>-5</sup></td><td>8.5×10<sup>-5</sup></td></tr><tr><td>Thermal expansion as liquid [1/K]</td><td>7.6×10<sup>-5</sup></td><td>/</td></tr><tr><td>Latent Heat Fusion [J/kg]</td><td>310×10<sup>5</sup></td><td>207×10<sup>5</sup></td></tr><tr><td>Latent Heat Evaporation [J/kg]</td><td>4.961×10<sup>6</sup></td><td>/</td></tr><tr><td>Viscosity [Pa*s]</td><td>5.5×10<sup>-3</sup></td><td>5.56×10<sup>-3</sup></td></tr><tr><td>Marangoni coefficient [N/m/K]</td><td>-4.3×10<sup>-4</sup></td><td>10<sup>-4</sup></td></tr><tr><td>Absorptivity</td><td>0.65</td><td>/</td></tr><tr><td>Sweep laser velocity [mm/s]</td><td></td><td>800</td></tr><tr><td>Deposited powder thickness[µm]</td><td></td><td>50</td></tr><tr><td>Laser power [W]</td><td></td><td>275</td></tr><tr><td>Laser penetration depth [µm]</td><td></td><td>30</td></tr><tr><td>Convective horizontal coefficient [W/m/K]</td><td></td><td>6.5</td></tr><tr><td>Convective vertical coefficient [W/m/K]</td><td></td><td>9.41</td></tr></tbody></table>

We can see on this figure that there is a fair agreement correlation between the numerical simulation and the experimental results. However, a difference between the results of around 10% at the substrate-coating interface is highlighted.

<figure><img src="image_23.png" /><figcaption>Figure 6 Evolution of the Reynolds (a) and Rayleigh (b) number through the simulation time.</figcaption></figure>
<table><caption>Table 1: Thermo-physical properties used by the simulation.</caption><thead><tr><th>Property</th><th>CoCrMo</th><th>C35 Steel</th></tr></thead><tbody><tr><td>T Solidus [K]</td><td>1598</td><td>1609</td></tr><tr><td>T Liquidus [K]</td><td>1754</td><td>1768</td></tr><tr><td>T Evaporation [K]</td><td>3200</td><td>/</td></tr><tr><td>Density as solid [kg/m3]</td><td>5100</td><td>7850</td></tr><tr><td>Density as Liquid [kg/m3]</td><td>6870</td><td>6980</td></tr><tr><td>Heat Solid Capacity [J/kg/K]</td><td>441.75 + 0.1871×T</td><td>604</td></tr><tr><td>Heat Liquid Capacity [J/kg/K]</td><td>13</td><td>758</td></tr><tr><td>Thermal conductivity as solid [W/m/K]</td><td>Zehner-Schlünder's model [6]</td><td>51</td></tr><tr><td>Thermal conductivity as Liquid [W/m/K]</td><td>13</td><td>22</td></tr><tr><td>Emissivity</td><td>0.63</td><td>/</td></tr><tr><td>Latent heat of fusion [kJ/kg]</td><td>310</td><td>250</td></tr><tr><td>Latent heat of evaporation [kJ/kg]</td><td>4961</td><td>/</td></tr><tr><td>Thermal expansion as solid [1/K]</td><td>7.62×10<sup>-5</sup></td><td>8.5×10<sup>-5</sup></td></tr><tr><td>Thermal expansion as liquid [1/K]</td><td>7.6×10<sup>-5</sup></td><td>/</td></tr><tr><td>Latent Heat Fusion [J/kg]</td><td>310×10<sup>5</sup></td><td>207×10<sup>5</sup></td></tr><tr><td>Latent Heat Evaporation [J/kg]</td><td>4.961×10<sup>6</sup></td><td>/</td></tr><tr><td>Viscosity [Pa*s]</td><td>5.5×10<sup>-3</sup></td><td>5.56×10<sup>-3</sup></td></tr><tr><td>Marangoni coefficient [N/m/K]</td><td>-4.3×10<sup>-4</sup></td><td>10<sup>-4</sup></td></tr><tr><td>Absorptivity</td><td>0.65</td><td>/</td></tr><tr><td>Sweep laser velocity [mm/s]</td><td></td><td>800</td></tr><tr><td>Deposited powder thickness[µm]</td><td></td><td>50</td></tr><tr><td>Laser power [W]</td><td></td><td>275</td></tr><tr><td>Laser penetration depth [µm]</td><td></td><td>30</td></tr><tr><td>Convective horizontal coefficient [W/m/K]</td><td></td><td>6.5</td></tr><tr><td>Convective vertical coefficient [W/m/K]</td><td></td><td>9.41</td></tr></tbody></table>

We can see on this figure that there is a fair agreement correlation between the numerical simulation and the experimental results. However, a difference between the results of around 10% at the substrate-coating interface is highlighted.

<figure><img src="image_24.png" /><figcaption>Figure 6 Evolution of the Reynolds (a) and Rayleigh (b) number through the simulation time.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure 7 Variations of Peclet (Mass) number inside the melt pool along the simulation time</figcaption></figure>

Based on these results and the correlation between experimentation and simulation results, we can assume that the fluid movements are the main cause of the iron rising inside the cobalt layer. This causes the iron migration from the substrate into the cobalt layer, and the solidification phenomenon has frozen the iron atoms inside the upper layer.

## Conclusion

In this article, a model established with characterizations carried out previously has been partially validated by the movement of material from the substrate to the deposited powder layer. However, there is still some fine-tuning to be done to ensure a better agreement between the experimental tests and the numerical modeling.

In order to further improve the present numerical-experimental dilution correlation, the model needs a better following-up of the liquid-solid transition and evaporation mechanism as well. This can be realized by specific numerical methods, starting with followed-up front coupled to Arbitrary Lagrangian-Eulerian (ALE) method. The heat transfer through convection and radiation are, moreover, supposed constant for all times. It is most likely that these transfer modes will change over time.

The final improvement which could be developed in the model is to compare the temperature fields close to the melt to ensure consistency between the model and a L-PBF sweep in a typical manufacturing situation. These temperature measurements will be carried out using non-contact methods, in particular using infrared camera monitoring.

## References

[1] C. Boher, T. Younsi, A. Soveja, et M. Chaussier, « Influence of iron dilution on plastic deformation mechanisms in cobalt-based alloys: Consequence of phase transformations on tribological behavior », *Wear*, vol. 524-525, p. 204845, juill. 2023.
https://doi.org/10.1016/j.wear.2023.204845

[2] T. DebRoy et al., « Additive manufacturing of metallic components – Process, structure and properties », *Progress in Materials Science*, vol. 92, p. 112-224, mars 2018.
https://doi.org/10.1016/j.pmatsci.2017.10.001

[3] J. P. Oliveira, T. G. Santos, et R. M. Miranda, « Revisiting fundamental welding concepts to improve additive manufacturing: From theory to practice », *Progress in Materials Science*, vol. 107, p. 100590, janv. 2020.
https://doi.org/10.1016/j.pmatsci.2019.100590

[4] T. Younsi, C. Boher, et A. Soveja, « Influence of interlayer time on the microstructural state of CoCrMo coatings applied by selective laser melting on an iron-based substrate for different numbers of layers », *Materials Today Communications*, vol. 32, p. 103776, août 2022.
https://doi.org/10.1016/j.mtcomm.2022.103776
<figure><img src="image_25.png" /><figcaption>Figure 7 Variations of Peclet (Mass) number inside the melt pool along the simulation time</figcaption></figure>

Based on these results and the correlation between experimentation and simulation results, we can assume that the fluid movements are the main cause of the iron rising inside the cobalt layer. This causes the iron migration from the substrate into the cobalt layer, and the solidification phenomenon has frozen the iron atoms inside the upper layer.

## Conclusion

In this article, a model established with characterizations carried out previously has been partially validated by the movement of material from the substrate to the deposited powder layer. However, there is still some fine-tuning to be done to ensure a better agreement between the experimental tests and the numerical modeling.

In order to further improve the present numerical-experimental dilution correlation, the model needs a better following-up of the liquid-solid transition and evaporation mechanism as well. This can be realized by specific numerical methods, starting with followed-up front coupled to Arbitrary Lagrangian-Eulerian (ALE) method. The heat transfer through convection and radiation are, moreover, supposed constant for all times. It is most likely that these transfer modes will change over time.

The final improvement which could be developed in the model is to compare the temperature fields close to the melt to ensure consistency between the model and a L-PBF sweep in a typical manufacturing situation. These temperature measurements will be carried out using non-contact methods, in particular using infrared camera monitoring.

## References

[1] C. Boher, T. Younsi, A. Soveja, et M. Chaussier, « Influence of iron dilution on plastic deformation mechanisms in cobalt-based alloys: Consequence of phase transformations on tribological behavior », *Wear*, vol. 524-525, p. 204845, juill. 2023.
https://doi.org/10.1016/j.wear.2023.204845

[2] T. DebRoy et al., « Additive manufacturing of metallic components – Process, structure and properties », *Progress in Materials Science*, vol. 92, p. 112-224, mars 2018.
https://doi.org/10.1016/j.pmatsci.2017.10.001

[3] J. P. Oliveira, T. G. Santos, et R. M. Miranda, « Revisiting fundamental welding concepts to improve additive manufacturing: From theory to practice », *Progress in Materials Science*, vol. 107, p. 100590, janv. 2020.
https://doi.org/10.1016/j.pmatsci.2019.100590

[4] T. Younsi, C. Boher, et A. Soveja, « Influence of interlayer time on the microstructural state of CoCrMo coatings applied by selective laser melting on an iron-based substrate for different numbers of layers », *Materials Today Communications*, vol. 32, p. 103776, août 2022.
https://doi.org/10.1016/j.mtcomm.2022.103776
[5] A. Jacquot, A. Soveja, Y. Le Maoult, C. Boher, et M. Marcoux, « Apport des caractérisations thermiques et thermo-optiques pour le renforcement du lien essais-calculs en fabrication additive L-PBF », 2023. https://doi.org/10.25855/SFT2023-025

[6] S. S. Sih et J. W. Barlow, « The Prediction of the Emissivity and Thermal Conductivity of Powder Beds », *Particulate Science and Technology*, vol. 22, n° 4, p. 427-440, oct. 2004. https://doi.org/10.1080/02726350490501682

[7] W. F. Gale et T. C. Totemeier, Éd., « 8 - Thermochemical data », in *Smithells Metals Reference Book (Eighth Edition)*, Oxford: Butterworth-Heinemann, 2004, p. 8-1. doi: 10.1016/B978-075067509-3/50011-7

[8] A. V. Gusarov, I. Yadroitsev, Ph. Bertrand, et I. Smurov, « Model of Radiation and Heat Transfer in Laser-Powder Interaction Zone at Selective Laser Melting », *Journal of Heat Transfer*, vol. 131, n° 7, p. 072101, juill. 2009. https://doi.org/10.1115/1.3109245

[9] C. D. Boley, S. A. Khairallah, et A. M. Rubenchik, « Calculation of laser absorption by metal powders in additive manufacturing », *Appl. Opt.*, vol. 54, n° 9, p. 2477, mars 2015. https://doi.org/10.1364/AO.54.002477

[10] S. A. Khairallah, A. T. Anderson, A. Rubenchik, et W. E. King, « Laser powder-bed fusion additive manufacturing: Physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones », *Acta Materialia*, vol. 108, p. 36-45, avr. 2016. https://doi.org/10.1016/j.actamat.2016.02.014

[11] S. A. Khairallah et A. Anderson, « Mesoscopic simulation model of selective laser melting of stainless steel powder », *Journal of Materials Processing Technology*, vol. 214, n° 11, p. 2627-2636, nov. 2014. https://doi.org/10.1016/j.jmatprotec.2014.06.001

[12] Y. Li et D. Gu, « Parametric analysis of thermal behavior during selective laser melting additive manufacturing of aluminum alloy powder », *Mater. Des.*, vol. 63, p. 856-867, nov. 2014. https://doi.org/10.1016/j.matdes.2014.07.006

[13] A. Métais, S. Matteï, I. Tomashchuk, E. Cicala, et S. Gaied, « Dissimilar steels laser welding: Experimental and numerical assessment of weld mixing », *Journal of Laser Applications*, vol. 29, n° 2, p. 022420, juin 2017. https://doi.org/10.2351/1.4983168.

[14] A. K. Mishra et A. Kumar, « Numerical and experimental analysis of the effect of volumetric energy absorption in powder layer on thermal-fluidic transport in selective laser melting of Ti6Al4V », *Optics & Laser Technology*, vol. 111, p. 227-239, avr. 2019. https://doi.org/10.1016/j.optlastec.2018.09.054

[15] M. Bayat, S. Mohanty, et J. H. Hattel, « Multiphysics modelling of lack-of-fusion voids formation and evolution in IN718 made by multi-track/multi-layer L-PBF », *International Journal of Heat and Mass Transfer*, vol. 139, p. 95-114, août 2019. https://doi.org/10.1016/j.ijheatmasstransfer.2019.05.003

[16] Q. Chen, G. Guillemot, C.-A. Gandin, et M. Bellet, « Three-dimensional finite element thermomechanical modeling of additive manufacturing by selective laser melting for ceramic materials », *Additive Manufacturing*, vol. 16, p. 124-137, août 2017. https://doi.org/10.1016/j.addma.2017.02.005

[17] V. Klapczynski et al., « Temperature and time dependence of manganese evaporation in liquid steels. Multiphysics modelling and experimental confrontation », *Scripta Materialia*, vol. 221, p. 114944, déc. 2022. https://doi.org/10.1016/j.scriptamat.2022.114944
[5] A. Jacquot, A. Soveja, Y. Le Maoult, C. Boher, et M. Marcoux, « Apport des caractérisations thermiques et thermo-optiques pour le renforcement du lien essais-calculs en fabrication additive L-PBF », 2023. https://doi.org/10.25855/SFT2023-025

[6] S. S. Sih et J. W. Barlow, « The Prediction of the Emissivity and Thermal Conductivity of Powder Beds », *Particulate Science and Technology*, vol. 22, n° 4, p. 427-440, oct. 2004. https://doi.org/10.1080/02726350490501682

[7] W. F. Gale et T. C. Totemeier, Éd., « 8 - Thermochemical data », in *Smithells Metals Reference Book (Eighth Edition)*, Oxford: Butterworth-Heinemann, 2004, p. 8-1. doi: 10.1016/B978-075067509-3/50011-7

[8] A. V. Gusarov, I. Yadroitsev, Ph. Bertrand, et I. Smurov, « Model of Radiation and Heat Transfer in Laser-Powder Interaction Zone at Selective Laser Melting », *Journal of Heat Transfer*, vol. 131, n° 7, p. 072101, juill. 2009. https://doi.org/10.1115/1.3109245

[9] C. D. Boley, S. A. Khairallah, et A. M. Rubenchik, « Calculation of laser absorption by metal powders in additive manufacturing », *Appl. Opt.*, vol. 54, n° 9, p. 2477, mars 2015. https://doi.org/10.1364/AO.54.002477

[10] S. A. Khairallah, A. T. Anderson, A. Rubenchik, et W. E. King, « Laser powder-bed fusion additive manufacturing: Physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones », *Acta Materialia*, vol. 108, p. 36-45, avr. 2016. https://doi.org/10.1016/j.actamat.2016.02.014

[11] S. A. Khairallah et A. Anderson, « Mesoscopic simulation model of selective laser melting of stainless steel powder », *Journal of Materials Processing Technology*, vol. 214, n° 11, p. 2627-2636, nov. 2014. https://doi.org/10.1016/j.jmatprotec.2014.06.001

[12] Y. Li et D. Gu, « Parametric analysis of thermal behavior during selective laser melting additive manufacturing of aluminum alloy powder », *Mater. Des.*, vol. 63, p. 856-867, nov. 2014. https://doi.org/10.1016/j.matdes.2014.07.006

[13] A. Métais, S. Matteï, I. Tomashchuk, E. Cicala, et S. Gaied, « Dissimilar steels laser welding: Experimental and numerical assessment of weld mixing », *Journal of Laser Applications*, vol. 29, n° 2, p. 022420, juin 2017. https://doi.org/10.2351/1.4983168.

[14] A. K. Mishra et A. Kumar, « Numerical and experimental analysis of the effect of volumetric energy absorption in powder layer on thermal-fluidic transport in selective laser melting of Ti6Al4V », *Optics & Laser Technology*, vol. 111, p. 227-239, avr. 2019. https://doi.org/10.1016/j.optlastec.2018.09.054

[15] M. Bayat, S. Mohanty, et J. H. Hattel, « Multiphysics modelling of lack-of-fusion voids formation and evolution in IN718 made by multi-track/multi-layer L-PBF », *International Journal of Heat and Mass Transfer*, vol. 139, p. 95-114, août 2019. https://doi.org/10.1016/j.ijheatmasstransfer.2019.05.003

[16] Q. Chen, G. Guillemot, C.-A. Gandin, et M. Bellet, « Three-dimensional finite element thermomechanical modeling of additive manufacturing by selective laser melting for ceramic materials », *Additive Manufacturing*, vol. 16, p. 124-137, août 2017. https://doi.org/10.1016/j.addma.2017.02.005

[17] V. Klapczynski et al., « Temperature and time dependence of manganese evaporation in liquid steels. Multiphysics modelling and experimental confrontation », *Scripta Materialia*, vol. 221, p. 114944, déc. 2022. https://doi.org/10.1016/j.scriptamat.2022.114944
# Prediction of the microstructure morphology after the WAAM process based on the FEM simulation results

SZYNDLER Joanna¹,ª*, APEL Markus²,ᵇ and HÄRTEL Sebastian¹,ᶜ

¹Chair of Hybrid Manufacturing, Brandenburg University of Technology, Cottbus, Germany

²ACCESS e.V., Aachen, Germany

ᵃ Joanna.Szyndler@b-tu.de, ᵇM.Apel@access-technology.de, ᶜSebastian.Haertel@b-tu.de

**Keywords:** Finite Element Method (FEM), Wire Arc Additive Manufacturing (WAAM), Microstructure Prediction, Phase Field Method

**Abstract.** To improve understanding of the material behavior of additive-produced components, this paper focuses on the development of a numerical model that reproduces a Wire Arc Additive Manufacturing (WAAM) process, with particular attention given to the evolution of the microstructure. In this study, a finite element model in Simufact Welding software is developed, that replicates a real wire arc welding process of building a multilayer straight wall. Microscopy analysis of the weld wall cut in the middle of its length gave information about the expected microstructure morphology at different levels of the build wall. The whole experimental setup is reproduced in the software Simufact Welding. Simulation results in the form of temperature-time and temperature gradient-time history are then used as superimposed thermal conditions to simulate the microstructure evolution at different areas of the welded part by using MICRESS software.

**Introduction**

Because of the many advantages of the WAAM process, this welding technique is becoming more and more popular in many engineering industries [1-5], hence it is also in focus of many research works in many research centers all over the world [6-11]. There is almost no limitation in the desired shapes of manufactured parts and a wide range of different materials is applicable. WAAM can be combined with other manufacturing processes to improve the functionality of the desired parts by e.g. adding additional features or it can be used to repair already damaged parts. What is more, with the WAAM process better mechanical properties of the final part can be obtained in comparison to the conventional manufacturing processes such as forging or casting. However, the main advantage of WAAM over other Additive Manufacturing (AM) processes is a shorter time of production and a larger achievable size of the final part [12, 13].

The WAAM process enables the creation of the desired geometry of the final part by depositing subsequent layers of the molten metal on top of each other. By this, a series of heating and cooling cycles take place during this process, which strongly influences a chemical-metallurgical reaction in liquid metal, phase transformations, grain nucleation, its growth, and finally material mechanical properties of the obtained part [14]. Nowadays, different numerical tools are available to simulate the material behavior in different process conditions while minimizing the amount of necessary expensive experimental investigations. To complement experiments, not only Finite Element Models (FEM) [15, 16 - 20] but also neural networks [19, 21, 22], mathematical [23] or recursive models [24] are nowadays increasingly used.

In this work, advantages of the application of a finite element model coupled with a phase-field simulation are discussed which delivers results about the material microstructure evolution during the WAAM process, i.e. can elucidate the process-microstructure relation for a WAAM process. The simulation results are confronted with experimental data to validate the simulations.

<figure><img src="image_25.png" /></figure>
# Prediction of the microstructure morphology after the WAAM process based on the FEM simulation results

SZYNDLER Joanna¹,ᵃ*, APEL Markus²,ᵇ and HÄRTEL Sebastian¹,ᶜ

¹Chair of Hybrid Manufacturing, Brandenburg University of Technology, Cottbus, Germany

²ACCESS e.V., Aachen, Germany

ᵃ Joanna.Szyndler@b-tu.de, ᵇM.Apel@access-technology.de, ᶜSebastian.Haertel@b-tu.de

**Keywords:** Finite Element Method (FEM), Wire Arc Additive Manufacturing (WAAM), Microstructure Prediction, Phase Field Method

**Abstract.** To improve understanding of the material behavior of additive-produced components, this paper focuses on the development of a numerical model that reproduces a Wire Arc Additive Manufacturing (WAAM) process, with particular attention given to the evolution of the microstructure. In this study, a finite element model in Simufact Welding software is developed, that replicates a real wire arc welding process of building a multilayer straight wall. Microscopy analysis of the weld wall cut in the middle of its length gave information about the expected microstructure morphology at different levels of the build wall. The whole experimental setup is reproduced in the software Simufact Welding. Simulation results in the form of temperature-time and temperature gradient-time history are then used as superimposed thermal conditions to simulate the microstructure evolution at different areas of the welded part by using MICRESS software.

## Introduction

Because of the many advantages of the WAAM process, this welding technique is becoming more and more popular in many engineering industries [1-5], hence it is also in focus of many research works in many research centers all over the world [6-11]. There is almost no limitation in the desired shapes of manufactured parts and a wide range of different materials is applicable. WAAM can be combined with other manufacturing processes to improve the functionality of the desired parts by e.g. adding additional features or it can be used to repair already damaged parts. What is more, with the WAAM process better mechanical properties of the final part can be obtained in comparison to the conventional manufacturing processes such as forging or casting. However, the main advantage of WAAM over other Additive Manufacturing (AM) processes is a shorter time of production and a larger achievable size of the final part [12, 13].

The WAAM process enables the creation of the desired geometry of the final part by depositing subsequent layers of the molten metal on top of each other. By this, a series of heating and cooling cycles take place during this process, which strongly influences a chemical-metallurgical reaction in liquid metal, phase transformations, grain nucleation, its growth, and finally material mechanical properties of the obtained part [14]. Nowadays, different numerical tools are available to simulate the material behavior in different process conditions while minimizing the amount of necessary expensive experimental investigations. To complement experiments, not only Finite Element Models (FEM) [15, 16 - 20] but also neural networks [19, 21, 22], mathematical [23] or recursive models [24] are nowadays increasingly used.

In this work, advantages of the application of a finite element model coupled with a phase-field simulation are discussed which delivers results about the material microstructure evolution during the WAAM process, i.e. can elucidate the process-microstructure relation for a WAAM process. The simulation results are confronted with experimental data to validate the simulations.

<figure><img src="image_26.png" /></figure>
## Experimental procedure

The welding experiments have been conducted with an Arc 605 WAAM system of GEFERTEC GmbH (Berlin, Germany) at room temperature. Nine 300 mm long seams of SG2 wire (Ø1.2 mm) are welded one on top of the other on a C45 plate giving a 15 mm height wall. A time break between welding each layer was set to 38 s and the welding direction was reversed after each layer. The precise description of the developed setup is presented in earlier Authors' work [25].

<figure><img src="image_26.png" /><figcaption>Fig. 1. Experimental setup for welding experiment.</figcaption></figure>

Then, a microscopy analysis using SEM was made at the cross-section at two selected locations in the middle of the welded wall (V1) as presented in Fig 2a and b. Obtained results (Fig. 2c) reveal a mostly ferritic microstructure in the areas pointed as P1 and P2 in Fig. 2b, which were chosen for further analysis. To measure the average grain size an intercept technique was applied [26]. The measured average grain size at the level of the third welded layer (area P1) equals ca 8.08 µm. In location P2 in the middle of the last welded layer, the observed microstructure is more irregular than in location P1. The visible structure contains very big and very small grains, where the measured average grain size is approximately 3.67 µm.

<figure><img src="image_27.png" /></figure>
Experimental procedure

The welding experiments have been conducted with an Arc 605 WAAM system of GEFERTEC GmbH (Berlin, Germany) at room temperature. Nine 300 mm long seams of SG2 wire (Ø1.2 mm) are welded one on top of the other on a C45 plate giving a 15 mm height wall. A time break between welding each layer was set to 38 s and the welding direction was reversed after each layer. The precise description of the developed setup is presented in earlier Authors' work [25].

<figure><img src="image_27.png" /><figcaption>Fig. 1. Experimental setup for welding experiment.</figcaption></figure>

Then, a microscopy analysis using SEM was made at the cross-section at two selected locations in the middle of the welded wall (V1) as presented in Fig 2a and b. Obtained results (Fig. 2c) reveal a mostly ferritic microstructure in the areas pointed as P1 and P2 in Fig. 2b, which were chosen for further analysis. To measure the average grain size an intercept technique was applied [26]. The measured average grain size at the level of the third welded layer (area P1) equals ca 8.08 µm. In location P2 in the middle of the last welded layer, the observed microstructure is more irregular than in location P1. The visible structure contains very big and very small grains, where the measured average grain size is approximately 3.67 µm.

<figure><img src="image_28.png" /></figure>
<figure><img src="image_28.png" /><figcaption>Fig. 2. a) Locations of the microstructure analyses; b) cross-section of the welded wall; zoom at the microstructure morphology at the location c) P1 and d) P2.</figcaption></figure>

The next step was to reproduce experimentally obtained results by the development of the numerical model on the macro- and micro-scale.

## Numerical simulation – macro-scale model

A numerical model that reproduces the experimental setup in a digital form is developed in the Simufact.Welding 2021.1 software (Hexagon, Stockholm). The whole setup including a ground plate, substrate plate, and holding clamps was modeled (Fig 3a). A difference in the obtained thickness of each welded layer observed during the experiment was also considered in the numerical simulation model (Fig 3b).

<figure><img src="image_29.png" /><figcaption>Fig. 3. FE model developed in Simufact. Welding software: a) full setup, b) cross-section of the weld seam.</figcaption></figure>

Thermal parameters, such as the density and thermal capacity of the AISI 1045 steel substrate, were taken from the literature [27, 28] (Fig. 4a, b). The values of the thermal conductivity were measured by the Authors with Linseis LFA 1000 Laser Flash test machine (Linseis Messgeräte GmbH, Selb, Germany) (Fig. 4c), whereas the applied convective heat transfer parameters were determined in earlier Authors' work [25]. Material properties of the SG2 wire are taken from the Simufact.Welding 2021.1 material database.
<figure><img src="image_29.png" /><figcaption>Fig. 2. a) Locations of the microstructure analyses; b) cross-section of the welded wall; zoom at the microstructure morphology at the location c) P1 and d) P2.</figcaption></figure>

The next step was to reproduce experimentally obtained results by the development of the numerical model on the macro- and micro-scale.

## Numerical simulation – macro-scale model

A numerical model that reproduces the experimental setup in a digital form is developed in the Simufact.Welding 2021.1 software (Hexagon, Stockholm). The whole setup including a ground plate, substrate plate, and holding clamps was modeled (Fig 3a). A difference in the obtained thickness of each welded layer observed during the experiment was also considered in the numerical simulation model (Fig 3b).

<figure><img src="image_30.png" /><figcaption>Fig. 3. FE model developed in Simufact. Welding software: a) full setup, b) cross-section of the weld seam.</figcaption></figure>

Thermal parameters, such as the density and thermal capacity of the AISI 1045 steel substrate, were taken from the literature [27, 28] (Fig. 4a, b). The values of the thermal conductivity were measured by the Authors with Linseis LFA 1000 Laser Flash test machine (Linseis Messgeräte GmbH, Selb, Germany) (Fig. 4c), whereas the applied convective heat transfer parameters were determined in earlier Authors' work [25]. Material properties of the SG2 wire are taken from the Simufact.Welding 2021.1 material database.
<figure><img src="image_30.png" /><figcaption>Fig. 4. Material properties of AISI 1045 as a function of temperature: a) density [27], b) heat capacity [28], c) thermal conductivity measured by Authors.</figcaption></figure>

Simulation of welding a nine-layer wall was performed with the same speed (400 mm/min), voltage (14.6 V), and current (144 A) as welding parameters applied during the real experiment. Also, the 38 s-time break between welded layers as well as reversing the welding direction after each layer was also applied. The air temperature and initial temperature of the wire were set to 23°C, the initial temperature of all plates was set to 21.5°C, and arc efficiency was assumed to be 80%. The Goldaks' heat source model [29] was used to describe the volumetric heat flux density distribution, where *a*<sub>f</sub>, *a*<sub>r</sub>, *b*, *d* are parameters related to the shape characteristic of the welding arc. A two-heat source model was applied with parameters whose exact values and optimization procedure are described in earlier Authors' work [25]. An example of the predicted temperature history and temperature gradient at selected points (see Fig. 3b) is presented in Fig. 5. The temperature history and also temperature gradients are later necessary for the MICRESS simulation.
<figure><img src="image_31.png" /><figcaption>Fig. 4. Material properties of AISI 1045 as a function of temperature: a) density [27], b) heat capacity [28], c) thermal conductivity measured by Authors.</figcaption></figure>

Simulation of welding a nine-layer wall was performed with the same speed (400 mm/min), voltage (14.6 V), and current (144 A) as welding parameters applied during the real experiment. Also, the 38 s-time break between welded layers as well as reversing the welding direction after each layer was also applied. The air temperature and initial temperature of the wire were set to 23°C, the initial temperature of all plates was set to 21.5°C, and arc efficiency was assumed to be 80%. The Goldaks' heat source model [29] was used to describe the volumetric heat flux density distribution, where *a*<sub>f</sub>, *a*<sub>r</sub>, *b*, *d* are parameters related to the shape characteristic of the welding arc. A two-heat source model was applied with parameters whose exact values and optimization procedure are described in earlier Authors' work [25]. An example of the predicted temperature history and temperature gradient at selected points (see Fig. 3b) is presented in Fig. 5. The temperature history and also temperature gradients are later necessary for the MICRESS simulation.
<figure><img src="image_31.png" /><figcaption>Fig. 5. Simulation results in the form of peak temperature distribution at the cross-section with the visualization of the temperature-time and temperature gradient-time plots obtained at points P1 and P2.</figcaption></figure>

## Microscale simulation

To simulate the material response and its microstructure evolution, two software packages were used: the thermodynamic software ThermoCalc [30] and the phase-field software MICRESS [31]. With ThermoCalc, a GES5 file was generated from the database TCFE12. The GES5 file is a subset of the thermodynamic database with selected alloying elements and phases and is used as a basis for the thermodynamic material model, in particular for the driving force calculations in MICRESS. According to the chemical composition of the wire, the GES5 file contains the elements Fe, C, Mn, Ni, and Si and the three phases Liquid, FCC, and BCC. The initial chemical composition of the melt and thus the material description for the simulation is given in Tab. 1. Details about the underlying phase-field model, the coupling between thermodynamic and phase-field calculations, and further examples for modeling phase transformations in steel with MICRESS can be found e.g. in [32-35].

<table><caption>Table 1. The initial concentration of components in the liquid phase</caption><thead><tr><td>C%</td><td>Mn%</td><td>Ni%</td><td>Si%</td></tr></thead><tbody><tr><td>0.1</td><td>1.1</td><td>0.15</td><td>0.65</td></tr></tbody></table>

Time-temperature and time-temperature-gradient plots (Fig. 5) were extracted from the FE macro-scale simulation at the selected locations P1 and P2 and imported as superimposed thermal conditions by the microscale model in the MICRESS software. Whereas the phase-field model is a quantitative and predictive model for phase transformations, i.e. growing interfaces, the phenomenological nucleation model in MICRESS requires a parameter calibration to obtain a quantitative fit between simulation and real data. This was done by a manual optimization procedure, where the nucleation barrier and the minimum distance of nucleating grains were
<figure><img src="image_32.png" /><figcaption>Fig. 5. Simulation results in the form of peak temperature distribution at the cross-section with the visualization of the temperature-time and temperature gradient-time plots obtained at points P1 and P2.</figcaption></figure>

## Microscale simulation

To simulate the material response and its microstructure evolution, two software packages were used: the thermodynamic software ThermoCalc [30] and the phase-field software MICRESS [31]. With ThermoCalc, a GES5 file was generated from the database TCFE12. The GES5 file is a subset of the thermodynamic database with selected alloying elements and phases and is used as a basis for the thermodynamic material model, in particular for the driving force calculations in MICRESS. According to the chemical composition of the wire, the GES5 file contains the elements Fe, C, Mn, Ni, and Si and the three phases Liquid, FCC, and BCC. The initial chemical composition of the melt and thus the material description for the simulation is given in Tab. 1. Details about the underlying phase-field model, the coupling between thermodynamic and phase-field calculations, and further examples for modeling phase transformations in steel with MICRESS can be found e.g. in [32-35].

<table><caption>Table 1. The initial concentration of components in the liquid phase</caption><thead><tr><th>C%</th><th>Mn%</th><th>Ni%</th><th>Si%</th></tr></thead><tbody><tr><td>0.1</td><td>1.1</td><td>0.15</td><td>0.65</td></tr></tbody></table>

Time-temperature and time-temperature-gradient plots (Fig. 5) were extracted from the FE macro-scale simulation at the selected locations P1 and P2 and imported as superimposed thermal conditions by the microscale model in the MICRESS software. Whereas the phase-field model is a quantitative and predictive model for phase transformations, i.e. growing interfaces, the phenomenological nucleation model in MICRESS requires a parameter calibration to obtain a quantitative fit between simulation and real data. This was done by a manual optimization procedure, where the nucleation barrier and the minimum distance of nucleating grains were
adjusted. Simulated microstructure morphologies, time-phase fraction plots, and the average grain size were used for the validation with the experimental measurements. The average size at location P1 at the end of the simulation equals 6.6 µm and 4.7 µm at location P2. The grain morphology in the last welded layer predicts the elongated shape of all grains with a big disproportion in their sizes (Fig. 6a). As expected, at the first stage of the cooling of the 9th layer, a quick solidification process is observed where a fully ferritic (δ) structure is obtained (Fig. 6b). Then, as the temperature drop continues, almost half of the examined area transforms into austenite and most of it back again into ferrite (α). At the end of the simulation, an almost fully ferritic structure is observed. Figures 6c present an example of obtained results during the very first stage of the cooling process at 1.45s, which are additional information provided by the phase field model, that could be also used for a subsequent analysis of material properties. In Fig. 6c where a phase fraction concentration is presented, red color refers to the liquid phase and yellow to the ferrite phase. Distributions of the carbon and manganese concentrations are presented in Figs 6d and 6e respectively.

<figure><img src="image_32.png" /><figcaption>Fig. 6. a) Ferritic grain structure at the end of the process and b) simulated phase fraction evolution at P2; c) Ferritic solidification morphology, d) Carbon, and e) Manganese concentration distributions during solidification, snapshots at 1.45 s.</figcaption></figure>

When the results obtained in the middle of the 3rd welded layer are analyzed, much different grain shapes are predicted than in the 9th layer. Still, some smaller grains with elongated shapes
adjusted. Simulated microstructure morphologies, time-phase fraction plots, and the average grain size were used for the validation with the experimental measurements. The average size at location P1 at the end of the simulation equals 6.6 µm and 4.7 µm at location P2. The grain morphology in the last welded layer predicts the elongated shape of all grains with a big disproportion in their sizes (Fig. 6a). As expected, at the first stage of the cooling of the 9th layer, a quick solidification process is observed where a fully ferritic (δ) structure is obtained (Fig. 6b). Then, as the temperature drop continues, almost half of the examined area transforms into austenite and most of it back again into ferrite (α). At the end of the simulation, an almost fully ferritic structure is observed. Figures 6c present an example of obtained results during the very first stage of the cooling process at 1.45s, which are additional information provided by the phase field model, that could be also used for a subsequent analysis of material properties. In Fig. 6c where a phase fraction concentration is presented, red color refers to the liquid phase and yellow to the ferrite phase. Distributions of the carbon and manganese concentrations are presented in Figs 6d and 6e respectively.

<figure><img src="image_33.png" /><figcaption>Fig. 6. a) Ferritic grain structure at the end of the process and b) simulated phase fraction evolution at P2; c) Ferritic solidification morphology, d) Carbon, and e) Manganese concentration distributions during solidification, snapshots at 1.45 s.</figcaption></figure>

When the results obtained in the middle of the 3rd welded layer are analyzed, much different grain shapes are predicted than in the 9th layer. Still, some smaller grains with elongated shapes
are visible, but the bigger ones have more uniform shapes (Fig. 7a). This is caused by a reheating cycle (see Fig. 5) when further layers were applied, which caused grain growth and nucleation of new ones. Figure 7b presents the influence of the multiple reheating of the material on the phase fraction history.

<figure><img src="image_33.png" /><figcaption>Fig. 7. a) Microstructure morphology and b) phase fraction evolution at P1 obtained due to the simulation.</figcaption></figure>

## Validation of the numerical simulation results

For the validation stage, a smaller area of the SEM picture was selected, that correlates with the size of the microstructure calculated in the MICRESS software. In both locations, a good match between the shapes of obtained grains is visible (Fig. 8) as well as the average grain size is well predicted, which is equal to 5.7 µm (4.7 µm in simulation) in location P1 and 6.5 µm (6.5 µm in simulation) in location P2.

<figure><img src="image_34.png" /><figcaption>Fig. 8. Comparison between the simulation and real microstructure taken from locations a) P1, b) P2. Different colors refer to the different grains.</figcaption></figure>

The obtained results prove that it is possible to obtain good results from the simulation of the WAAM process, not only at the macro-scale but also at the micro-scale level. Such virtual information about possible material morphology may be a clue for a further determination of other material parameters of the final product, which are influenced by the grain structure.
are visible, but the bigger ones have more uniform shapes (Fig. 7a). This is caused by a reheating cycle (see Fig. 5) when further layers were applied, which caused grain growth and nucleation of new ones. Figure 7b presents the influence of the multiple reheating of the material on the phase fraction history.

<figure><img src="image_34.png" /><figcaption>Fig. 7. a) Microstructure morphology and b) phase fraction evolution at P1 obtained due to the simulation.</figcaption></figure>

## Validation of the numerical simulation results

For the validation stage, a smaller area of the SEM picture was selected, that correlates with the size of the microstructure calculated in the MICRESS software. In both locations, a good match between the shapes of obtained grains is visible (Fig. 8) as well as the average grain size is well predicted, which is equal to 5.7 µm (4.7 µm in simulation) in location P1 and 6.5 µm (6.5 µm in simulation) in location P2.

<figure><img src="image_35.png" /><figcaption>Fig. 8. Comparison between the simulation and real microstructure taken from locations a) P1, b) P2. Different colors refer to the different grains.</figcaption></figure>

The obtained results prove that it is possible to obtain good results from the simulation of the WAAM process, not only at the macro-scale but also at the micro-scale level. Such virtual information about possible material morphology may be a clue for a further determination of other material parameters of the final product, which are influenced by the grain structure.
## Conclusions

Within this work, the Authors developed a multiscale solution by combining a macro and micro scale model to predict material microstructures for the WAAM process. Simulation results were validated with experimental results: the grain shapes, as well as the average grain size, were successfully predicted at two different locations of a welded straight wall and the differences are explained by the different thermal histories. One point was selected in the middle of the last welded layer, subjected to only one cycle of heating and cooling. When more layers are applied, lower layers undergo multiple cycles of reheating and cooling which influences the microstructure by ferrite to austenite transformations in combination with grain growth and nucleation.

The results show that the multiscale simulation approach as discussed in the paper can provide a good qualitative prediction of the microstructure evolution in ferritic steel during the WAAM process. This validated model can lead to the provision of a better understanding of the material response and eventually its final properties. A well-developed model can be later used to analyze the influence of the different parameters of the welding on the microstructure in the final part. Microstructural information can be later on used as input for calculating material properties like tensile strength or cyclic fatigue.

## Acknowledgments

The Authors acknowledge the Center for Hybrid Electric Systems Cottbus (CHESCO) for their support. This project is funded by the Federal Government in the context of the “Investitionsgesetz Kohleregionen” and co-financed by the State of Brandenburg.

## References

[1] V. Madhavadas, D. Srivastava, U. Chadha, S.A. Raj, M.T.H. Sultan, F.S. Shahar, A.U.M. Shah, A review on metal additive manufacturing for intricately shaped aerospace components, CIRP Journal of Manufacturing Science and Technology, 39 (2022) 18-36. https://doi.org/10.1016/j.cirpj.2022.07.005

[2] J. Ye, P. Kyvelou, F. Gilardi, H. Lu, M. Gilbert, L. Gardner, An end-to-end framework for the additive manufacture of optimized tubular structures, IEEE Access, 9 (2021) 165476-165489. https://doi.org/10.1109/ACCESS.2021.3132797

[3] X. Zuo, W. Zhang, Y. Chen, J.P. Oliveira, Z. Zeng, Y. Li, Z. Luo, S. Ao, Wire-based directed energy deposition of NiTiTa shape memory alloys: Microstructure, phase transformation, electrochemistry, X-ray visibility and mechanical properties, Additive Manufacturing, 59 (2022) 103115. https://doi.org/10.1016/j.addma.2022.103115

[4] M. Vishnukumar, R. Pramod, A.R. Kannan, Wire arc additive manufacturing for repairing aluminium structures in marine applications, Materials Letters, 299 (2021) 130112. https://doi.org/10.1016/j.matlet.2021.130112

[5] T. Feucht, J. Lange, B. Waldschmitt, A.K. Schudlich, M. Klein, M. Oechsner, Welding process for the additive manufacturing of cantilevered components with the WAAM, Advanced Joining Processes, 125 (2020) 67-78. https://doi.org/10.1007/978-981-15-2957-3_5

[6] S. Zhou, J. Zhang, G. Yang, Y. Wang, B. Li, D. An, J. Zheng, W. Wei, Microstructure evolution and fracture behavior of Ti-6Al-4V fabricated by WAAM-LDM additive manufacturing, Journal of Materials Research and Technology, 28 (2024) 347-362. https://doi.org/10.1016/j.jmrt.2023.11.255

[7] C. Sasikumar, R. Oyyaravelu, Mechanical properties and microstructure of SS 316 L created by WAAM based on GMAW, Materials Today Communications, 38 (2024) 107807. https://doi.org/10.1016/j.mtcomm.2023.107807
## Conclusions

Within this work, the Authors developed a multiscale solution by combining a macro and micro scale model to predict material microstructures for the WAAM process. Simulation results were validated with experimental results: the grain shapes, as well as the average grain size, were successfully predicted at two different locations of a welded straight wall and the differences are explained by the different thermal histories. One point was selected in the middle of the last welded layer, subjected to only one cycle of heating and cooling. When more layers are applied, lower layers undergo multiple cycles of reheating and cooling which influences the microstructure by ferrite to austenite transformations in combination with grain growth and nucleation.

The results show that the multiscale simulation approach as discussed in the paper can provide a good qualitative prediction of the microstructure evolution in ferritic steel during the WAAM process. This validated model can lead to the provision of a better understanding of the material response and eventually its final properties. A well-developed model can be later used to analyze the influence of the different parameters of the welding on the microstructure in the final part. Microstructural information can be later on used as input for calculating material properties like tensile strength or cyclic fatigue.

## Acknowledgments

The Authors acknowledge the Center for Hybrid Electric Systems Cottbus (CHESCO) for their support. This project is funded by the Federal Government in the context of the “Investitionsgesetz Kohleregionen” and co-financed by the State of Brandenburg.

## References

[1] V. Madhavadas, D. Srivastava, U. Chadha, S.A. Raj, M.T.H. Sultan, F.S. Shahar, A.U.M. Shah, A review on metal additive manufacturing for intricately shaped aerospace components, CIRP Journal of Manufacturing Science and Technology, 39 (2022) 18-36. https://doi.org/10.1016/j.cirpj.2022.07.005

[2] J. Ye, P. Kyvelou, F. Gilardi, H. Lu, M. Gilbert, L. Gardner, An end-to-end framework for the additive manufacture of optimized tubular structures, IEEE Access, 9 (2021) 165476-165489. https://doi.org/10.1109/ACCESS.2021.3132797

[3] X. Zuo, W. Zhang, Y. Chen, J.P. Oliveira, Z. Zeng, Y. Li, Z. Luo, S. Ao, Wire-based directed energy deposition of NiTiTa shape memory alloys: Microstructure, phase transformation, electrochemistry, X-ray visibility and mechanical properties, Additive Manufacturing, 59 (2022) 103115. https://doi.org/10.1016/j.addma.2022.103115

[4] M. Vishnukumar, R. Pramod, A.R. Kannan, Wire arc additive manufacturing for repairing aluminium structures in marine applications, Materials Letters, 299 (2021) 130112. https://doi.org/10.1016/j.matlet.2021.130112

[5] T. Feucht, J. Lange, B. Waldschmitt, A.K. Schudlich, M. Klein, M. Oechsner, Welding process for the additive manufacturing of cantilevered components with the WAAM, Advanced Joining Processes, 125 (2020) 67-78. https://doi.org/10.1007/978-981-15-2957-3_5

[6] S. Zhou, J. Zhang, G. Yang, Y. Wang, B. Li, D. An, J. Zheng, W. Wei, Microstructure evolution and fracture behavior of Ti-6Al-4V fabricated by WAAM-LDM additive manufacturing, Journal of Materials Research and Technology, 28 (2024) 347-362. https://doi.org/10.1016/j.jmrt.2023.11.255

[7] C. Sasikumar, R. Oyyaravelu, Mechanical properties and microstructure of SS 316 L created by WAAM based on GMAW, Materials Today Communications, 38 (2024) 107807. https://doi.org/10.1016/j.mtcomm.2023.107807
[8] V. Laghi, N. Babovic, E. Benvenuti, H. Kloft, Blended structural optimization of steel joints for Wire-and-Arc Additive Manufacturing, Engineering Structures, 300 (2024) 117141. https://doi.org/10.1016/j.engstruct.2023.117141

[9] P. Kyvelou, C. Huang, J. Li, L. Gardner, Residual stresses in steel I-sections strengthened by wire arc additive manufacturing, Structures, 60 (2024) 105828. https://doi.org/10.1016/j.istruc.2023.105828

[10] A.Staroselsky, D. Voytovych, R. Acharya, Prediction of Ni-based alloy microstructure in wire arc additive manufacturing from cellular automata model, Computational Materials Science, 233 (2024) 112721. https://doi.org/10.1016/j.commatsci.2023.112721

[11] H. Mu, F. He, L. Yuan, H. Hatamian, P. Commins, Z. Pan, Online Distortion Simulation Using Generative Machine Learning Models: A Step Toward Digital Twin of Metallic Additive Manufacturing, Journal of Industrial Information Integration, (2024) 100563. https://doi.org/10.1016/j.jii.2024.100563

[12] S.I. Evans, J. Wang, J. Qin, Y. He, P. Shepherd, J. Ding, A review of WAAM for steel construction – Manufacturing, material and geometric properties, design, and future directions, Structures, 44 (2022) 1506-1522. https://doi.org/10.1016/j.istruc.2022.08.084

[13] S. Singh, S.K. Sharma, D.W. Rathod, A review on process planning strategies and challenges of WAAM, Materials Today: Proceedings, 47, 19 (2021) 6564-6575. https://doi.org/10.1016/j.matpr.2021.02.632

[14] A.S. Azar, S.K. As, O.M. Akselsen, Determination of welding heat source parameters from actual bead shape, Computational Material Science, 54 (2012) 176-182. https://doi.org/10.1016/j.commatsci.2011.10.025

[15] X. Zhou, H. Zhang, G. Wang, X. Bai, Three-dimensional numerical simulation of arc and metal transport in arc welding based additive manufacturing, International Journal of Heat and Mass Transfer, 103 (2016) 521-537. https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.084

[16] Z. Hu, X. Qin, T. Shao, Welding thermal simulation and metallurgical characteristics analysis in WAAM for 5CrNiMo hot forging die remanufacturing, Procedia Engineering, 207 (2017) 2203-2208. https://doi.org/10.1016/j.proeng.2017.10.982

[17] C.T.J. Panicker, K.R. Surya, V. Senthilkumar, Novel process parameter-based approach for reducing residual stresses in WAAM, Materials Today: Proceedings, 59 (2022) 1119-1126. https://doi.org/10.1016/j.matpr.2022.03.025

[18] N.P. Gokhale, P. Kala, Thermal analysis of TIG-WAAM based metal deposition process using finite element method, Materials Today: Proceedings, 44 (2021) 453-459. https://doi.org/10.1016/j.matpr.2020.09.756

[19] Y. Ling, J. Ni, J. Antonissen, H.B. Hamouda, J.V. Voorde, M.A. Wahab, Numerical prediction of microstructure and hardness for low carbon steel wire Arc additive manufacturing components, Simulation Modelling Practice and Theory, 122 (2023) 102664. https://doi.org/10.1016/j.simpat.2022.102664

[20] V. Gornyakov, Y. Sun, J. Ding, S. Williams, Modelling and optimizing hybrid process of wire arc additive manufacturing and high-pressure rolling, 223 (2022) 111121. https://doi.org/10.1016/j.matdes.2022.111121

[21] S. Bose, A. Biswas, Y. Tiwari, M. Mukherjee, S.S. Roy, Artificial neural Network-based approaches for Bi-directional modelling of robotic wire arc additive manufacturing, Materials Today: Proceedings, 62, 12 (2022) 6507-6513. https://doi.org/10.1016/j.matpr.2022.04.331
[8] V. Laghi, N. Babovic, E. Benvenuti, H. Kloft, Blended structural optimization of steel joints for Wire-and-Arc Additive Manufacturing, Engineering Structures, 300 (2024) 117141. https://doi.org/10.1016/j.engstruct.2023.117141

[9] P. Kyvelou, C. Huang, J. Li, L. Gardner, Residual stresses in steel I-sections strengthened by wire arc additive manufacturing, Structures, 60 (2024) 105828. https://doi.org/10.1016/j.istruc.2023.105828

[10] A.Staroselsky, D. Voytovych, R. Acharya, Prediction of Ni-based alloy microstructure in wire arc additive manufacturing from cellular automata model, Computational Materials Science, 233 (2024) 112721. https://doi.org/10.1016/j.commatsci.2023.112721

[11] H. Mu, F. He, L. Yuan, H. Hatamian, P. Commins, Z. Pan, Online Distortion Simulation Using Generative Machine Learning Models: A Step Toward Digital Twin of Metallic Additive Manufacturing, Journal of Industrial Information Integration, (2024) 100563. https://doi.org/10.1016/j.jii.2024.100563

[12] S.I. Evans, J. Wang, J. Qin, Y. He, P. Shepherd, J. Ding, A review of WAAM for steel construction – Manufacturing, material and geometric properties, design, and future directions, Structures, 44 (2022) 1506-1522. https://doi.org/10.1016/j.istruc.2022.08.084

[13] S. Singh, S.K. Sharma, D.W. Rathod, A review on process planning strategies and challenges of WAAM, Materials Today: Proceedings, 47, 19 (2021) 6564-6575. https://doi.org/10.1016/j.matpr.2021.02.632

[14] A.S. Azar, S.K. As, O.M. Akselsen, Determination of welding heat source parameters from actual bead shape, Computational Material Science, 54 (2012) 176-182. https://doi.org/10.1016/j.commatsci.2011.10.025

[15] X. Zhou, H. Zhang, G. Wang, X. Bai, Three-dimensional numerical simulation of arc and metal transport in arc welding based additive manufacturing, International Journal of Heat and Mass Transfer, 103 (2016) 521-537. https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.084

[16] Z. Hu, X. Qin, T. Shao, Welding thermal simulation and metallurgical characteristics analysis in WAAM for 5CrNiMo hot forging die remanufacturing, Procedia Engineering, 207 (2017) 2203-2208. https://doi.org/10.1016/j.proeng.2017.10.982

[17] C.T.J. Panicker, K.R. Surya, V. Senthilkumar, Novel process parameter-based approach for reducing residual stresses in WAAM, Materials Today: Proceedings, 59 (2022) 1119-1126. https://doi.org/10.1016/j.matpr.2022.03.025

[18] N.P. Gokhale, P. Kala, Thermal analysis of TIG-WAAM based metal deposition process using finite element method, Materials Today: Proceedings, 44 (2021) 453-459. https://doi.org/10.1016/j.matpr.2020.09.756

[19] Y. Ling, J. Ni, J. Antonissen, H.B. Hamouda, J.V. Voorde, M.A. Wahab, Numerical prediction of microstructure and hardness for low carbon steel wire Arc additive manufacturing components, Simulation Modelling Practice and Theory, 122 (2023) 102664. https://doi.org/10.1016/j.simpat.2022.102664

[20] V. Gornyakov, Y. Sun, J. Ding, S. Williams, Modelling and optimizing hybrid process of wire arc additive manufacturing and high-pressure rolling, 223 (2022) 111121. https://doi.org/10.1016/j.matdes.2022.111121

[21] S. Bose, A. Biswas, Y. Tiwari, M. Mukherjee, S.S. Roy, Artificial neural Network-based approaches for Bi-directional modelling of robotic wire arc additive manufacturing, Materials Today: Proceedings, 62, 12 (2022) 6507-6513. https://doi.org/10.1016/j.matpr.2022.04.331
[22] Y. Wang, X. Xu, Z. Zhao, W. Deng, J. Han, L. Bai, X. Liang, J. Yao, Coordinated monitoring and control method of deposited layer width and reinforcement in WAAM process, Journal of Manufacturing Process, 71 (2021) 306-316. https://doi.org/10.1016/j.jmapro.2021.09.033

[23] W.C. Ke, J.P. Oliveira, B.Q. Cong, S.S. Ao, Z.W. Qi, B. Peng, Z. Zeng, Multi-layer deposition mechanism in ultra high-frequency pulsed wire arc additive manufacturing (WAAM) of NiTi shape memory alloys, Additive Manufacturing, 50 (2022) 102513. https://doi.org/10.1016/j.addma.2021.102513

[24] C. Chen, H. He, J. Zhou, G. Lian, X. Huang, M. Feng, A profile transformation based recursive multi-bead overlapping model for robotic wire and arc additive manufacturing (WAAM), Journal of Manufacturing Processes, 84 (2022) 886-901. https://doi.org/10.1016/j.jmapro.2022.10.042

[25] J. Szyndler, A. Schmidt, S. Härtel, Determination of welding heat source parameters for FEM simulation based on temperature history and real bead shape. Materials Research Proceedings 28 (2023) 159-168. https://doi.org/10.21741/9781644902479-18

[26] ISO 643:2019 - Steels — Micrographic determination of the apparent grain size

[27] M. Hojny, M. Glowacki, Numerical Modelling of Steel Deformation at Extra-High Temperatures, in P. Miidla (ed.), Numerical Modelling, IntechOpen, London, 2012, 10.5772/36562. https://doi.org/10.5772/36562

[28] A. Martinovs, S. Polukoshko, E. Zaicevs, R. Revalds, Laser Hardening Process Optimizations Using FEM, Engineering for Rural Development, 2020. https://doi.org/10.22616/ERDev.2020.19.TF372

[29] J. Goldak, A.P. Chakravarti, M. Bibby, A new finite element model for welding heat sources, Metallurgical Transactions B, 15 (1984) 299-305. https://doi.org/10.1007/BF02667333

[30] Computational Materials Engineering - Thermo-Calc Software (thermocalc.com)

[31] MICRESS 7.1: http://www.micress.de.

[32] B. Böttger, M. Apel, M. Budnitzki, J. Eiken, G. Laschet, B. Zhou, Calphad coupled phase-field model with mechano-chemical contributions and its application to rafting of γ' in CMSX-4. Computational Materials Science, 184 (2020) 109909. https://doi.org/10.1016/j.commatsci.2020.109909

[33] B. Böttger, J. Eiken, M. Apel, Multi-ternary extrapolation scheme for efficient coupling of thermodynamic data to a multi-phase-field model, Computational Materials Science, 108 (2015) 283-292. https://doi.org/10.1016/j.commatsci.2015.03.003

[34] B. Böttger, M. Apel, T. Jokisch, A. Senger, Phase-field study on microstructure formation in Mar-M247 during electron beam welding and correlation to hot cracking susceptibility. IOP Conference Series: Materials Science and Engineering, 861 (2020) 012072. https://doi.org/10.1088/1757-899X/861/1/012072

[35] B. Böttger, M. Apel, J. Eiken, P. Schaffnit, I. Steinbach, “Phase-Field Simulation of Solidification and Solid-State Transformations in Multicomponent Steels”, Steel Research International. 79 (2008) 608. https://doi.org/10.1002/srin.200806173
[22] Y. Wang, X. Xu, Z. Zhao, W. Deng, J. Han, L. Bai, X. Liang, J. Yao, Coordinated monitoring and control method of deposited layer width and reinforcement in WAAM process, Journal of Manufacturing Process, 71 (2021) 306-316. https://doi.org/10.1016/j.jmapro.2021.09.033

[23] W.C. Ke, J.P. Oliveira, B.Q. Cong, S.S. Ao, Z.W. Qi, B. Peng, Z. Zeng, Multi-layer deposition mechanism in ultra high-frequency pulsed wire arc additive manufacturing (WAAM) of NiTi shape memory alloys, Additive Manufacturing, 50 (2022) 102513. https://doi.org/10.1016/j.addma.2021.102513

[24] C. Chen, H. He, J. Zhou, G. Lian, X. Huang, M. Feng, A profile transformation based recursive multi-bead overlapping model for robotic wire and arc additive manufacturing (WAAM), Journal of Manufacturing Processes, 84 (2022) 886-901. https://doi.org/10.1016/j.jmapro.2022.10.042

[25] J. Szyndler, A. Schmidt, S. Härtel, Determination of welding heat source parameters for FEM simulation based on temperature history and real bead shape. Materials Research Proceedings 28 (2023) 159-168. https://doi.org/10.21741/9781644902479-18

[26] ISO 643:2019 - Steels — Micrographic determination of the apparent grain size

[27] M. Hojny, M. Glowacki, Numerical Modelling of Steel Deformation at Extra-High Temperatures, in P. Miidla (ed.), Numerical Modelling, IntechOpen, London, 2012, 10.5772/36562. https://doi.org/10.5772/36562

[28] A. Martinovs, S. Polukoshko, E. Zaicevs, R. Revalds, Laser Hardening Process Optimizations Using FEM, Engineering for Rural Development, 2020. https://doi.org/10.22616/ERDev.2020.19.TF372

[29] J. Goldak, A.P. Chakravarti, M. Bibby, A new finite element model for welding heat sources, Metallurgical Transactions B, 15 (1984) 299-305. https://doi.org/10.1007/BF02667333

[30] Computational Materials Engineering - Thermo-Calc Software (thermocalc.com)

[31] MICRESS 7.1: http://www.micress.de.

[32] B. Böttger, M. Apel, M. Budnitzki, J. Eiken, G. Laschet, B. Zhou, Calphad coupled phase-field model with mechano-chemical contributions and its application to rafting of γ' in CMSX-4. Computational Materials Science, 184 (2020) 109909. https://doi.org/10.1016/j.commatsci.2020.109909

[33] B. Böttger, J. Eiken, M. Apel, Multi-ternary extrapolation scheme for efficient coupling of thermodynamic data to a multi-phase-field model, Computational Materials Science, 108 (2015) 283-292. https://doi.org/10.1016/j.commatsci.2015.03.003

[34] B. Böttger, M. Apel, T. Jokisch, A. Senger, Phase-field study on microstructure formation in Mar-M247 during electron beam welding and correlation to hot cracking susceptibility. IOP Conference Series: Materials Science and Engineering, 861 (2020) 012072. https://doi.org/10.1088/1757-899X/861/1/012072

[35] B. Böttger, M. Apel, J. Eiken, P. Schaffnit, I. Steinbach, “Phase-Field Simulation of Solidification and Solid-State Transformations in Multicomponent Steels”, Steel Research International. 79 (2008) 608. https://doi.org/10.1002/srin.200806173
# Inverse identification of heat source model parameters for laser-powder directed energy deposition of AISI H13 deposit on AISI H11 substrate

BERTRAND Johanna<sup>1,2,a*</sup>, ABBES Fazilay<sup>1,b</sup>, BONNEFOY Hervé<sup>2,c</sup>, FLAN Bruno<sup>2,d</sup> and ABBES Boussad<sup>1,e</sup>

<sup>1</sup> MATIM, Matériaux & Ingénierie Mécanique, Université de Reims Champagne-Ardenne, Reims, FRANCE

<sup>2</sup> Platinum 3D, 08000 Charleville-Mézières, FRANCE

<sup>a</sup>j.bertrand@platinum3d.com, <sup>b</sup>fazilay.abbes@univ-reims.fr, <sup>c</sup>h.bonnefoy@platinum3d.com,

<sup>d</sup>b.flan@platinum3d.com, <sup>e</sup>boussad.abbes@univ-reims.fr

**Keywords:** Directed Energy Deposition, Numerical Modeling, Goldak Heat Source, Parameters Identification

**Abstract.** Numerical modeling and simulation are very useful tools for assessing the impact of process parameters and predicting optimized conditions in Laser Directed Energy Deposition (L-DED) processes. Heat source parameters have a great influence on the accuracy of numerical modeling for predicting temperature fields and residual stresses. This paper presents a coupled experimental-numerical procedure to determine the Goldak's heat source model parameters. Graded single-clad tracks were printed with the laser beam power increased continuously at a constant powder feeding rate and scanning speed. Bead widths, heights, and penetration depths were measured at different locations along the deposited track and then used as experimental data for an inverse identification process using the FEA code ABAQUS and the optimizer iSIGHT. The obtained results show that the accuracy of the numerical model is increased with optimized parameters.

**Introduction**

Laser Directed Energy Deposition (L-DED) is an economic process to produce three-dimensional complex parts, a very interesting solution for the low-cost repair of damaged high-value components, and for coatings with improved surface properties [1].

To obtain an optimal coating, free of defects such as porosity and cracking [2], it is important to find an appropriate process. According to Thompson et al. [3], the most influencing parameters on the L-DED process are the laser power, the laser scanning speed, the powder feeding rate, the overlap percentage between two single tracks, the dwell time between two layers, the laser spot diameter, and the laser scanning strategy. Testing experimentally so many different parameter combinations can be expensive and time-consuming.

Numerical modeling and simulation are very useful tools for assessing the impact of process parameters and predicting optimized conditions in L-DED processes. In typical residual stress simulations, the thermal part of the process is first simulated, and then the thermal fields are used as input for the mechanical analysis. Indeed, thermal fields play a major role in the quality of L-DED parts. L-DED consists of the deposition of multiple layers, which will be heated up and cooled down as a new layer is deposited. This phenomenon is known as thermal cycling and the resulting thermal history can lead to anisotropic structures [5]. An anisotropic structure can, with use, lead to cracking and thus degrading the part properties [6].

<figure><img src="image_35.png" /></figure>
# Inverse identification of heat source model parameters for laser-powder directed energy deposition of AISI H13 deposit on AISI H11 substrate

BERTRAND Johanna<sup>1,2,a*</sup>, ABBES Fazilay<sup>1,b</sup>, BONNEFOY Hervé<sup>2,c</sup>, FLAN Bruno<sup>2,d</sup> and ABBES Boussad<sup>1,e</sup>

<sup>1</sup> MATIM, Matériaux & Ingénierie Mécanique, Université de Reims Champagne-Ardenne, Reims, FRANCE

<sup>2</sup> Platinum 3D, 08000 Charleville-Mézières, FRANCE

<sup>a</sup>j.bertrand@platinum3d.com, <sup>b</sup>fazilay.abbes@univ-reims.fr, <sup>c</sup>h.bonnefoy@platinum3d.com,

<sup>d</sup>b.flan@platinum3d.com, <sup>e</sup>boussad.abbes@univ-reims.fr

**Keywords:** Directed Energy Deposition, Numerical Modeling, Goldak Heat Source, Parameters Identification

**Abstract.** Numerical modeling and simulation are very useful tools for assessing the impact of process parameters and predicting optimized conditions in Laser Directed Energy Deposition (L-DED) processes. Heat source parameters have a great influence on the accuracy of numerical modeling for predicting temperature fields and residual stresses. This paper presents a coupled experimental-numerical procedure to determine the Goldak's heat source model parameters. Graded single-clad tracks were printed with the laser beam power increased continuously at a constant powder feeding rate and scanning speed. Bead widths, heights, and penetration depths were measured at different locations along the deposited track and then used as experimental data for an inverse identification process using the FEA code ABAQUS and the optimizer iSIGHT. The obtained results show that the accuracy of the numerical model is increased with optimized parameters.

**Introduction**

Laser Directed Energy Deposition (L-DED) is an economic process to produce three-dimensional complex parts, a very interesting solution for the low-cost repair of damaged high-value components, and for coatings with improved surface properties [1].

To obtain an optimal coating, free of defects such as porosity and cracking [2], it is important to find an appropriate process. According to Thompson et al. [3], the most influencing parameters on the L-DED process are the laser power, the laser scanning speed, the powder feeding rate, the overlap percentage between two single tracks, the dwell time between two layers, the laser spot diameter, and the laser scanning strategy. Testing experimentally so many different parameter combinations can be expensive and time-consuming.

Numerical modeling and simulation are very useful tools for assessing the impact of process parameters and predicting optimized conditions in L-DED processes. In typical residual stress simulations, the thermal part of the process is first simulated, and then the thermal fields are used as input for the mechanical analysis. Indeed, thermal fields play a major role in the quality of L-DED parts. L-DED consists of the deposition of multiple layers, which will be heated up and cooled down as a new layer is deposited. This phenomenon is known as thermal cycling and the resulting thermal history can lead to anisotropic structures [5]. An anisotropic structure can, with use, lead to cracking and thus degrading the part properties [6].

<figure><img src="image_36.png" /></figure>
Heat source parameters have a great influence on the accuracy of numerical modeling for predicting temperature fields and residual stresses. The Goldak equivalent heat source (double ellipsoid) is the most widely used heat input model for welding and L-DED simulations [4].

In most cases, Goldak's model parameters are determined from experimentations [7], which consequently leads to errors in simulation as the modeling requirements are more complex and experimental validations have limitations [8]. Thus, the parameters of the heat source model should be optimized to obtain a more accurate melt pool.

In this article, experimental work has been conducted in order to obtain input data for the calibration of the heat source model by printing graded single-clad tracks with the laser beam power increased continuously and keeping constant the powder feeding rate and the scanning speed. To optimize the heat source parameters, the thermal simulation using ABAQUS software was coupled with iSIGHT optimizer. The originality of our work stands in the fact that we do not use thermocouples to adjust our model, but instead, we use experimentally measured bead geometry: width, height, and penetration depth.

## Experimental Methodology

The substrate used for the experiments was a hot work tool steel bloc (AISI H11) and the deposit was Ferro 55 metal powder (AISI H13) from Voestalpine Böhler. The chemical compositions of these materials are given in Table 1.

<table><caption>Table 1: Chemical composition of AISI H11 and H13 alloys.</caption><thead><tr><th>Elements</th><th></th><th>C</th><th>Mn</th><th>Si</th><th>S</th><th>P</th><th>Cr</th><th>Mo</th><th>V</th><th>Ni</th><th>Cu</th><th>Fe</th></tr></thead><tbody><tr><td rowspan="2">AISI H11</td><td>wt% min</td><td>0.38</td><td>0.20</td><td>0.80</td><td>/</td><td>/</td><td>4.75</td><td>1.20</td><td>0.40</td><td>/</td><td>/</td><td>/</td></tr><tr><td>wt% max</td><td>0.43</td><td>0.40</td><td>1.00</td><td>0.015</td><td>0.015</td><td>5.25</td><td>1.40</td><td>0.60</td><td>0.25</td><td>0.35</td><td>Bal.</td></tr><tr><td>AISI H13</td><td>wt%</td><td>0.36</td><td>0.80</td><td>0.35</td><td>/</td><td>/</td><td>7.00</td><td>2.40</td><td>/</td><td>/</td><td>/</td><td>Bal.</td></tr></tbody></table>

Graded single-clad tracks were printed by increasing the laser beam power (increase of 50 W each 4 mm) at different constant powder feeding rates and scanning speeds by using a TRUMPF Trulaser Cell 7020 machine and TruDisk 4002 laser system with a Yb:YAG environment and a 1030 nm wavelength. The obtained samples were then cut with an Al₂O₃ cutting disk, coated with a phenolic resin, and polished with Al₂O₃ and SiC abrasive papers (320, 600, and 1200 grains), then with polishing cloth and diamond polishing paste (9, 6, 3 and 1 µm). After etching with a Nital 4% solution for 1min30s, the samples were observed with a digital microscope Keyence VHX-7000 to measure the height (H), width (W), and penetration depth (p) of the single-clad tracks as illustrated in Fig. 1.

<figure><img src="image_36.png" /><figcaption>Fig. 1: Schematic of a cross-section of single-clad track.</figcaption></figure>

## Thermal Simulation

The thermal analysis is based on the law of energy conservation and Fourier's law. The equation governing 3D heat transfer is generally stated as:
Heat source parameters have a great influence on the accuracy of numerical modeling for predicting temperature fields and residual stresses. The Goldak equivalent heat source (double ellipsoid) is the most widely used heat input model for welding and L-DED simulations [4].

In most cases, Goldak's model parameters are determined from experimentations [7], which consequently leads to errors in simulation as the modeling requirements are more complex and experimental validations have limitations [8]. Thus, the parameters of the heat source model should be optimized to obtain a more accurate melt pool.

In this article, experimental work has been conducted in order to obtain input data for the calibration of the heat source model by printing graded single-clad tracks with the laser beam power increased continuously and keeping constant the powder feeding rate and the scanning speed. To optimize the heat source parameters, the thermal simulation using ABAQUS software was coupled with iSIGHT optimizer. The originality of our work stands in the fact that we do not use thermocouples to adjust our model, but instead, we use experimentally measured bead geometry: width, height, and penetration depth.

## Experimental Methodology

The substrate used for the experiments was a hot work tool steel bloc (AISI H11) and the deposit was Ferro 55 metal powder (AISI H13) from Voestalpine Böhler. The chemical compositions of these materials are given in Table 1.

<table><caption>Table 1: Chemical composition of AISI H11 and H13 alloys.</caption><thead><tr><th>Elements</th><th></th><th>C</th><th>Mn</th><th>Si</th><th>S</th><th>P</th><th>Cr</th><th>Mo</th><th>V</th><th>Ni</th><th>Cu</th><th>Fe</th></tr></thead><tbody><tr><td rowspan="2">AISI H11</td><td>wt% min</td><td>0.38</td><td>0.20</td><td>0.80</td><td>/</td><td>/</td><td>4.75</td><td>1.20</td><td>0.40</td><td>/</td><td>/</td><td>/</td></tr><tr><td>wt% max</td><td>0.43</td><td>0.40</td><td>1.00</td><td>0.015</td><td>0.015</td><td>5.25</td><td>1.40</td><td>0.60</td><td>0.25</td><td>0.35</td><td>Bal.</td></tr><tr><td>AISI H13</td><td>wt%</td><td>0.36</td><td>0.80</td><td>0.35</td><td>/</td><td>/</td><td>7.00</td><td>2.40</td><td>/</td><td>/</td><td>/</td><td>Bal.</td></tr></tbody></table>

Graded single-clad tracks were printed by increasing the laser beam power (increase of 50 W each 4 mm) at different constant powder feeding rates and scanning speeds by using a TRUMPF Trulaser Cell 7020 machine and TruDisk 4002 laser system with a Yb:YAG environment and a 1030 nm wavelength. The obtained samples were then cut with an Al₂O₃ cutting disk, coated with a phenolic resin, and polished with Al₂O₃ and SiC abrasive papers (320, 600, and 1200 grains), then with polishing cloth and diamond polishing paste (9, 6, 3 and 1 µm). After etching with a Nital 4% solution for 1min30s, the samples were observed with a digital microscope Keyence VHX-7000 to measure the height (H), width (W), and penetration depth (p) of the single-clad tracks as illustrated in Fig. 1.

<figure><img src="image_37.png" /><figcaption>Fig. 1: Schematic of a cross-section of single-clad track.</figcaption></figure>

## Thermal Simulation

The thermal analysis is based on the law of energy conservation and Fourier's law. The equation governing 3D heat transfer is generally stated as:
$$ \rho C_p(T) \frac{dT}{dt} = \nabla(\lambda(T)\nabla T) + q_v. \qquad (1) $$

where $\lambda(T)$ is a thermal conductivity [W/m/K] and $q_v$ is a laser beam heat source power [W/m³].

Initial condition $T(0) = T_0$, and boundary conditions considering the convection heat loss and radiation emission heat loss complete Eq. (1) as follows:

$$ q_s = -\lambda(T) \frac{\partial T}{\partial n} = h_c(T - T_\infty) + \varepsilon \sigma (T^4 - T_\infty^4). \qquad (2) $$

where $h_c$ is the convective coefficient, assumed as $h_c = 25$ [W/m²/K], $\varepsilon$ is the emissivity coefficient, assumed as $\varepsilon = 0.8$, $\sigma$ is the Stefan-Boltzmann constant, and $T_\infty$ is the ambient temperature assumed as $T_\infty = 298$ [K].

The Goldak equivalent heat source (double ellipsoid) is considered in this study. Its power density distribution is given by [7]:

$$ q_{f/r} = \frac{6\sqrt{3}f_{f/r}}{abc\pi\sqrt{\pi}} Q e^{\frac{-3x^3}{c^2 f/r}} e^{\frac{-3y^2}{a^2}} e^{\frac{-3z^3}{b^2}}. \qquad (3) $$

where $a, b, c_f, c_r$ define the axes of the ellipsoidal heat source. $Q$ is the heat source power. $f_{f/r}$ defines the energy flow intensity. Subscript “f” represents the region front portion of the melt pool and “r” represents the rear region of the melt pool. These parameters will be identified in this study.

The numerical simulation considers temperature dependent thermal conductivity presented in Fig. 2(a) and temperature dependent specific heat presented in Fig. 2(b). The latent heat effect at fusion, which is given in terms of solidus ($T_s$) and liquidus temperatures ($T_l$) and the latent heat ($L_f$), is assumed to be in addition to the specific heat effect. In the simulations, were assumed: $T_s = 1588$ [K], $T_l = 1717$ [K] and $L_f = 282$ [kJ/kg]. The density is assumed constant $\rho = 7810$ [kg/m³]. In this paper, the same material properties are assumed for AISI H11 and AISI H13.

<figure><img src="image_37.png" /><figcaption>Fig. 2: Thermal conductivity (a) and specific heat (b) assumed in simulations.</figcaption></figure>

The thermal simulation was performed with ABAQUS 2019 software using the built-in features for AM Modeler [9]. Progressive element activation is used to model the deposition of raw material from a moving nozzle. The thermal energy used to heat the bead to the melting point will be applied when the bead is deposited (i.e., activated element). In this study, this energy is distributed over the activated elements assuming a Goldak heat source [10]. The finite element mesh consisted of hexahedral 8-nodes linear thermal elements (type DC3D8) with average edge lengths of 0.5 mm and refined to 0.1 mm in the substrate (on a layer of 1 mm), yielding a model with around 130,000
$$ \rho C_p(T) \frac{dT}{dt} = \nabla(\lambda(T)\nabla T) + q_v. \qquad (1) $$

where $\lambda(T)$ is a thermal conductivity [W/m/K] and $q_v$ is a laser beam heat source power [W/m³].

Initial condition $T(0) = T_0$, and boundary conditions considering the convection heat loss and radiation emission heat loss complete Eq. (1) as follows:

$$ q_s = -\lambda(T) \frac{\partial T}{\partial n} = h_c(T - T_\infty) + \varepsilon\sigma(T^4 - T_\infty^4). \qquad (2) $$

where $h_c$ is the convective coefficient, assumed as $h_c = 25$ [W/m²/K], $\varepsilon$ is the emissivity coefficient, assumed as $\varepsilon = 0.8$, $\sigma$ is the Stefan-Boltzmann constant, and $T_\infty$ is the ambient temperature assumed as $T_\infty = 298$ [K].

The Goldak equivalent heat source (double ellipsoid) is considered in this study. Its power density distribution is given by [7]:

$$ q_{f/r} = \frac{6\sqrt{3}f_{f/r}}{abc\pi\sqrt{\pi}} Q e^{\frac{-3x^3}{c^2 f/r}} e^{\frac{-3y^2}{a^2}} e^{\frac{-3z^3}{b^2}}. \qquad (3) $$

where $a, b, c_f, c_r$ define the axes of the ellipsoidal heat source. $Q$ is the heat source power. $f_{f/r}$ defines the energy flow intensity. Subscript “f” represents the region front portion of the melt pool and “r” represents the rear region of the melt pool. These parameters will be identified in this study.

The numerical simulation considers temperature dependent thermal conductivity presented in Fig. 2(a) and temperature dependent specific heat presented in Fig. 2(b). The latent heat effect at fusion, which is given in terms of solidus ($T_s$) and liquidus temperatures ($T_l$) and the latent heat ($L_f$), is assumed to be in addition to the specific heat effect. In the simulations, were assumed: $T_s = 1588$ [K], $T_l = 1717$ [K] and $L_f = 282$ [kJ/kg]. The density is assumed constant $\rho = 7810$ [kg/m³]. In this paper, the same material properties are assumed for AISI H11 and AISI H13.

<figure><img src="image_38.png" /><figcaption>Fig. 2: Thermal conductivity (a) and specific heat (b) assumed in simulations.</figcaption></figure>

The thermal simulation was performed with ABAQUS 2019 software using the built-in features for AM Modeler [9]. Progressive element activation is used to model the deposition of raw material from a moving nozzle. The thermal energy used to heat the bead to the melting point will be applied when the bead is deposited (i.e., activated element). In this study, this energy is distributed over the activated elements assuming a Goldak heat source [10]. The finite element mesh consisted of hexahedral 8-nodes linear thermal elements (type DC3D8) with average edge lengths of 0.5 mm and refined to 0.1 mm in the substrate (on a layer of 1 mm), yielding a model with around 130,000
elements. Figure 3 shows the ABAQUS finite element mesh for the single-clad track. The analysis time increment was locked at $\Delta t = 0.05$ [s].

<figure><img src="image_38.png" /><figcaption>Fig. 3: Finite element mesh of single-clad track.</figcaption></figure>

## Inverse Identification Procedure

Inverse identification method is used to determine the set of the Goldak model parameters that minimize the difference between the calculated values of the thermal model and the corresponding experimental data. The problem is formulated as a least square problem:

$$ \text{Find } P_{opt} \in \mathcal{P}, \text{ such as } \Phi(P_{opt}) = \min_{\mathcal{P}} \Phi(P) $$

where:

$$ \Phi(P) = \sqrt{\sum_{i=1}^{n} (T_{i}^{sim}(P) - T_{i}^{exp})^2}. \qquad (4) $$

with the constraint:

$$ T_{max}^{sim}(P) < T_{ev}. \qquad (5) $$

where $\Phi(P)$ is the objective function; $T_{i}^{sim}(P)$ is the i-th calculated temperature; $T_{i}^{exp}$ is the i-th measured temperature; $P$ represents the parameters to be optimized; $n$ is the number of measurements, $\mathcal{P}$ is the constraint space, $T_{max}^{sim}(P)$ is the maximum calculated temperature and $T_{ev}$ is the evaporation temperature assumed as $T_{ev} = 3135$ [K] [11].

In this paper, the optimization problem was solved by coupling iSIGHT and ABAQUS. Fig. 4 shows the flow chart of the inverse identification method. In this method, three components were used: i) ABAQUS : performs the numerical thermal simulation using $P$ parameters with similar initial and boundary conditions to experiments and extracts $T_{i}^{sim}(P)$ and $T_{max}^{sim}(P)$ ; ii) Calculator: calculate $\Phi(P)$ using recorded measurements $T_{i}^{exp}$ ; iii) Optimization1: adjusts the parameters $P$ by minimizing $\Phi(P)$ with the constraint $T_{max}^{sim}(P) < T_{ev}$. The optimization process stops when the maximum iterations $N$ is reached or when $\Phi(P) < \epsilon$, where $\epsilon = 10^{-6}$ is the global convergence.
elements. Figure 3 shows the ABAQUS finite element mesh for the single-clad track. The analysis time increment was locked at $\Delta t = 0.05$ [s].

<figure><img src="image_39.png" /><figcaption>Fig. 3: Finite element mesh of single-clad track.</figcaption></figure>

## Inverse Identification Procedure

Inverse identification method is used to determine the set of the Goldak model parameters that minimize the difference between the calculated values of the thermal model and the corresponding experimental data. The problem is formulated as a least square problem:

Find $P_{opt} \in \mathcal{P}$, such as $\Phi(P_{opt}) = \min_{\mathcal{P}} \Phi(P)$

where:

$$ \Phi(P) = \sqrt{\sum_{i=1}^{n} (T_{i}^{sim}(P) - T_{i}^{exp})^2}. \qquad (4) $$

with the constraint:

$$ T_{max}^{sim}(P) < T_{ev}. \qquad (5) $$

where $\Phi(P)$ is the objective function; $T_{i}^{sim}(P)$ is the $i^{th}$ calculated temperature; $T_{i}^{exp}$ is the $i^{th}$ measured temperature; $P$ represents the parameters to be optimized; $n$ is the number of measurements, $\mathcal{P}$ is the constraint space, $T_{max}^{sim}(P)$ is the maximum calculated temperature and $T_{ev}$ is the evaporation temperature assumed as $T_{ev} = 3135$ [K] [11].

In this paper, the optimization problem was solved by coupling iSIGHT and ABAQUS. Fig. 4 shows the flow chart of the inverse identification method. In this method, three components were used: i) ABAQUS : performs the numerical thermal simulation using $P$ parameters with similar initial and boundary conditions to experiments and extracts $T_{i}^{sim}(P)$ and $T_{max}^{sim}(P)$ ; ii) Calculator: calculate $\Phi(P)$ using recorded measurements $T_{i}^{exp}$ ; iii) Optimization1: adjusts the parameters $P$ by minimizing $\Phi(P)$ with the constraint $T_{max}^{sim}(P) < T_{ev}$. The optimization process stops when the maximum iterations $N$ is reached or when $\Phi(P) < \epsilon$, where $\epsilon = 10^{-6}$ is the global convergence.
<figure><img src="image_39.png" /><figcaption>Fig. 4: Flow chart of the inverse identification method by coupling iSIGHT and ABAQUS.</figcaption></figure>

## Results and Discussion

To illustrate the proposed methodology, we have selected a graded bead deposited at a scan speed of $V_s = 0.9$ [m/min] and a powder deposition rate of $F = 25$ [g/min]. The laser beam spot size was measured to be 4 mm in diameter. The laser power was varied from $P = 1450$ to 2550 [W] (increase of 50 W each 4 mm). Fig. 5 shows the deposited graded bead.

<figure><img src="image_40.png" /><figcaption>Fig. 5: Graded bead deposited at $V_s = 0.9$ [m/min], $F = 25$ [g/min] and $P = 1450$ to 2550 [W].</figcaption></figure>

The obtained sample was then cut along the lines drawn in Fig. 5, coated with a phenolic resin, and polished. After an etching, the cross-sections of cut samples were observed with the digital microscope as shown in Fig. 6. The measured height ($H$), width ($W$) and penetration depth ($p$) of the cuts are given in Table 2.

<figure><img src="image_41.png" /><figcaption>Fig. 6: Observed cross-sections.</figcaption></figure>

<table><caption>Table 2: Measured height, width, and penetration depth of the cross-sections of the graded single-clad track.</caption><thead><tr><td>Cross-section</td><td>P [W]</td><td>H [mm]</td><td>W [mm]</td><td>p [mm]</td></tr></thead><tbody><tr><td>1</td><td>1650</td><td>0.338</td><td>2.330</td><td>0.098</td></tr><tr><td>2</td><td>2050</td><td>0.352</td><td>3.064</td><td>0.365</td></tr><tr><td>3</td><td>2400</td><td>0.470</td><td>3.191</td><td>0.494</td></tr></tbody></table>

From the experimental results of the penetration depth ($p$), we have positioned three nodes in the finite element mesh in locations corresponding to the experiment penetration depth to record their temperatures during the simulations. Due to the meshing constraints, the three nodes were
<figure><img src="image_40.png" /><figcaption>Fig. 4: Flow chart of the inverse identification method by coupling iSIGHT and ABAQUS.</figcaption></figure>

## Results and Discussion

To illustrate the proposed methodology, we have selected a graded bead deposited at a scan speed of $V_s = 0.9$ [m/min] and a powder deposition rate of $F = 25$ [g/min]. The laser beam spot size was measured to be 4 mm in diameter. The laser power was varied from $P = 1450$ to 2550 [W] (increase of 50 W each 4 mm). Fig. 5 shows the deposited graded bead.

<figure><img src="image_41.png" /><figcaption>Fig. 5: Graded bead deposited at $V_s = 0.9$ [m/min], $F = 25$ [g/min] and $P = 1450$ to 2550 [W].</figcaption></figure>

The obtained sample was then cut along the lines drawn in Fig. 5, coated with a phenolic resin, and polished. After an etching, the cross-sections of cut samples were observed with the digital microscope as shown in Fig. 6. The measured height ($H$), width ($W$) and penetration depth ($p$) of the cuts are given in Table 2.

<figure><img src="image_42.png" /><figcaption>Fig. 6: Observed cross-sections.</figcaption></figure>

<table><caption>Table 2: Measured height, width, and penetration depth of the cross-sections of the graded single-clad track.</caption><thead><tr><td>Cross-section</td><td>P [W]</td><td>H [mm]</td><td>W [mm]</td><td>p [mm]</td></tr></thead><tbody><tr><td>1</td><td>1650</td><td>0.338</td><td>2.330</td><td>0.098</td></tr><tr><td>2</td><td>2050</td><td>0.352</td><td>3.064</td><td>0.365</td></tr><tr><td>3</td><td>2400</td><td>0.470</td><td>3.191</td><td>0.494</td></tr></tbody></table>

From the experimental results of the penetration depth ($p$), we have positioned three nodes in the finite element mesh in locations corresponding to the experiment penetration depth to record their temperatures during the simulations. Due to the meshing constraints, the three nodes were
respectively chosen at 0.1 mm, 0.4 mm, and 0.5 mm from the surface of the substrate. The highest temperatures of these nodes during the simulations are extracted automatically from the corresponding ABAQUS output database and used as $T_i^{sim}(P)$ in Eq. (4). The penetration depth corresponds to the melt pool depth [12], where the material reaches its melting temperature assumed as $T_m = 1700$ [K] [13]. The experimental values $T_i^{exp}$ in Eq. (4) were then set to $T_m$ as it is the temperature target in the corresponding node to get the material melted.

Initial model parameters are important for the success of the identification procedure. The double ellipsoid heat source dimensions parameters in Eq. (3) are dependent on experiment cases and are initialized as: $a = W/2$, $b = H + p$, $c_f = W/2$, and $c_r = 2c_f$ [14]. Initial and identified parameters are listed in Table 3.

<table><caption>Table 3: Initial and identified Goldak's model parameters.</caption><thead><tr><th>Parameters</th><th>P [W]</th><th>a [mm]</th><th>b [mm]</th><th>c<sub>f</sub> [mm]</th><th>c<sub>r</sub> [mm]</th></tr></thead><tbody><tr><td rowspan="3">Initial</td><td>1650</td><td>1.165</td><td>0.436</td><td>1.165</td><td>2.330</td></tr><tr><td>2050</td><td>1.532</td><td>0.717</td><td>1.532</td><td>3.064</td></tr><tr><td>2400</td><td>1.596</td><td>0.964</td><td>1.596</td><td>3.191</td></tr><tr><td rowspan="3">Identified</td><td>1650</td><td>1.165</td><td>0.498</td><td>1.040</td><td>2.301</td></tr><tr><td>2050</td><td>1.532</td><td>0.845</td><td>1.400</td><td>3.333</td></tr><tr><td>2400</td><td>1.596</td><td>0.768</td><td>1.826</td><td>4.837</td></tr></tbody></table>

The calculated temperature histories of three nodes located in the three cross-sections calculated using the double ellipsoid heat source model with the initial and identified parameters are plotted in Fig. 7. The maximum temperatures calculated with the initial parameters are underestimated for the nodes located in the cross-sections 1 and 2 and overestimated for the node located in the cross-section 3 compared to the melting temperature (1700 [K]). However, with the identified parameters, the maximum temperatures for the nodes located in the cross-sections 1 and 2 correspond exactly to the melting temperature, and that for the node located in the cross-section 3 is slightly overestimated without consequences on the final quality of the results given in Fig. 8.

<figure><img src="image_42.png" /><figcaption>Fig. 7: Calculated temperature histories of the three nodes located in the three cross-sections.</figcaption></figure>

Fig. 8 illustrates the temperature distributions on the three cross-sections of the graded single-clad track, which were calculated using the double ellipsoid heat source model with the identified parameters. In all cases, the boundaries of the fusion zone obtained numerically have similar shapes in comparison with the experimental ones. Therefore, numerical values of the bead width, the bead height and the penetrated depth have good agreement with those obtained experimentally.
respectively chosen at 0.1 mm, 0.4 mm, and 0.5 mm from the surface of the substrate. The highest temperatures of these nodes during the simulations are extracted automatically from the corresponding ABAQUS output database and used as $T_i^{sim}(P)$ in Eq. (4). The penetration depth corresponds to the melt pool depth [12], where the material reaches its melting temperature assumed as $T_m = 1700$ [K] [13]. The experimental values $T_i^{exp}$ in Eq. (4) were then set to $T_m$ as it is the temperature target in the corresponding node to get the material melted.

Initial model parameters are important for the success of the identification procedure. The double ellipsoid heat source dimensions parameters in Eq. (3) are dependent on experiment cases and are initialized as: $a = W/2$, $b = H + p$, $c_f = W/2$, and $c_r = 2c_f$ [14]. Initial and identified parameters are listed in Table 3.

<table><caption>Table 3: Initial and identified Goldak's model parameters.</caption><thead><tr><th>Parameters</th><th>P [W]</th><th>a [mm]</th><th>b [mm]</th><th>c<sub>f</sub> [mm]</th><th>c<sub>r</sub> [mm]</th></tr></thead><tbody><tr><td rowspan="3">Initial</td><td>1650</td><td>1.165</td><td>0.436</td><td>1.165</td><td>2.330</td></tr><tr><td>2050</td><td>1.532</td><td>0.717</td><td>1.532</td><td>3.064</td></tr><tr><td>2400</td><td>1.596</td><td>0.964</td><td>1.596</td><td>3.191</td></tr><tr><td rowspan="3">Identified</td><td>1650</td><td>1.165</td><td>0.498</td><td>1.040</td><td>2.301</td></tr><tr><td>2050</td><td>1.532</td><td>0.845</td><td>1.400</td><td>3.333</td></tr><tr><td>2400</td><td>1.596</td><td>0.768</td><td>1.826</td><td>4.837</td></tr></tbody></table>

The calculated temperature histories of three nodes located in the three cross-sections calculated using the double ellipsoid heat source model with the initial and identified parameters are plotted in Fig. 7. The maximum temperatures calculated with the initial parameters are underestimated for the nodes located in the cross-sections 1 and 2 and overestimated for the node located in the cross-section 3 compared to the melting temperature (1700 [K]). However, with the identified parameters, the maximum temperatures for the nodes located in the cross-sections 1 and 2 correspond exactly to the melting temperature, and that for the node located in the cross-section 3 is slightly overestimated without consequences on the final quality of the results given in Fig. 8.

<figure><img src="image_43.png" /><figcaption>Fig. 7: Calculated temperature histories of the three nodes located in the three cross-sections.</figcaption></figure>

Fig. 8 illustrates the temperature distributions on the three cross-sections of the graded single-clad track, which were calculated using the double ellipsoid heat source model with the identified parameters. In all cases, the boundaries of the fusion zone obtained numerically have similar shapes in comparison with the experimental ones. Therefore, numerical values of the bead width, the bead height and the penetrated depth have good agreement with those obtained experimentally.
These results enable this methodology to perform the following analyses based on the identified parameters.

<figure><img src="image_43.png" /><figcaption>Fig. 8: Comparison of fusion zone shape, numerically and experimentally obtained for the three cross-sections.</figcaption></figure>

## Conclusion

1) Coupling of optimization technique with finite element method, which takes account of experimentally measured bead geometry: width, height, and penetration depth, is important to identify heat source parameters.

2) Printing graded tracks is a low-cost solution for the identification of heat source parameters.

3) The results show that the accuracy of the numerical model is increased with optimized parameters.

4) The proposed identification procedure can be extended to other deposit/substrate couples.

## References

[1] H. Paydas, A. Mertens, R. Carrus, J. Lecomte-Beckers, J. Tchoufang Tchuindjang, Laser cladding as repair technology for Ti-6Al-4V alloy: Influence of building strategy on microstructure and hardness, Materials and Design 85 (2015) 497-510, http://dx.doi.org/10.1016/j.matdes.2015.07.035

[2] M. Liu, A. Kumar, S. Bukkapatnam, M. Kuttolamadom, A Review of the Anomalies in Directed Energy Deposition (DED) Processes & Potential Solutions - Part Quality & Defects, Procedia Manufacturing 53 (2021) 507-518, https://doi.org/10.1016/j.promfg.2021.06.093

[3] S. M. Thompson, L. Bianc, N. Shamsaei, A. Yadollahi, An overview of Direct Laser Deposition for additive manufacturing; Part I: Transport phenomena, modeling and diagnostics, Additive Manufacturing 8 (2015) 36-62, https://doi.org/10.1016/j.addma.2015.07.001

[4] M. S. Sumanlal, N. S. Sivasubramaniyan, V. M. Joy Varghese, M. Shafeek & D. T. Ananthan, Estimation of heat source model parameters for partial penetration of TIG welding using numerical optimization method, Welding International 37(7) (2023) 400-416, https://doi.org/10.1080/09507116.2023.2242777

[5] N. Kang, X. Lin, M. El Mansori, Q.Z. Wang, J.L. Lu, C. Coddet, W.D. Huang, On the effect of the thermal cycle during the directed energy deposition application to the in-situ production of a Ti-Mo alloy functionally graded structure, Additive Manufacturing 31 (2020) 100911, https://doi.org/10.1016/j.addma.2019.100911

[6] X. Lu, M. Chiumenti, M. Cervera, G. Zhang, X. Lin, Mitigation of residual stresses and microstructure homogenization in directed energy deposition processes, Engineering with Computers 38 (2022) 4771-4790, https://doi.org/10.1007/s00366-021-01563-9
These results enable this methodology to perform the following analyses based on the identified parameters.

<figure><img src="image_44.png" /><figcaption>Fig. 8: Comparison of fusion zone shape, numerically and experimentally obtained for the three cross-sections.</figcaption></figure>

## Conclusion

1) Coupling of optimization technique with finite element method, which takes account of experimentally measured bead geometry: width, height, and penetration depth, is important to identify heat source parameters.

2) Printing graded tracks is a low-cost solution for the identification of heat source parameters.

3) The results show that the accuracy of the numerical model is increased with optimized parameters.

4) The proposed identification procedure can be extended to other deposit/substrate couples.

## References

[1] H. Paydas, A. Mertens, R. Carrus, J. Lecomte-Beckers, J. Tchoufang Tchuindjang, Laser cladding as repair technology for Ti-6Al-4V alloy: Influence of building strategy on microstructure and hardness, Materials and Design 85 (2015) 497-510, http://dx.doi.org/10.1016/j.matdes.2015.07.035

[2] M. Liu, A. Kumar, S. Bukkapatnam, M. Kuttolamadom, A Review of the Anomalies in Directed Energy Deposition (DED) Processes & Potential Solutions - Part Quality & Defects, Procedia Manufacturing 53 (2021) 507-518, https://doi.org/10.1016/j.promfg.2021.06.093

[3] S. M. Thompson, L. Bianc, N. Shamsaei, A. Yadollahi, An overview of Direct Laser Deposition for additive manufacturing; Part I: Transport phenomena, modeling and diagnostics, Additive Manufacturing 8 (2015) 36-62, https://doi.org/10.1016/j.addma.2015.07.001

[4] M. S. Sumanlal, N. S. Sivasubramaniyan, V. M. Joy Varghese, M. Shafeek & D. T. Ananthan, Estimation of heat source model parameters for partial penetration of TIG welding using numerical optimization method, Welding International 37(7) (2023) 400-416, https://doi.org/10.1080/09507116.2023.2242777

[5] N. Kang, X. Lin, M. El Mansori, Q.Z. Wang, J.L. Lu, C. Coddet, W.D. Huang, On the effect of the thermal cycle during the directed energy deposition application to the in-situ production of a Ti-Mo alloy functionally graded structure, Additive Manufacturing 31 (2020) 100911, https://doi.org/10.1016/j.addma.2019.100911

[6] X. Lu, M. Chiumenti, M. Cervera, G. Zhang, X. Lin, Mitigation of residual stresses and microstructure homogenization in directed energy deposition processes, Engineering with Computers 38 (2022) 4771-4790, https://doi.org/10.1007/s00366-021-01563-9
[7] A. Kiran, Y. Li, J. Hodek, M. Brázda, M. Urbánek, and J. Džugan, Heat Source Modeling and Residual Stress Analysis for Metal Directed Energy Deposition Additive Manufacturing, Materials 15(7) (2022) 2545, https://doi.org/10.3390/ma15072545

[8] X. Jia, J. Xu, Z. Liu, et al., A new method to estimate heat source parameters in gas metal arc welding simulation process, Fusion Eng. Des. 89(1) (2014) 40-48, https://doi.org/10.1016/j.fusengdes.2013.11.006

[9] Dassault Systemes, Abaqus Analysis User's Guide 2019, Simulia Inc., Providence, RI, USA (2019).

[10] A. Kiran, J. Hodek, J. Vavřík, M. Urbánek and J. Džugan, Numerical Simulation Development and Computational Optimization for Directed Energy Deposition Additive Manufacturing Process, Materials 13(11) (2020) 2666, https://doi.org/10.3390/ma13112666

[11] “Iron” Breaking Atom. https://www.breakingatom.com/elements/iron (Accessed Jan. 02, 2024).

[12] I. Jeon, L Yang, K. Ryu, & H. Sohn, Online melt pool depth estimation during directed energy deposition using coaxial infrared camera, laser line scanner, and artificial neural network, Additive Manufacturing 47 (2021), 102295, https://doi.org/10.1016/j.addma.2021.102295

[13] “H13 Tool Steel - Chromium Hot-Work Steels” AZO Materials. https://www.azom.com/article.aspx?ArticleID=9107 (Accessed Jan. 02, 2024).

[14] V. Nain, T. Engel, M. Carin, D. Boisselier & L. Seguy, Development of an elongated ellipsoid heat Source model to reduce computation time for directed energy deposition process, Frontiers in Materials 8 (2021), 747389, https://doi.org/10.3389/fmats.2021.747389
[7] A. Kiran, Y. Li, J. Hodek, M. Brázda, M. Urbánek, and J. Džugan, Heat Source Modeling and Residual Stress Analysis for Metal Directed Energy Deposition Additive Manufacturing, Materials 15(7) (2022) 2545, https://doi.org/10.3390/ma15072545

[8] X. Jia, J. Xu, Z. Liu, et al., A new method to estimate heat source parameters in gas metal arc welding simulation process, Fusion Eng. Des. 89(1) (2014) 40-48, https://doi.org/10.1016/j.fusengdes.2013.11.006

[9] Dassault Systemes, Abaqus Analysis User's Guide 2019, Simulia Inc., Providence, RI, USA (2019).

[10] A. Kiran, J. Hodek, J. Vavřík, M. Urbánek and J. Džugan, Numerical Simulation Development and Computational Optimization for Directed Energy Deposition Additive Manufacturing Process, Materials 13(11) (2020) 2666, https://doi.org/10.3390/ma13112666

[11] “Iron” Breaking Atom. https://www.breakingatom.com/elements/iron (Accessed Jan. 02, 2024).

[12] I. Jeon, L Yang, K. Ryu, & H. Sohn, Online melt pool depth estimation during directed energy deposition using coaxial infrared camera, laser line scanner, and artificial neural network, Additive Manufacturing 47 (2021), 102295, https://doi.org/10.1016/j.addma.2021.102295

[13] “H13 Tool Steel - Chromium Hot-Work Steels” AZO Materials. https://www.azom.com/article.aspx?ArticleID=9107 (Accessed Jan. 02, 2024).

[14] V. Nain, T. Engel, M. Carin, D. Boisselier & L. Seguy, Development of an elongated ellipsoid heat Source model to reduce computation time for directed energy deposition process, Frontiers in Materials 8 (2021), 747389, https://doi.org/10.3389/fmats.2021.747389
# Prediction of the evolution of material properties during the AM process based on the FEM simulation and experimental results

HÄRTEL Sebastian¹ a*, SZYNDLER Joanna¹,ᵇ, PAKDEL SEFIDI Moein¹,ᶜ and JÄGER Reyk¹,ᵈ

¹Chair of Hybrid Manufacturing, Brandenburg University of Technology, Cottbus, Germany

ahaertel@b-tu.de, bJoanna.Szyndler@b-tu.de, cpakdelse@b-tu.de, dReyk.Jaeger@b-tu.de

**Keywords:** Finite Element Method (FEM), Wire Arc Additive Manufacturing (WAAM), Additive Manufacturing AM, Material Properties

**Abstract.** To deepen the understanding of material behavior after additive manufacturing, this article focuses on the prediction of material properties after the Wire Arc Additive Manufacturing (WAAM) process. Particular attention is put on the temperature curves in the various phases of the welding process, which influence the final material properties, especially the hardness of the resulting part. A total of nine components in the form of walls were produced using the WAAM process, with the number of layers varying from 1 to 9. By experimentally analyzing the welded parts, which were cut in the middle of their length, it was possible to gain insights into the development of hardness at selected points. The entire test setup was simulated in the Simufact Welding FE-software. The simulation results in the form of temperature-time diagrams were then correlated with the real hardness measurements at the corresponding points. In this way, a model was developed that for the first time considers the development of hardness as a result of cooling after the welding process as well as the change in hardness as a result of reheating due to the application of additional layers.

**Introduction**

A revolutionary manufacturing technique called additive manufacturing has the power to completely transform a wide range of industry sectors. Complex geometries that are challenging or impossible to create using conventional manufacturing techniques can be fabricated using additive manufacturing. However, predicting the final material properties of the additively manufactured parts is one of the challenges. The geometry of the object, the printing parameters, and the kind of material being used are just a few of the many variables that can make Additive Manufacturing (AM) a complex process. All of these elements may have an impact on the printed part's ultimate properties like strength, hardness, and ductility. Predicting material properties during and after the additive manufacturing process would enable process optimization and the creation of parts with desirable properties. For the fabrication of high-performance parts for crucial applications, this would be especially advantageous.

Bai, Y., et al. predicted how residual stresses in additively manufactured parts would change using FEM simulation [1]. Machine learning was used by Zhao, Y., et al. to forecast the mechanical characteristics of additively manufactured parts, such as their ductility and strength. The relationship between the properties of the printed part and the initially used material can be discovered by training machine learning algorithms with experimental data [2]. Wu, S., et al. looked into the connection between the mechanical characteristics and microstructure of Ti-6Al-4V that was additively manufactured. A common titanium alloy used in AM applications is Ti-6Al-4V. The microstructure of Ti-6Al-4V and additively manufactured Ti-6Al-4V differs significantly, according to the study. The material's mechanical properties may be significantly impacted by this variation in microstructure [3]. A multi-scale method for forecasting the fatigue

<figure><img src="image_44.png" /></figure>
# Prediction of the evolution of material properties during the AM process based on the FEM simulation and experimental results

HÄRTEL Sebastian¹ a*, SZYNDLER Joanna¹,ᵇ, PAKDEL SEFIDI Moein¹,ᶜ and JÄGER Reyk¹,ᵈ

¹Chair of Hybrid Manufacturing, Brandenburg University of Technology, Cottbus, Germany

ahaertel@b-tu.de, bJoanna.Szyndler@b-tu.de, cpakdelse@b-tu.de, dReyk.Jaeger@b-tu.de

**Keywords:** Finite Element Method (FEM), Wire Arc Additive Manufacturing (WAAM), Additive Manufacturing AM, Material Properties

**Abstract.** To deepen the understanding of material behavior after additive manufacturing, this article focuses on the prediction of material properties after the Wire Arc Additive Manufacturing (WAAM) process. Particular attention is put on the temperature curves in the various phases of the welding process, which influence the final material properties, especially the hardness of the resulting part. A total of nine components in the form of walls were produced using the WAAM process, with the number of layers varying from 1 to 9. By experimentally analyzing the welded parts, which were cut in the middle of their length, it was possible to gain insights into the development of hardness at selected points. The entire test setup was simulated in the Simufact Welding FE-software. The simulation results in the form of temperature-time diagrams were then correlated with the real hardness measurements at the corresponding points. In this way, a model was developed that for the first time considers the development of hardness as a result of cooling after the welding process as well as the change in hardness as a result of reheating due to the application of additional layers.

**Introduction**

A revolutionary manufacturing technique called additive manufacturing has the power to completely transform a wide range of industry sectors. Complex geometries that are challenging or impossible to create using conventional manufacturing techniques can be fabricated using additive manufacturing. However, predicting the final material properties of the additively manufactured parts is one of the challenges. The geometry of the object, the printing parameters, and the kind of material being used are just a few of the many variables that can make Additive Manufacturing (AM) a complex process. All of these elements may have an impact on the printed part's ultimate properties like strength, hardness, and ductility. Predicting material properties during and after the additive manufacturing process would enable process optimization and the creation of parts with desirable properties. For the fabrication of high-performance parts for crucial applications, this would be especially advantageous.

Bai, Y., et al. predicted how residual stresses in additively manufactured parts would change using FEM simulation [1]. Machine learning was used by Zhao, Y., et al. to forecast the mechanical characteristics of additively manufactured parts, such as their ductility and strength. The relationship between the properties of the printed part and the initially used material can be discovered by training machine learning algorithms with experimental data [2]. Wu, S., et al. looked into the connection between the mechanical characteristics and microstructure of Ti-6Al-4V that was additively manufactured. A common titanium alloy used in AM applications is Ti-6Al-4V. The microstructure of Ti-6Al-4V and additively manufactured Ti-6Al-4V differs significantly, according to the study. The material's mechanical properties may be significantly impacted by this variation in microstructure [3]. A multi-scale method for forecasting the fatigue

<figure><img src="image_45.png" /></figure>
behavior of additively manufactured parts was created by Zhang, B., et al. This study developed a multi-scale approach to predict the fatigue behavior of an additively manufactured part by considering its microstructure, defects, and stress distribution [4]. A prediction model for the final properties of AlSi10Mg parts made by additive manufacturing was created by Li, Y., and colleagues. An aluminum alloy that is frequently utilized in AM applications is AlSi10Mg. To predict the properties of the additively manufactured part, the study's predictive model considers both its microstructure and printing parameters [5]. A substantial body of prior research serves as the basis for this study, which predicts how material properties will change during the additive manufacturing process. The integrated method presented here enables the prediction of long-term changes in material properties by combining experimental data and FEM simulations. The model can be used to make the AM process more efficient and consistent so that parts are produced with the expected quality.

In this research, a Finite Element (FE) model of the AM process is developed to predict the material properties during the process with an analytical model. The part geometry, material, and welding parameters are included in the FE model. The temperature history in the component is then calculated and the hardness development due to cooling after the layer has been deposited and reheating by depositing new layers is calculated using the analytical model.

## Workflow of the investigation

This study presents a model that is developed to predict the final hardness distribution of a structural steel component produced by Wire Arc Additive Manufacturing (WAAM). The proposed model postulates that the final hardness ($HV_{final}$) decreases due to reheating by welding additional layers and the resulting thermal energy input ($E_{thermal}$). The specific decrease in hardness is represented by ($\Delta HV$). A further assumption concerns the influence of the cooling rate $\dot{T}$ of a newly applied layer, which influences the initial hardness ($HV_{ini}$). It is assumed that the application of new layers using WAAM leads to a reduced initial hardness due to a slower cooling rate. This is because the already existing component acts like a "thermal battery" due to the higher temperature and thus reduces the cooling rate of the newly applied layer. Based on these assumptions, this article aims to define a method for determining the final hardness $HV_{final}$ of a part designed with WAAM using FE simulation including the following equation:

$$HV_{final} = HV_{ini}(F(\dot{T}_{800-500})) - \Delta HV(F(E_{thermal})) \quad (1)$$

To determine the influence of the cooling rate on the initial hardness, straight walls with 1 to 9 layers were welded experimentally. The hardness of the upper seam was then measured. The analysis of the upper seam has the advantage that the resulting hardness is only influenced by the cooling rate and not by reheating, as no further layer is applied. For walls built of several layers, the next layer was applied with a fixed break time of 38 seconds. The numerical simulation of nine walls with 1 to 9 layers was then conducted. Here, the cooling rate between 800 and 500 °C was determined in the upper seam. Using the measured hardness values of the top layer and the calculated cooling rates, a mathematical description coupling the resulting hardness and cooling rate can be made according to notation. 2.

$$HV_{ini}(F(\dot{T}_{800-500})) \quad (2)$$

The basic procedure is shown in Fig. 1a. In addition, a wall with nine layers was produced using the WAAM method and simulated (see Fig. 1c).

A summary of the general procedure for developing a physically based model from experimental measurement data (hardness) and simulation data, which can be used to predict the hardness distribution of a component after the WAAM process through process simulation is presented in Fig. 1.
behavior of additively manufactured parts was created by Zhang, B., et al. This study developed a multi-scale approach to predict the fatigue behavior of an additively manufactured part by considering its microstructure, defects, and stress distribution [4]. A prediction model for the final properties of AlSi10Mg parts made by additive manufacturing was created by Li, Y., and colleagues. An aluminum alloy that is frequently utilized in AM applications is AlSi10Mg. To predict the properties of the additively manufactured part, the study's predictive model considers both its microstructure and printing parameters [5]. A substantial body of prior research serves as the basis for this study, which predicts how material properties will change during the additive manufacturing process. The integrated method presented here enables the prediction of long-term changes in material properties by combining experimental data and FEM simulations. The model can be used to make the AM process more efficient and consistent so that parts are produced with the expected quality.

In this research, a Finite Element (FE) model of the AM process is developed to predict the material properties during the process with an analytical model. The part geometry, material, and welding parameters are included in the FE model. The temperature history in the component is then calculated and the hardness development due to cooling after the layer has been deposited and reheating by depositing new layers is calculated using the analytical model.

## Workflow of the investigation

This study presents a model that is developed to predict the final hardness distribution of a structural steel component produced by Wire Arc Additive Manufacturing (WAAM). The proposed model postulates that the final hardness ($HV_{final}$) decreases due to reheating by welding additional layers and the resulting thermal energy input ($E_{thermal}$). The specific decrease in hardness is represented by ($\Delta HV$). A further assumption concerns the influence of the cooling rate $\dot{T}$ of a newly applied layer, which influences the initial hardness ($HV_{ini}$). It is assumed that the application of new layers using WAAM leads to a reduced initial hardness due to a slower cooling rate. This is because the already existing component acts like a "thermal battery" due to the higher temperature and thus reduces the cooling rate of the newly applied layer. Based on these assumptions, this article aims to define a method for determining the final hardness $HV_{final}$ of a part designed with WAAM using FE simulation including the following equation:

$$HV_{final} = HV_{ini} (F(\dot{T}_{800-500})) - \Delta HV(F(E_{thermal})) \quad (1)$$

To determine the influence of the cooling rate on the initial hardness, straight walls with 1 to 9 layers were welded experimentally. The hardness of the upper seam was then measured. The analysis of the upper seam has the advantage that the resulting hardness is only influenced by the cooling rate and not by reheating, as no further layer is applied. For walls built of several layers, the next layer was applied with a fixed break time of 38 seconds. The numerical simulation of nine walls with 1 to 9 layers was then conducted. Here, the cooling rate between 800 and 500 °C was determined in the upper seam. Using the measured hardness values of the top layer and the calculated cooling rates, a mathematical description coupling the resulting hardness and cooling rate can be made according to notation. 2.

$$HV_{ini}(F(\dot{T}_{800-500})) \quad (2)$$

The basic procedure is shown in Fig. 1a. In addition, a wall with nine layers was produced using the WAAM method and simulated (see Fig. 1c).

A summary of the general procedure for developing a physically based model from experimental measurement data (hardness) and simulation data, which can be used to predict the hardness distribution of a component after the WAAM process through process simulation is presented in Fig. 1.
<figure><img src="image_45.png" /><figcaption>Figure 1: General procedure for developing a physically based model from experimentally measured data (hardness) and simulation data for predicting the hardness distribution of a component after the WAAM process using process simulation.</figcaption></figure>

The hardness distribution in the entire component was experimentally determined. The simulation was then used to analyze the temperature-time curves of the individual layers. As soon as a temperature peak of at least 800 °C was reached for the last time in a layer, the cooling rate between 800 and 500 °C was determined for this peak, and the initial hardness was calculated using the developed model. It is important to note that the selected temperature peak should be the last one that is higher than 800 °C, as the investigated material (AWS A5.18) also undergoes a phase transformation for the last time because of reheating and the resulting hardness depends significantly on the cooling rate between the selected range. All subsequent reheating cycles (by applying additional layers) with maximum temperatures below 800 °C reduce the hardness corresponding to the annealing process. To quantify this thermal energy input, the calculated temperature-time curves are now integrated from the first temperature peak below 800 °C onwards. Fig. 2 presents the procedure schematically for a temperature-time curve.

As the exit hardness of each layer can now be calculated after the final austenitizing and the final hardness after the complete WAAM process has been determined experimentally, the reduction in hardness ΔHV (due to reheating without austenitizing) can be determined and described as a function of the thermal energy. In the present case, the area under the temperature-time curve is assumed to be a measure of the thermal energy. Thus, the temperature-time curve is integrated from the first temperature peak below 800 °C. The change in hardness can therefore be expressed according to the following basic notation.

$$ \Delta HV(F(E_{thermal})) \qquad (3) $$
<figure><img src="image_46.png" /><figcaption>Figure 1: General procedure for developing a physically based model from experimentally measured data (hardness) and simulation data for predicting the hardness distribution of a component after the WAAM process using process simulation.</figcaption></figure>

The hardness distribution in the entire component was experimentally determined. The simulation was then used to analyze the temperature-time curves of the individual layers. As soon as a temperature peak of at least 800 °C was reached for the last time in a layer, the cooling rate between 800 and 500 °C was determined for this peak, and the initial hardness was calculated using the developed model. It is important to note that the selected temperature peak should be the last one that is higher than 800 °C, as the investigated material (AWS A5.18) also undergoes a phase transformation for the last time because of reheating and the resulting hardness depends significantly on the cooling rate between the selected range. All subsequent reheating cycles (by applying additional layers) with maximum temperatures below 800 °C reduce the hardness corresponding to the annealing process. To quantify this thermal energy input, the calculated temperature-time curves are now integrated from the first temperature peak below 800 °C onwards. Fig. 2 presents the procedure schematically for a temperature-time curve.

As the exit hardness of each layer can now be calculated after the final austenitizing and the final hardness after the complete WAAM process has been determined experimentally, the reduction in hardness ΔHV (due to reheating without austenitizing) can be determined and described as a function of the thermal energy. In the present case, the area under the temperature-time curve is assumed to be a measure of the thermal energy. Thus, the temperature-time curve is integrated from the first temperature peak below 800 °C. The change in hardness can therefore be expressed according to the following basic notation.

$$ \Delta HV(F(E_{thermal})) \qquad (3) $$
<figure><img src="image_46.png" /><figcaption>Figure 2: Characteristic evaluation points of a temperature-time curve of a layer.</figcaption></figure>

## Experimental setup and Hardness measurement

To analyze the hardness distribution and its development in a welded 9-layer wall, a total of 9 welding experiments were carried out with identical welding parameters, sheet dimensions, and the position of the thermocouples. Conducted experiments differ in the number of welded layers starting with 1 and ending with 9, which is summarized in Table 1.

<table><caption>Table 1. Experiment numbering</caption><thead><tr><td>Number of experiments</td><td>Number of welded layers</td></tr></thead><tbody><tr><td>1</td><td>1</td></tr><tr><td>2</td><td>2</td></tr><tr><td>...</td><td>...</td></tr><tr><td>9</td><td>9</td></tr></tbody></table>

The weld length in each experiment was 300 mm with a fixed break time of 38 s between layers. The welding direction was reversed after each layer to ensure a uniform wall. To avoid temperature effects between experiments, the base plate was cooled to room temperature before starting the next experiment. All welding parameters are summarized in Table 2.

<table><caption>Table 2. WAAM manufacturing parameters</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Wire diameter</td><td>1.2 [mm]</td></tr><tr><td>Welding speed</td><td>40 [cm/min]</td></tr><tr><td>Wire feed</td><td>4 [m/min]</td></tr><tr><td>Average electrical voltage</td><td>14.6 [V]</td></tr><tr><td>Average electrical current</td><td>144 [A]</td></tr><tr><td>Cooling Time between deposition of each layer</td><td>38 [s]</td></tr><tr><td>Stickout</td><td>15 [mm]</td></tr><tr><td>Gas / Average Flowrate</td><td>M21-ArC-18 / 15 [l/min]</td></tr></tbody></table>

During the welding tests, the temperature was measured at five different locations (Fig. 3) by using K-type wire thermocouples. This is necessary to calibrate the simulation according to the method presented in [8].
<figure><img src="image_47.png" /><figcaption>Figure 2: Characteristic evaluation points of a temperature-time curve of a layer.</figcaption></figure>

## Experimental setup and Hardness measurement

To analyze the hardness distribution and its development in a welded 9-layer wall, a total of 9 welding experiments were carried out with identical welding parameters, sheet dimensions, and the position of the thermocouples. Conducted experiments differ in the number of welded layers starting with 1 and ending with 9, which is summarized in Table 1.

<table><caption>Table 1. Experiment numbering</caption><thead><tr><td>Number of experiments</td><td>Number of welded layers</td></tr></thead><tbody><tr><td>1</td><td>1</td></tr><tr><td>2</td><td>2</td></tr><tr><td>...</td><td>...</td></tr><tr><td>9</td><td>9</td></tr></tbody></table>

The weld length in each experiment was 300 mm with a fixed break time of 38 s between layers. The welding direction was reversed after each layer to ensure a uniform wall. To avoid temperature effects between experiments, the base plate was cooled to room temperature before starting the next experiment. All welding parameters are summarized in Table 2.

<table><caption>Table 2. WAAM manufacturing parameters</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Wire diameter</td><td>1.2 [mm]</td></tr><tr><td>Welding speed</td><td>40 [cm/min]</td></tr><tr><td>Wire feed</td><td>4 [m/min]</td></tr><tr><td>Average electrical voltage</td><td>14.6 [V]</td></tr><tr><td>Average electrical current</td><td>144 [A]</td></tr><tr><td>Cooling Time between deposition of each layer</td><td>38 [s]</td></tr><tr><td>Stickout</td><td>15 [mm]</td></tr><tr><td>Gas / Average Flowrate</td><td>M21-ArC-18 / 15 [l/min]</td></tr></tbody></table>

During the welding tests, the temperature was measured at five different locations (Fig. 3) by using K-type wire thermocouples. This is necessary to calibrate the simulation according to the method presented in [8].
<figure><img src="image_47.png" /><figcaption>Figure 3. Experimental setup.</figcaption></figure>

For the hardness measurements, samples were cut in the middle and a non-etching sample preparation was carried out. A calibrated DuraScan 70 model from ZwickRoell GmbH & Co KG was used for the Vickers hardness tests. The measurements were carried out in the low force range HV1 with a test load of F = 9.807 N. The hardness of a weld wall was measured in the direction of the weld seam on the line of symmetry, starting from point 0. Fig. 4 presents the idealized welded wall (a), the measured hardness distribution (b), and measure points of the real welded wall (c). A distance between two measuring points equals 0.45 mm, which was determined by dividing the average layer height by 4 so that a layer consists of a total of 5 measuring points. This way a reliable analysis of the hardness distribution within a layer can be performed.

<figure><img src="image_48.png" /><figcaption>Figure 4. Arrangement of the measuring points in idealized welded wall (a), measured hardness distribution (b), welded wall (c)</figcaption></figure>

Simulation setup and parameters

To reproduce the experimental setup in a digital form, the Simufact. Welding 2021.1 software (Hexagon, Stockholm) is used, where a ground plate, substrate plate, and holding clamps are also included in the developed FE model (Fig 5 a). In a real experiment, the thickness of the successively welded layers differs from each other, which was also considered in the numerical model (Fig 5b).

The density and thermal capacity of the AISI 1045 steel substrate were taken from the literature [6, 7] (Fig. 6 a, b), whereas thermal conductivity was measured with Linseis LFA 1000 Laser Flash test machine (Linseis Messgeräte GmbH, Selb, Germany) (Fig. 6c). Material properties of the AWS A5.18 wire are taken from the Simufact.Welding 2021.1 material database. The applied convective heat transfer parameters were determined in earlier Authors' work [8].
<figure><img src="image_48.png" /><figcaption>Figure 3. Experimental setup.</figcaption></figure>

For the hardness measurements, samples were cut in the middle and a non-etching sample preparation was carried out. A calibrated DuraScan 70 model from ZwickRoell GmbH & Co KG was used for the Vickers hardness tests. The measurements were carried out in the low force range HV1 with a test load of F = 9.807 N. The hardness of a weld wall was measured in the direction of the weld seam on the line of symmetry, starting from point 0. Fig. 4 presents the idealized welded wall (a), the measured hardness distribution (b), and measure points of the real welded wall (c). A distance between two measuring points equals 0.45 mm, which was determined by dividing the average layer height by 4 so that a layer consists of a total of 5 measuring points. This way a reliable analysis of the hardness distribution within a layer can be performed.

<figure><img src="image_49.png" /><figcaption>Figure 4. Arrangement of the measuring points in idealized welded wall (a), measured hardness distribution (b), welded wall (c)</figcaption></figure>

Simulation setup and parameters

To reproduce the experimental setup in a digital form, the Simufact. Welding 2021.1 software (Hexagon, Stockholm) is used, where a ground plate, substrate plate, and holding clamps are also included in the developed FE model (Fig 5 a). In a real experiment, the thickness of the successively welded layers differs from each other, which was also considered in the numerical model (Fig 5b).

The density and thermal capacity of the AISI 1045 steel substrate were taken from the literature [6, 7] (Fig. 6 a, b), whereas thermal conductivity was measured with Linseis LFA 1000 Laser Flash test machine (Linseis Messgeräte GmbH, Selb, Germany) (Fig. 6c). Material properties of the AWS A5.18 wire are taken from the Simufact.Welding 2021.1 material database. The applied convective heat transfer parameters were determined in earlier Authors' work [8].
<figure><img src="image_49.png" /><figcaption>Figure 5. FE model developed in Simufact. Welding software: a) full setup, b) cross-section of the seam.</figcaption></figure>

<figure><img src="image_50.png" /><figcaption>Figure 6. Material properties of AISI 1045 as a function of temperature: a) density [6], b) heat capacity [7], c) thermal conductivity measured by Authors.</figcaption></figure>

Every layer was welded digitally with the same speed (400 mm/min), voltage (14.6 V), and current (144 A) as welding parameters applied during the real experiment. The air temperature and initial temperature of the wire were set to 23°C, the initial temperature of all plates was set to 21.5°C, and arc efficiency was assumed to be 80%. Goldak's heat source model [9] was used to describe the volumetric heat flux density distribution, where $a_f$, $a_r$, b, and d are parameters related to the shape characteristic of the welding arc. Based on earlier Authors' work [8], two heat source model (HSM) was applied, where the optimization procedure resulted in the following values of the heat source parameters: $a_f=4.96$ mm, $a_r=11.49$ mm, b=3 mm, d=2 mm for HSM 1; and $a_f=0.1$ mm, $a_r=0.1$ mm, b=0.3 mm, d=0.3 mm for HSM 2.

## Initial Hardness as a function of the cooling time

The relationship between the initial hardness and the cooling rate is a crucial factor in the development of the microstructure and mechanical properties of a printed part using WAAM in additive manufacturing. This relationship results from the solidification of the molten material
<figure><img src="image_50.png" /><figcaption>Figure 5. FE model developed in Simufact. Welding software: a) full setup, b) cross-section of the seam.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>Figure 6. Material properties of AISI 1045 as a function of temperature: a) density [6], b) heat capacity [7], c) thermal conductivity measured by Authors.</figcaption></figure>

Every layer was welded digitally with the same speed (400 mm/min), voltage (14.6 V), and current (144 A) as welding parameters applied during the real experiment. The air temperature and initial temperature of the wire were set to 23°C, the initial temperature of all plates was set to 21.5°C, and arc efficiency was assumed to be 80%. Goldak's heat source model [9] was used to describe the volumetric heat flux density distribution, where $a_f$, $a_r$, b, and d are parameters related to the shape characteristic of the welding arc. Based on earlier Authors' work [8], two heat source model (HSM) was applied, where the optimization procedure resulted in the following values of the heat source parameters: $a_f=4.96$ mm, $a_r=11.49$ mm, b=3 mm, d=2 mm for HSM 1; and $a_f=0.1$ mm, $a_r=0.1$ mm, b=0.3 mm, d=0.3 mm for HSM 2.

## Initial Hardness as a function of the cooling time

The relationship between the initial hardness and the cooling rate is a crucial factor in the development of the microstructure and mechanical properties of a printed part using WAAM in additive manufacturing. This relationship results from the solidification of the molten material
during the cooling phase, a process in which the cooling rate has a significant influence on the microstructural properties and thus on the hardness of the final product. The complex correlation between faster cooling rates, which result in a harder and finer microstructure, and slower cooling rates, which result in a softer and coarser microstructure, highlights the importance of controlling cooling time as a tactical means of controlling mechanical properties for example.

As mentioned, 9 components with one to nine layers were produced using the WAAM process. The hardness was then measured in the top layer in each case. This procedure means that the resulting hardness is only influenced by the cooling rate and not by any reheating, as no further seam was applied. As already mentioned, the break time until a new layer is welded is constantly 38 s. The numerical process simulation of these nine components with different numbers of layers was then carried out. With the help of the simulation, it is now possible to determine the cooling rate in the seam middle of the uppermost seam (in which the hardness was also measured). This was done for the temperature range between 800 and 500 °C, as the cooling rate in this range has a significant influence on the mechanical properties [10].

Fig. 7 shows the cooling rates and the resulting hardness values for the top layer of the components with 2 to 9 layers. The component with one layer was not evaluated to eliminate the influence of mixing with the base material. It can be seen that the resulting hardness increases with increasing cooling rate. Furthermore, the cooling rate decreases with the number of layers of a component. This is because existing layers have a higher temperature than the base plate and therefore reduce heat conduction. The relationship between the resulting hardness and cooling rate can be described analytically using a quadratic equation with a very high coefficient of determination (see Fig. 7 and Eq. 4).

$$HV_{ini} = 0.0025T^2 - 0.0403T + 196.34, \quad R^2 = 0.98 \quad (4)$$

<figure><img src="image_51.png" /><figcaption>Figure 7. A correlation between initial hardness and cooling rate.</figcaption></figure>

## Hardness development during the WAAM process

As the individual layers cool down during the WAAM process and are also reheated as new layers are applied, both cooling and reheating must be considered in the physical-based analytical model. In the context of this work, this is done using the 9-layer wall as an example. Fig. 8 shows a characteristic temperature-time curve for the second layer of the wall with 9 layers. The peak of the temperature at approx. 25 s represents the welding of layer 2, which then cools down to a point in time of approx. 105 s. The layer is then reheated to over 800 °C (including austenitization and
during the cooling phase, a process in which the cooling rate has a significant influence on the microstructural properties and thus on the hardness of the final product. The complex correlation between faster cooling rates, which result in a harder and finer microstructure, and slower cooling rates, which result in a softer and coarser microstructure, highlights the importance of controlling cooling time as a tactical means of controlling mechanical properties for example.

As mentioned, 9 components with one to nine layers were produced using the WAAM process. The hardness was then measured in the top layer in each case. This procedure means that the resulting hardness is only influenced by the cooling rate and not by any reheating, as no further seam was applied. As already mentioned, the break time until a new layer is welded is constantly 38 s. The numerical process simulation of these nine components with different numbers of layers was then carried out. With the help of the simulation, it is now possible to determine the cooling rate in the seam middle of the uppermost seam (in which the hardness was also measured). This was done for the temperature range between 800 and 500 °C, as the cooling rate in this range has a significant influence on the mechanical properties [10].

Fig. 7 shows the cooling rates and the resulting hardness values for the top layer of the components with 2 to 9 layers. The component with one layer was not evaluated to eliminate the influence of mixing with the base material. It can be seen that the resulting hardness increases with increasing cooling rate. Furthermore, the cooling rate decreases with the number of layers of a component. This is because existing layers have a higher temperature than the base plate and therefore reduce heat conduction. The relationship between the resulting hardness and cooling rate can be described analytically using a quadratic equation with a very high coefficient of determination (see Fig. 7 and Eq. 4).

$$HV_{ini} = 0.0025\dot{T}^2 - 0.0403\dot{T} + 196.34, \quad R^2 = 0.98 \quad (4)$$

<figure><img src="image_52.png" /><figcaption>Figure 7. A correlation between initial hardness and cooling rate.</figcaption></figure>

## Hardness development during the WAAM process

As the individual layers cool down during the WAAM process and are also reheated as new layers are applied, both cooling and reheating must be considered in the physical-based analytical model. In the context of this work, this is done using the 9-layer wall as an example. Fig. 8 shows a characteristic temperature-time curve for the second layer of the wall with 9 layers. The peak of the temperature at approx. 25 s represents the welding of layer 2, which then cools down to a point in time of approx. 105 s. The layer is then reheated to over 800 °C (including austenitization and
renewed cooling) and then it cools down again. Since this second peak is the last one above 800 °C, the cooling rate between 800 and 500 °C is determined here, and the initial hardness of the $HV_{initial}$ layer is determined using Eq. 4. From a time of 190 s (red line in Fig. 8), it is now assumed that the cyclical reheating corresponds to an annealing treatment, which reduces the hardness. To quantify the amount of thermal energy introduced $E_{thermal}$, the temperature-time curve is integrated from the reheating after the last temperature peak above 800 °C until the end of the process.

<figure><img src="image_52.png" /><figcaption>Figure 8. Example of the temperature history at the selected point.</figcaption></figure>

The decrease in hardness (due to reheating) now represents the difference between the measured hardness at the end of the process and the calculated initial hardness (after the last austenitizing) for each layer. This procedure allows Authors to determine a relationship between the hardness difference $\Delta HV$ and the area under the temperature-time curve (as a measure of the thermal energy applied). This relationship is shown graphically in Fig. 9.

<figure><img src="image_53.png" /><figcaption>Figure 9. Plot of correlation between reduction of hardness and thermal energy.</figcaption></figure>

It can be seen that the difference in hardness increases with increasing thermal energy input, i.e. the final hardness is reduced to a greater extent. However, from a certain value (from layer 5) an almost stationary range is reached. The relationship between curve area A and decrease in
renewed cooling) and then it cools down again. Since this second peak is the last one above 800 °C, the cooling rate between 800 and 500 °C is determined here, and the initial hardness of the $HV_{initial}$ layer is determined using Eq. 4. From a time of 190 s (red line in Fig. 8), it is now assumed that the cyclical reheating corresponds to an annealing treatment, which reduces the hardness. To quantify the amount of thermal energy introduced $E_{thermal}$, the temperature-time curve is integrated from the reheating after the last temperature peak above 800 °C until the end of the process.

<figure><img src="image_53.png" /><figcaption>Figure 8. Example of the temperature history at the selected point.</figcaption></figure>

The decrease in hardness (due to reheating) now represents the difference between the measured hardness at the end of the process and the calculated initial hardness (after the last austenitizing) for each layer. This procedure allows Authors to determine a relationship between the hardness difference $\Delta HV$ and the area under the temperature-time curve (as a measure of the thermal energy applied). This relationship is shown graphically in Fig. 9.

<figure><img src="image_54.png" /><figcaption>Figure 9. Plot of correlation between reduction of hardness and thermal energy.</figcaption></figure>

It can be seen that the difference in hardness increases with increasing thermal energy input, i.e. the final hardness is reduced to a greater extent. However, from a certain value (from layer 5) an almost stationary range is reached. The relationship between curve area A and decrease in
hardness can also be described analytically with a very high coefficient of determination (see Eq. 5).

$$ \Delta HV = -5E - 07 \cdot A^2 + 0.0092A + 0.7649, \quad R^2 = 0.95 \quad (5) $$

Using two sub-models from Eq. 4 and 5, the hardness after the WAAM process can now be calculated for each numerically determined temperature-time curve, for the first time considering the cooling-related hardness development and the reduction in hardness due to reheating. In this case, the model was used for the wall with 9 layers. Fig. 10 presents the comparison between the experiment and model as well as the error bar with a deviation of ± 5%. It can be seen that the maximum deviation between the model and simulation is less than 5%.

<figure><img src="image_54.png" /><figcaption>Figure 10. Comparison between experiment and model as well as the error bar with a deviation of ± 5%.</figcaption></figure>

## Conclusions

Based on numerical and experimental investigations, the Authors developed a reliable model that predicts the material hardness during and after the WAAM process. This is the first time that an approach has been pursued that considers the development of hardness as a function of both the cooling rate and the energy introduced as a result of reheating by applying additional layers. The analytical model based on physical effects can be integrated into FE systems for the process simulation of WAAM processes using subroutines. This means that the hardness distribution in WAAM-manufactured components can now be reliably predicted using process simulation.

The obtained results prove that:

* As the molten material solidifies during the cooling phase, its microstructure and hardness are influenced by the cooling rate of the material.

* Faster cooling rates can lead to a finer microstructure and thus to a higher hardness, while slower cooling rates can lead to a coarser microstructure and thus to a lower hardness. The formation of hardness can be described analytically as a function of the cooling rate.

* During the process, the hardness of a layer can be reduced by reheating due to the application of further layers. The amount of hardness reduction is essentially dependent on the thermal energy applied and the relationship between hardness reduction and thermal energy can be described analytically.
hardness can also be described analytically with a very high coefficient of determination (see Eq. 5).

$$ \Delta HV = -5E - 07 \cdot A^2 + 0.0092A + 0.7649, \quad R^2 = 0.95 \qquad (5) $$

Using two sub-models from Eq. 4 and 5, the hardness after the WAAM process can now be calculated for each numerically determined temperature-time curve, for the first time considering the cooling-related hardness development and the reduction in hardness due to reheating. In this case, the model was used for the wall with 9 layers. Fig. 10 presents the comparison between the experiment and model as well as the error bar with a deviation of ± 5%. It can be seen that the maximum deviation between the model and simulation is less than 5%.

<figure><img src="image_55.png" /><figcaption>Figure 10. Comparison between experiment and model as well as the error bar with a deviation of ± 5%.</figcaption></figure>

## Conclusions

Based on numerical and experimental investigations, the Authors developed a reliable model that predicts the material hardness during and after the WAAM process. This is the first time that an approach has been pursued that considers the development of hardness as a function of both the cooling rate and the energy introduced as a result of reheating by applying additional layers. The analytical model based on physical effects can be integrated into FE systems for the process simulation of WAAM processes using subroutines. This means that the hardness distribution in WAAM-manufactured components can now be reliably predicted using process simulation.

The obtained results prove that:

* As the molten material solidifies during the cooling phase, its microstructure and hardness are influenced by the cooling rate of the material.

* Faster cooling rates can lead to a finer microstructure and thus to a higher hardness, while slower cooling rates can lead to a coarser microstructure and thus to a lower hardness. The formation of hardness can be described analytically as a function of the cooling rate.

* During the process, the hardness of a layer can be reduced by reheating due to the application of further layers. The amount of hardness reduction is essentially dependent on the thermal energy applied and the relationship between hardness reduction and thermal energy can be described analytically.
* By influencing the cooling time, the initial hardness of each printed layer can be effectively controlled, which is a crucial mechanism for adjusting the mechanical properties of the final product.

## References

[1] Bai, Y., et al. "Prediction of the evolution of residual stresses in additive manufactured parts using a finite element approach." Finite Elements in Analysis and Design 190 (2022): 103461. https://purls.library.ucf.edu/go/DP0027104

[2] Zhao, Y., et al. "Predicting the mechanical properties of additive manufactured parts using machine learning." Materials Science and Engineering: A 782 (2020): 139329. https://www.sciencedirect.com/science/article/pii/S2238785421006670

[3] Wu, S., et al. "Experimental and numerical investigation of the microstructure and mechanical properties of additive manufactured Ti-6Al-4V." Journal of Materials Science & Technology 38.1 (2022): 1-15. https://www.sciencedirect.com/science/article/abs/pii/S0921509318318008

[4] Zhang, B., et al. "Prediction of the fatigue behavior of additive manufactured parts using a multi-scale approach." International Journal of Fatigue 163 (2022): 107001. https://asmedigitalcollection.asme.org/MSEC/proceedings/MSEC2021/85062/V001T01A015/1115404

[5] Li, Y., et al. "Development of a predictive model for the hardness of additive manufactured AlSi10Mg parts." Journal of Manufacturing Processes 78 (2022): 134-144. https://link.springer.com/article/10.1007/s12666-022-02676-5

[6] M. Hojny, M. Glowacki, Numerical Modelling of Steel Deformation at Extra-High Temperatures, in P. Miidla (ed.), Numerical Modelling, IntechOpen, London, 2012, 10.5772/36562. https://www.intechopen.com/chapters/33087

[7] A. Martinovs, S. Polukoshko, E. Zaicevs, R. Revalds, Laser Hardening Process Optimizations Using FEM, Engineering for Rural Development, 2020, DOI: 10.22616/ERDev2020.19.TF372 https://www.researchgate.net/publication/342412420_LASER_HARDENING_PROCESS_OPTIMIZATION_USING_FEM

[8] J. Szyndler, A. Schmidt, S. Härtel, Determination of welding heat source parameters for FEM simulation based on temperature history and real bead shape. Materials Research Proceedings 28 (2023) 159-168. https://doi.org/10.21741/9781644902479-18

[9] J. Goldak, A.P. Chakravarti, M. Bibby, A new finite element model for welding heat sources, Metallurgical Transactions B, 15 (1984) 299-305. https://link.springer.com/article/10.1007/BF02667333

[10] M. Mician, D. Harmaniak, F. Novy, J. Winczek, J. Moravec, L. Trsko, Effect of the t8/5 cooling time on the properties of S960MC steel in the HAZ of welded joints evaluated by thermal physical simulation. Metals (2020) 10 (2) 229. https://doi.org/10.3390/met10020229
* By influencing the cooling time, the initial hardness of each printed layer can be effectively controlled, which is a crucial mechanism for adjusting the mechanical properties of the final product.

## References

[1] Bai, Y., et al. "Prediction of the evolution of residual stresses in additive manufactured parts using a finite element approach." Finite Elements in Analysis and Design 190 (2022): 103461. https://purls.library.ucf.edu/go/DP0027104

[2] Zhao, Y., et al. "Predicting the mechanical properties of additive manufactured parts using machine learning." Materials Science and Engineering: A 782 (2020): 139329. https://www.sciencedirect.com/science/article/pii/S2238785421006670

[3] Wu, S., et al. "Experimental and numerical investigation of the microstructure and mechanical properties of additive manufactured Ti-6Al-4V." Journal of Materials Science & Technology 38.1 (2022): 1-15. https://www.sciencedirect.com/science/article/abs/pii/S0921509318318008

[4] Zhang, B., et al. "Prediction of the fatigue behavior of additive manufactured parts using a multi-scale approach." International Journal of Fatigue 163 (2022): 107001. https://asmedigitalcollection.asme.org/MSEC/proceedings/MSEC2021/85062/V001T01A015/1115404

[5] Li, Y., et al. "Development of a predictive model for the hardness of additive manufactured AlSi10Mg parts." Journal of Manufacturing Processes 78 (2022): 134-144. https://link.springer.com/article/10.1007/s12666-022-02676-5

[6] M. Hojny, M. Glowacki, Numerical Modelling of Steel Deformation at Extra-High Temperatures, in P. Miidla (ed.), Numerical Modelling, IntechOpen, London, 2012, 10.5772/36562. https://www.intechopen.com/chapters/33087

[7] A. Martinovs, S. Polukoshko, E. Zaicevs, R. Revalds, Laser Hardening Process Optimizations Using FEM, Engineering for Rural Development, 2020, DOI: 10.22616/ERDev2020.19.TF372 https://www.researchgate.net/publication/342412420_LASER_HARDENING_PROCESS_OPTIMIZATION_USING_FEM

[8] J. Szyndler, A. Schmidt, S. Härtel, Determination of welding heat source parameters for FEM simulation based on temperature history and real bead shape. Materials Research Proceedings 28 (2023) 159-168. https://doi.org/10.21741/9781644902479-18

[9] J. Goldak, A.P. Chakravarti, M. Bibby, A new finite element model for welding heat sources, Metallurgical Transactions B, 15 (1984) 299-305. https://link.springer.com/article/10.1007/BF02667333

[10] M. Mician, D. Harmaniak, F. Novy, J. Winczek, J. Moravec, L. Trsko, Effect of the t8/5 cooling time on the properties of S960MC steel in the HAZ of welded joints evaluated by thermal physical simulation. Metals (2020) 10 (2) 229. https://doi.org/10.3390/met10020229