# Fatigue life analysis of hot forming dies produced by the L-PBF and WA-DED additive technologies

ALIMOV Artem¹,ᵃ*, SVIRIDOV Alexander¹,ᵇ and HÄRTEL Sebastian¹,ᶜ

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

ᵃalimov@b-tu.de, ᵇsviridov@b-tu.de, ᶜhaertel@b-tu.de

**Keywords:** Fatigue, LCF, Additive Manufacturing, Hot Forming Dies, L-PBF, WA-DED, WAAM

**Abstract.** Additive manufacturing (AM) technologies are widely used for the production of individual complex precise parts and small-scale production. AM has been used in many applications for the manufacturing of sheet metal forming dies. The production of hot bulk forming dies by additive manufacturing technology has not been extensively studied in the literature. Traditional hot working steels exhibit many problems during the AM process, such as crack formation due to complex thermal processing cycles. Maraging/precipitation-hardened (PH) steels may be a good alternative because of their ability to undergo additive manufacturing. One of the main causes of hot forming die failures is thermomechanical fatigue. The fatigue performance of 17-4PH was investigated primarily during high-cycle fatigue (HCF) and low-cycle fatigue (LCF) modes without heat treatment as well as with H1100 heat treatment. The LCF of dies with the heat treatment designation H900, which is most appropriate for die production, has not been intensively investigated. In this work, the mechanical properties and fatigue life of 17-4PH precipitation-hardened steel produced by laser powder bed fusion (L-PBF) and wire arc direct energy deposition (WA-DED) additive technologies were experimentally investigated, and numerical models for high-temperature low cycle fatigue life were developed for 17-4PH steel. Based on the FEM simulation in the commercial program QForm UK, the fatigue life of the produced dies was predicted.

**Introduction**

Additive manufacturing (AM) technologies represent methods of layer-by-layer production of three-dimensional objects from a CAD model [1]. The main application fields of AM include the aerospace [2], energy [3] and automotive industries [4] as well as biomedicine [5]. Previously reported AMs were used primarily for rapid prototyping, for the production of individual parts or for small batch production [6], but the rapid development of AMs in recent years has extended the application of these materials to the production of casting molds [7] and sheet metal forming dies [8]. The additive manufacturing of hot bulk forging dies has not been covered as extensively in the literature. Huskic et al. [9] and Junker et al. [10] reported the additive manufacturing of forming dies from 1.2709 and 1.2343 tool steels, respectively. Typically, tool steels tend to develop hot cracks in additive manufacturing [11] and require special measures such as preheating the build plate to high temperatures of up to 800°C for defect-free production.

Currently, the most commonly used additive manufacturing methods are powder bed fusion (PBF) and direct energy deposition (DED) methods [12]. The energy source can be an electric arc, laser, plasma or electron beam. Laser Powder Bed Fusion (L-PBF) technology fabricates complex 3D components from powder by layer-by-layer selective laser melting. It offers one of the highest accuracies among AM technologies [13]. The main disadvantage of L-PBF is the limited workspace and relatively low performance. Wire-arc DED (WA-DED) has the highest productivity among the AM methods. Due to the use of an electric arc, WA-DED is believed to

<figure><img src="image_2022.png" /></figure>
cause high heat loads during processing [14] and a large heat-affected area [15]; therefore, WA-DED can be utilized only limitedly for the die production.

In an earlier work [16] the feasibility of using lightweight hot forming dies produced from 17-4 steel via L-PBF and WA-DED technologies and their operability was demonstrated. However, the fatigue strength of additive manufactured dies made from 17-4 PH steel requires further study because this is one of the main failure causes of hot forging dies [17]. Hot forging dies have a rather short service life due to high thermomechanical loads [18]; therefore, low cycle fatigue (LCF) is the determining criterion. The fatigue performance of 17-4PH was investigated primarily during high-cycle (HCF) [19] and low-cycle (LCF) fatigue modes without heat treatment [20] as well as with H1100 heat treatment [21]. The LCF of dies with the heat treatment designation H900, which is most appropriate for die production, has not been intensively investigated. In this paper, the mechanical and fatigue properties of AM-processed 17-4PH steel after H900 designation heat treatment are experimentally investigated, the coefficients of the Manson-Coffin-Basquin model are determined, and the prediction of cycles to failure is carried out using QForm UK software.

## Materials and Methods

**Materials.** The material used in the L-PBF process was a 17-4PH powder with a 20–45 µm particle size provided by m4p material solutions GmbH, Magdeburg, Germany.

The material used in the WA-DED process was a 1,2 mm diameter wire made of 17-4PH provided by Metal Technology Canterbo GmbH, Meerbusch, Germany. The chemical composition of the material used is presented in Table 1.

<table><caption>Table 1. The chemical composition of the 17-4PH steel</caption><thead><tr><th>Element</th><th>Cr</th><th>Ni</th><th>Cu</th><th>Nb</th><th>C</th><th>Si</th><th>Mn</th><th>Fe</th></tr></thead><tbody><tr><td rowspan="2">Nom.</td><td>15,0–</td><td>3,0–</td><td>3,0–</td><td>0,15–</td><td rowspan="2">&lt;0,07</td><td rowspan="2">&lt;1,00</td><td rowspan="2">&lt;0,70</td><td rowspan="6">Bal.</td></tr><tr><td>17,0</td><td>5,0</td><td>5,0</td><td>0,45</td></tr><tr><td>L-PBF</td><td>16,0</td><td>4,0</td><td>4,0</td><td>0,29</td><td>0,03</td><td>0,60</td><td>0,70</td></tr><tr><td>WA-DED</td><td>16,37</td><td>4,78</td><td>3,63</td><td>0,23</td><td>0,019</td><td>0,46</td><td>0,64</td></tr></tbody></table>

**Production of the specimens.** The WA-DED process was carried out with a Guttroff robotic welding system using a Fronius TPSi 500 welding source. ArC2 (Argon with 2% CO₂) was used as a shielding gas with a flow rate of 15 L/min. AISI 304 stainless steel plates with dimensions of 50x150x10 mm were used as the substrate and were not preheated during the production process. The dwell time after each deposited layer was 30, 60 and 90 s for layers 1, 2 and 3, respectively, and further 200 s for the remaining layers. One sample block was produced at a time, alternating the starting point and welding direction for each layer. The main parameters of the WA-DED process were the same as those for die production [16] and are listed in Table 2.

<table><caption>Table 2. The main parameters of the WA-DED process</caption><tr><td>Wire feed speed, (m/min)</td><td>7,0</td></tr><tr><td>Stickout, (mm)</td><td>15,0</td></tr><tr><td>Welding speed, (mm/min)</td><td>400,0</td></tr><tr><td>Welding power, (W)</td><td>3600</td></tr><tr><td>Weld seam width, (mm)</td><td>7,5</td></tr><tr><td>Layer thickness, (mm)</td><td>3,2</td></tr><tr><td>Weld seam overlap, (mm)</td><td>4,2</td></tr></table>

The L-PBF process was carried out on an EOS M290 with the parameters provided by the machine manufacturer and were the same as those used for die production [16]. The main parameters of the L-PBF process are listed in Table 3.
cause high heat loads during processing [14] and a large heat-affected area [15]; therefore, WA-DED can be utilized only limitedly for the die production.

In an earlier work [16] the feasibility of using lightweight hot forming dies produced from 17-4 steel via L-PBF and WA-DED technologies and their operability was demonstrated. However, the fatigue strength of additive manufactured dies made from 17-4 PH steel requires further study because this is one of the main failure causes of hot forging dies [17]. Hot forging dies have a rather short service life due to high thermomechanical loads [18]; therefore, low cycle fatigue (LCF) is the determining criterion. The fatigue performance of 17-4PH was investigated primarily during high-cycle (HCF) [19] and low-cycle (LCF) fatigue modes without heat treatment [20] as well as with H1100 heat treatment [21]. The LCF of dies with the heat treatment designation H900, which is most appropriate for die production, has not been intensively investigated. In this paper, the mechanical and fatigue properties of AM-processed 17-4PH steel after H900 designation heat treatment are experimentally investigated, the coefficients of the Manson-Coffin-Basquin model are determined, and the prediction of cycles to failure is carried out using QForm UK software.

## Materials and Methods

**Materials.** The material used in the L-PBF process was a 17-4PH powder with a 20–45 µm particle size provided by m4p material solutions GmbH, Magdeburg, Germany.

The material used in the WA-DED process was a 1,2 mm diameter wire made of 17-4PH provided by Metal Technology Canterbo GmbH, Meerbusch, Germany. The chemical composition of the material used is presented in Table 1.

<table><caption>Table 1. The chemical composition of the 17-4PH steel</caption><thead><tr><th>Element</th><th>Cr</th><th>Ni</th><th>Cu</th><th>Nb</th><th>C</th><th>Si</th><th>Mn</th><th>Fe</th></tr></thead><tbody><tr><td rowspan="2">Nom.</td><td>15,0–</td><td>3,0–</td><td>3,0–</td><td>0,15–</td><td rowspan="2">&lt;0,07</td><td rowspan="2">&lt;1,00</td><td rowspan="2">&lt;0,70</td><td rowspan="6">Bal.</td></tr><tr><td>17,0</td><td>5,0</td><td>5,0</td><td>0,45</td></tr><tr><td>L-PBF</td><td>16,0</td><td>4,0</td><td>4,0</td><td>0,29</td><td>0,03</td><td>0,60</td><td>0,70</td></tr><tr><td>WA-DED</td><td>16,37</td><td>4,78</td><td>3,63</td><td>0,23</td><td>0,019</td><td>0,46</td><td>0,64</td></tr></tbody></table>

**Production of the specimens.** The WA-DED process was carried out with a Guttroff robotic welding system using a Fronius TPSi 500 welding source. ArC2 (Argon with 2% CO₂) was used as a shielding gas with a flow rate of 15 L/min. AISI 304 stainless steel plates with dimensions of 50x150x10 mm were used as the substrate and were not preheated during the production process. The dwell time after each deposited layer was 30, 60 and 90 s for layers 1, 2 and 3, respectively, and further 200 s for the remaining layers. One sample block was produced at a time, alternating the starting point and welding direction for each layer. The main parameters of the WA-DED process were the same as those for die production [16] and are listed in Table 2.

<table><caption>Table 2. The main parameters of the WA-DED process</caption><tr><td>Wire feed speed, (m/min)</td><td>7,0</td></tr><tr><td>Stickout, (mm)</td><td>15,0</td></tr><tr><td>Welding speed, (mm/min)</td><td>400,0</td></tr><tr><td>Welding power, (W)</td><td>3600</td></tr><tr><td>Weld seam width, (mm)</td><td>7,5</td></tr><tr><td>Layer thickness, (mm)</td><td>3,2</td></tr><tr><td>Weld seam overlap, (mm)</td><td>4,2</td></tr></table>

The L-PBF process was carried out on an EOS M290 with the parameters provided by the machine manufacturer and were the same as those used for die production [16]. The main parameters of the L-PBF process are listed in Table 3.
<table><caption>Table 3. The main parameters of the L-PBF process</caption><tr><td>Exposure speed, (mm/s)</td><td>750,0</td></tr><tr><td>Hatching distance, (mm)</td><td>0,11</td></tr><tr><td>Layer thickness, (µm)</td><td>40</td></tr><tr><td>Laser power, (W)</td><td>220</td></tr></table>

The WA-DED-deposited block (Fig. 1, a) was flat milled, and then, Ø10x60 mm cylindrical samples (Fig. 1, c) were extracted longitudinally (0°) and transversally (90°) to the deposition direction using electrical discharge machining. In L-PBF production, 32 samples each longitudinally (0°) and transversally (90°) to the deposition direction were produced simultaneously (Fig. 1, b).

<figure><img src="image_2031.png" /><figcaption>Fig. 1. WA-DED-produced block (a), L-PBF-produced cylindrical samples (b)</figcaption></figure>

The samples were heat treated horizontally in a chamber furnace to the H900 designation to maximize the mechanical properties. No visible deformation of the samples was observed during the heat treatment. The heat treatment scheme is shown in Fig. 2.

<figure><img src="image_2032.png" /><figcaption>Fig. 2. The heat treatment scheme for 17-4PH steel under the H900 designation</figcaption></figure>

The final stage of specimen fabrication involved turning the specimens according to the final dimensions (Fig. 3).

<figure><img src="image_2033.png" /><figcaption>Fig. 3. Geometry of the tensile (a) and fatigue (b) test specimens</figcaption></figure>

Microstructural characterization was performed using a Keyence VHX-7000 digital microscope. The microstructure of WA-DED and L-PBF-produced specimens after the heat treatment to the designation H900 is shown in Fig. 4.
<table><caption>Table 3. The main parameters of the L-PBF process</caption><tr><td>Exposure speed, (mm/s)</td><td>750,0</td></tr><tr><td>Hatching distance, (mm)</td><td>0,11</td></tr><tr><td>Layer thickness, (µm)</td><td>40</td></tr><tr><td>Laser power, (W)</td><td>220</td></tr></table>

The WA-DED-deposited block (Fig. 1, a) was flat milled, and then, Ø10x60 mm cylindrical samples (Fig. 1, c) were extracted longitudinally (0°) and transversally (90°) to the deposition direction using electrical discharge machining. In L-PBF production, 32 samples each longitudinally (0°) and transversally (90°) to the deposition direction were produced simultaneously (Fig. 1, b).

<figure><img src="image_2023.png" /><figcaption>Fig. 1. WA-DED-produced block (a), L-PBF-produced cylindrical samples (b)</figcaption></figure>

The samples were heat treated horizontally in a chamber furnace to the H900 designation to maximize the mechanical properties. No visible deformation of the samples was observed during the heat treatment. The heat treatment scheme is shown in Fig. 2.

<figure><img src="image_2024.png" /><figcaption>Fig. 2. The heat treatment scheme for 17-4PH steel under the H900 designation</figcaption></figure>

The final stage of specimen fabrication involved turning the specimens according to the final dimensions (Fig. 3).

<figure><img src="image_2025.png" /><figcaption>Fig. 3. Geometry of the tensile (a) and fatigue (b) test specimens</figcaption></figure>

Microstructural characterization was performed using a Keyence VHX-7000 digital microscope. The microstructure of WA-DED and L-PBF-produced specimens after the heat treatment to the designation H900 is shown in Fig. 4.
<figure><img src="image_2034.png" /><figcaption>Fig. 4. Microstructure of the WA-DED (a) L-PBF-produced (b) specimens</figcaption></figure>

Surface analysis was carried out with a Keyence VK-X1000 3D laser confocal scanning microscope. Surface profile analysis of the specimen gauge length showed that the roughnesses of all the samples after turning were nearly identical and equal to Ra 4.35 ± 0.5.

Test equipment and parameters. Static tensile tests and fatigue tests were carried out with an Instron 8801 ServoHydraulic testing system. The specimen was fixed in M8 threaded holes of adapters made of Waspaloy stainless steel and tightened with AISI 304 steel nuts with a 20 Nm torque. The adapters were clamped in hydraulic grips using notched prismatic wedge inserts. The samples were heated in a three-zone split resistance furnace with automatic temperature control using three thermocouples. The test setup is shown in Fig. 6.

<figure><img src="image_2035.png" /><figcaption>Fig. 5. Dynamic testing machine Instron 8801 (a), three-zone split furnace (b) and a broken specimen fixed with adapters (c)</figcaption></figure>
<figure><img src="image_2026.png" /><figcaption>Fig. 4. Microstructure of the WA-DED (a) L-PBF-produced (b) specimens</figcaption></figure>

Surface analysis was carried out with a Keyence VK-X1000 3D laser confocal scanning microscope. Surface profile analysis of the specimen gauge length showed that the roughnesses of all the samples after turning were nearly identical and equal to Ra 4.35 ± 0.5.

Test equipment and parameters. Static tensile tests and fatigue tests were carried out with an Instron 8801 ServoHydraulic testing system. The specimen was fixed in M8 threaded holes of adapters made of Waspaloy stainless steel and tightened with AISI 304 steel nuts with a 20 Nm torque. The adapters were clamped in hydraulic grips using notched prismatic wedge inserts. The samples were heated in a three-zone split resistance furnace with automatic temperature control using three thermocouples. The test setup is shown in Fig. 6.

<figure><img src="image_2027.png" /><figcaption>Fig. 5. Dynamic testing machine Instron 8801 (a), three-zone split furnace (b) and a broken specimen fixed with adapters (c)</figcaption></figure>
The main test parameters are shown in Table 4.

<table><caption>Table 4. The main parameters of the tensile and fatigue tests</caption><tr><td>Test temperature, (°C)</td><td>23,2 ± 2,0<br>200,0 ± 5,0</td></tr><tr><td colspan="2"><u>Static tensile tests</u></td></tr><tr><td>Test speed, (mm/s)</td><td>1,0</td></tr><tr><td colspan="2"><u>Fatigue tests</u></td></tr><tr><td>Control</td><td>σ<sub>a</sub></td></tr><tr><td>Frequency, (Hz)</td><td>20,0</td></tr><tr><td>Cycle asymmetry coefficient R</td><td>-1</td></tr></table>

**FEM simulation.** Simulations were performed with the commercial FEM software QForm UK 11.0.1 (QForm Group FZ LLC, Fujairah, United Arab Emirates) in a general forming module. The forging process and simulation parameters are described in detail in [16].

Manson-Coffin-Basquin equation with Morrow's mean stress correction [22] implemented in QForm UK [23] was used for the prediction of fatigue life:

$$ \frac{\Delta\epsilon}{2} = \frac{\sigma'_{f} - \sigma_{m}}{E} (2N_{f})^{b} + \epsilon'_{f} (2N_{f})^{c} \quad (1) $$

Due to the absence of any considerable plastic strains up to the last few cycles before failure, their influence was neglected. Cyclic hardening and softening effects were also not recognized during fatigue testing and thus were not considered. The equation (1) was used in the following form:

$$ \frac{\Delta\sigma}{2} = (\sigma'_{f} - \sigma_{m})(2N_{f})^{b} \quad (2) $$

A special module for cyclic tool heating was used to obtain an equilibrium temperature during the batch forging process.

The section cuts of the geometry models used in the forging simulation with WA-DED (b) and L-PBF (b) dies are shown in Fig. 7.

<figure><img src="image_2036.png" /><figcaption>Fig. 6. The section cuts of the geometry models used in the forging simulation with WA-DED (b) and L-PBF (b) die</figcaption></figure>

## Results and discussion

**Analysis of die thermomechanical loads.** To evaluate fatigue life, it is necessary to determine the temperature and loads to which the hot forging die is subjected. During the continuous forging process, the die does not cool to the initial temperature; therefore, the QForm tool cycling module was used to analyse the steady-state die temperature. The temperature distribution over the die impressions at the end of the forging is shown in Fig. 8.
The main test parameters are shown in Table 4.

<table><caption>Table 4. The main parameters of the tensile and fatigue tests</caption><tr><td>Test temperature, (°C)</td><td>23,2 ± 2,0<br>200,0 ± 5,0</td></tr><tr><td colspan="2"><u>Static tensile tests</u></td></tr><tr><td>Test speed, (mm/s)</td><td>1,0</td></tr><tr><td colspan="2"><u>Fatigue tests</u></td></tr><tr><td>Control</td><td>σ<sub>a</sub></td></tr><tr><td>Frequency, (Hz)</td><td>20,0</td></tr><tr><td>Cycle asymmetry coefficient R</td><td>-1</td></tr></table>

**FEM simulation.** Simulations were performed with the commercial FEM software QForm UK 11.0.1 (QForm Group FZ LLC, Fujairah, United Arab Emirates) in a general forming module. The forging process and simulation parameters are described in detail in [16].

Manson-Coffin-Basquin equation with Morrow's mean stress correction [22] implemented in QForm UK [23] was used for the prediction of fatigue life:

$$ \frac{\Delta\epsilon}{2} = \frac{\sigma'_{f} - \sigma_{m}}{E} (2N_{f})^{b} + \epsilon'_{f} (2N_{f})^{c} \quad (1) $$

Due to the absence of any considerable plastic strains up to the last few cycles before failure, their influence was neglected. Cyclic hardening and softening effects were also not recognized during fatigue testing and thus were not considered. The equation (1) was used in the following form:

$$ \frac{\Delta\sigma}{2} = (\sigma'_{f} - \sigma_{m})(2N_{f})^{b} \quad (2) $$

A special module for cyclic tool heating was used to obtain an equilibrium temperature during the batch forging process.

The section cuts of the geometry models used in the forging simulation with WA-DED (b) and L-PBF (b) dies are shown in Fig. 7.

<figure><img src="image_2028.png" /><figcaption>Fig. 6. The section cuts of the geometry models used in the forging simulation with WA-DED (b) and L-PBF (b) dies</figcaption></figure>

## Results and discussion

**Analysis of die thermomechanical loads.** To evaluate fatigue life, it is necessary to determine the temperature and loads to which the hot forging die is subjected. During the continuous forging process, the die does not cool to the initial temperature; therefore, the QForm tool cycling module was used to analyse the steady-state die temperature. The temperature distribution over the die impressions at the end of the forging is shown in Fig. 8.
<figure><img src="image_2037.png" /><figcaption>Fig. 7. The surface temperature distributions of the WA-DED (a) and L-PBF (b) produced dies</figcaption></figure>

Due to the different designs of the WA-DED and L-PBF dies, the temperature is slightly different and amounts to an average of 200°C. This temperature will be used to analyze mechanical and fatigue properties of WA-DED and L-PBF produced specimens.

The maximum effective stresses occur at the bottom point of the ram stroke at the final moment of forging and are shown in Fig. 9. The effective stress in the WA-DED die is approximately 950 MPa, and that in the L-PBF die is approximately 1100 MPa.

<figure><img src="image_2038.png" /><figcaption>Fig. 8. The effective stress surface distribution during forging with WA-DED (a) and L-PBF produced dies</figcaption></figure>

Analysis of the first principal stress distribution (Fig. 10) shows that these stresses are mostly tensile. Thus, the corners of the impression cavities are the critical areas.

<figure><img src="image_2039.png" /><figcaption>Fig. 9. 1st principal stress surface distribution during forging with WA-DED (a) and L-PBF produced dies</figcaption></figure>
<figure><img src="image_2029.png" /><figcaption>Fig. 7. The surface temperature distributions of the WA-DED (a) and L-PBF (b) produced dies</figcaption></figure>

Due to the different designs of the WA-DED and L-PBF dies, the temperature is slightly different and amounts to an average of 200°C. This temperature will be used to analyze mechanical and fatigue properties of WA-DED and L-PBF produced specimens.

The maximum effective stresses occur at the bottom point of the ram stroke at the final moment of forging and are shown in Fig. 9. The effective stress in the WA-DED die is approximately 950 MPa, and that in the L-PBF die is approximately 1100 MPa.

<figure><img src="image_2030.png" /><figcaption>Fig. 8. The effective stress surface distribution during forging with WA-DED (a) and L-PBF produced dies</figcaption></figure>

Analysis of the first principal stress distribution (Fig. 10) shows that these stresses are mostly tensile. Thus, the corners of the impression cavities are the critical areas.

<figure><img src="image_2031.png" /><figcaption>Fig. 9. 1st principal stress surface distribution during forging with WA-DED (a) and L-PBF produced dies</figcaption></figure>
A preliminary estimation of the tool fatigue life was carried out with H11 steel as the die material, the parameters of which were set according to the work of Velay et al. [24]. The number of cycles to failure predicted using H11 tool steel shown in Fig. 11 and equals approximately 400000 and 100000 for WA-DED and L-PBF dies respectively. Due to the lower effective stresses in the WA-DED die, the predicted number of cycles to failure is significantly higher than that of the L-PBF die. However, this prediction does not take into account the fatigue properties of AM-produced materials, which will be investigated further.

<figure><img src="image_2040.png" /><figcaption>Fig. 10. Cycles to failure using H11 tool steel for WA-DED (a) and L-PBF (b) produced dies</figcaption></figure>

**Tensile and fatigue testing.** Flow curves at RT and 200°C for the WA-DED- and L-PBF-produced specimens after the H900 heat treatment are shown in Fig. 12.

<figure><img src="image_2041.png" /><figcaption>Fig. 11. Flow curves at RT and 200°C for the WA-DED- and L-PBF-produced specimens</figcaption></figure>

Compared with the specimens made by L-PBF, the specimens made by WA-DED exhibit significantly lower ultimate tensile strength at room temperature, but they still fulfil the H900 destination requirements. The yield strength of the WA-DED made specimens no longer met the requirements of the H900 designation. At 200°C, neither the ultimate tensile strength nor yield strength of the samples made by WA-DED meet the requirements of the H900 designation. The specimens made by L-PBF meet the requirements of the standard H900 heat treatment conditions both at room temperature and at 200°C. The mechanical property comparison results of the WA-DED- and L-PBF-fabricated specimens are shown in Fig. 13.
A preliminary estimation of the tool fatigue life was carried out with H11 steel as the die material, the parameters of which were set according to the work of Velay et al. [24]. The number of cycles to failure predicted using H11 tool steel shown in Fig. 11 and equals approximately 400000 and 100000 for WA-DED and L-PBF dies respectively. Due to the lower effective stresses in the WA-DED die, the predicted number of cycles to failure is significantly higher than that of the L-PBF die. However, this prediction does not take into account the fatigue properties of AM-produced materials, which will be investigated further.

<figure><img src="image_2032.png" /><figcaption>Fig. 10. Cycles to failure using H11 tool steel for WA-DED (a) and L-PBF (b) produced dies</figcaption></figure>

**Tensile and fatigue testing.** Flow curves at RT and 200°C for the WA-DED- and L-PBF-produced specimens after the H900 heat treatment are shown in Fig. 12.

<figure><img src="image_2033.png" /><figcaption>Fig. 11. Flow curves at RT and 200°C for the WA-DED- and L-PBF-produced specimens</figcaption></figure>

Compared with the specimens made by L-PBF, the specimens made by WA-DED exhibit significantly lower ultimate tensile strength at room temperature, but they still fulfil the H900 destination requirements. The yield strength of the WA-DED made specimens no longer met the requirements of the H900 designation. At 200°C, neither the ultimate tensile strength nor yield strength of the samples made by WA-DED meet the requirements of the H900 designation. The specimens made by L-PBF meet the requirements of the standard H900 heat treatment conditions both at room temperature and at 200°C. The mechanical property comparison results of the WA-DED- and L-PBF-fabricated specimens are shown in Fig. 13.
<figure><img src="image_2042.png" /><figcaption>Fig. 12. Comparison of the ultimate tensile strength (a) and yield stress (b) at RT (blue columns) and 200°C (orange columns) for the WA-DED- and L-PBF-produced specimens</figcaption></figure>

At both room temperature and 200°C, the L-PBF specimens exhibited significantly higher fatigue properties than did the WA-DED specimens. Wöhler diagrams of the WA-DED and L-PBF produced specimens are shown in Fig. 14.

<figure><img src="image_2043.png" /><figcaption>Fig. 13. Wöhler diagrams of the WA-DED- and L-PBF-produced specimens</figcaption></figure>

**Die fatigue life analysis.** The parameters of the tool fatigue life model were determined using the least squares method and are presented in Table 3.

<table><caption>Table 5. The parameters of the tool fatigue life model</caption><thead><tr><td>Parameter</td><td>WA-DED</td><td>L-PBF</td></tr></thead><tbody><tr><td>σ'<sub>f</sub>, (MPa)</td><td>1628,191</td><td>1739,161</td></tr><tr><td>b</td><td>-0,09</td><td>-0,059</td></tr></tbody></table>

Fatigue analysis (Fig. 15) showed that the die made by WA-DED has a significantly lower fatigue strength than the die made by L-PBF. The number of cycles to failure of the WA-DED die is approximately 10000, whereby the predicted failure area includes all cavities as well as an edge on the workpiece bar where no fatigue failure was initially predicted. The L-PBF-fabricated die exhibited significantly higher fatigue strength than the WA-DED- produced die, with the area of predicted fatigue failure being localized at the expected location in the die impression cavity. The predicted number of cycles to failure for the WA-DED die is approximately 6000. Despite the higher effective stresses (Fig. 9) due to the lightweight construction of L-PBF dies, the lifetime is significantly longer and amounts to approximately 50000 cycles to failure. This can be explained by higher process-related material properties of L-PBF specimens, which are considerably higher those of WA-DED material even after the H900 heat treatment, what has been proven by static tensile tests as well as fatigue tests.
<figure><img src="image_2034.png" /><figcaption>Fig. 12. Comparison of the ultimate tensile strength (a) and yield stress (b) at RT (blue columns) and 200°C (orange columns) for the WA-DED- and L-PBF-produced specimens</figcaption></figure>

At both room temperature and 200°C, the L-PBF specimens exhibited significantly higher fatigue properties than did the WA-DED specimens. Wöhler diagrams of the WA-DED and L-PBF produced specimens are shown in Fig. 14.

<figure><img src="image_2035.png" /><figcaption>Fig. 13. Wöhler diagrams of the WA-DED- and L-PBF-produced specimens</figcaption></figure>

**Die fatigue life analysis.** The parameters of the tool fatigue life model were determined using the least squares method and are presented in Table 3.

<table><caption>Table 5. The parameters of the tool fatigue life model</caption><thead><tr><td>Parameter</td><td>WA-DED</td><td>L-PBF</td></tr></thead><tbody><tr><td>σ'<sub>f</sub>, (MPa)</td><td>1628,191</td><td>1739,161</td></tr><tr><td>b</td><td>-0,09</td><td>-0,059</td></tr></tbody></table>

Fatigue analysis (Fig. 15) showed that the die made by WA-DED has a significantly lower fatigue strength than the die made by L-PBF. The number of cycles to failure of the WA-DED die is approximately 10000, whereby the predicted failure area includes all cavities as well as an edge on the workpiece bar where no fatigue failure was initially predicted. The L-PBF-fabricated die exhibited significantly higher fatigue strength than the WA-DED- produced die, with the area of predicted fatigue failure being localized at the expected location in the die impression cavity. The predicted number of cycles to failure for the WA-DED die is approximately 6000. Despite the higher effective stresses (Fig. 9) due to the lightweight construction of L-PBF dies, the lifetime is significantly longer and amounts to approximately 50000 cycles to failure. This can be explained by higher process-related material properties of L-PBF specimens, which are considerably higher those of WA-DED material even after the H900 heat treatment, what has been proven by static tensile tests as well as fatigue tests.
<figure><img src="image_2044.png" /><figcaption>Fig. 14. Predicted cycles to failure of the WA-DED (a) and L-PBF (b) produced dies</figcaption></figure>

## Conclusions

1. 17-4PH steel is suitable for the production of hot forming dies by AM technologies.

2. L-PBF-produced tools exhibited higher mechanical and fatigue properties than WA-DED-produced tools.

3. Even the resource-efficient production of L-PBF lightweight forming dies with 56% less weight (compared to WA-DED solid dies) leads to a significantly longer lifetime.

4. A numerical model for 17-4PH fatigue life of WA-DED and L-PBF-produced forming dies was developed and implemented in QForm UK software.

5. The fatigue life of AM-produced dies has been predicted and evaluated for small batch hot forging.

## References

[1] I. Gibson, D. Rosen, B. Stucker, and M. Khorasani, *Additive Manufacturing Technologies*. Cham: Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-56127-7

[2] A. Gisario, M. Kazarian, F. Martina, and M. Mehrpouya, "Metal additive manufacturing in the commercial aviation industry: A review," *Journal of Manufacturing Systems*, 53 (2019) 124–149. https://doi.org/10.1016/j.jmsy.2019.08.005

[3] C. Sun, Y. Wang, M. D. McMurtrey, N. D. Jerred, F. Liou, and J. Li, "Additive manufacturing for energy: A review," *Applied Energy*, 282 (2021) 116041. https://doi.org/10.1016/j.apenergy.2020.116041

[4] S. G. Scholz, R. J. Howlett, and R. Setchi, Eds., *Sustainable Design and Manufacturing: Proceedings of the 8th International Conference on Sustainable Design and Manufacturing (KES-SDM 2021)*, in *Smart Innovation, Systems and Technologies*, vol. 262. Singapore: Springer Singapore, 2022. https://doi.org/10.1007/978-981-16-6128-0

[5] A. J. Sheoran, H. Kumar, P. K. Arora, and G. Moona, "Bio-Medical applications of Additive Manufacturing: A Review," *Procedia Manufacturing*, 51 (2020) 663–670. https://doi.org/10.1016/j.promfg.2020.10.093

[6] G. Liu *et al.*, "Additive manufacturing of structural materials," *Materials Science and Engineering: R: Reports*, 145 (2021) 100596. https://doi.org/10.1016/j.mser.2020.100596

[7] M. Gao, L. Li, Q. Wang, Z. Ma, X. Li, and Z. Liu, "Integration of Additive Manufacturing in Casting: Advances, Challenges, and Prospects," *Int. J. of Precis. Eng. and Manuf.-Green Tech.*, 9 (2022) 305–322. https://doi.org/10.1007/s40684-021-00323-w

[8] D. Chantzis *et al.*, "Review on additive manufacturing of tooling for hot stamping," *Int J Adv Manuf Technol*, 109 (2020) 87-107. https://doi.org/10.1007/s00170-020-05622-1
<figure><img src="image_2036.png" /><figcaption>Fig. 14. Predicted cycles to failure of the WA-DED (a) and L-PBF (b) produced dies</figcaption></figure>

## Conclusions

1. 17-4PH steel is suitable for the production of hot forming dies by AM technologies.

2. L-PBF-produced tools exhibited higher mechanical and fatigue properties than WA-DED-produced tools.

3. Even the resource-efficient production of L-PBF lightweight forming dies with 56% less weight (compared to WA-DED solid dies) leads to a significantly longer lifetime.

4. A numerical model for 17-4PH fatigue life of WA-DED and L-PBF-produced forming dies was developed and implemented in QForm UK software.

5. The fatigue life of AM-produced dies has been predicted and evaluated for small batch hot forging.

## References

[1] I. Gibson, D. Rosen, B. Stucker, and M. Khorasani, *Additive Manufacturing Technologies*. Cham: Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-56127-7

[2] A. Gisario, M. Kazarian, F. Martina, and M. Mehrpouya, "Metal additive manufacturing in the commercial aviation industry: A review," *Journal of Manufacturing Systems*, 53 (2019) 124-149. https://doi.org/10.1016/j.jmsy.2019.08.005

[3] C. Sun, Y. Wang, M. D. McMurtrey, N. D. Jerred, F. Liou, and J. Li, "Additive manufacturing for energy: A review," *Applied Energy*, 282 (2021) 116041. https://doi.org/10.1016/j.apenergy.2020.116041

[4] S. G. Scholz, R. J. Howlett, and R. Setchi, Eds., *Sustainable Design and Manufacturing: Proceedings of the 8th International Conference on Sustainable Design and Manufacturing (KES-SDM 2021)*, in *Smart Innovation, Systems and Technologies*, vol. 262. Singapore: Springer Singapore, 2022. https://doi.org/10.1007/978-981-16-6128-0

[5] A. J. Sheoran, H. Kumar, P. K. Arora, and G. Moona, "Bio-Medical applications of Additive Manufacturing: A Review," *Procedia Manufacturing*, 51 (2020) 663-670. https://doi.org/10.1016/j.promfg.2020.10.093

[6] G. Liu et al., "Additive manufacturing of structural materials," *Materials Science and Engineering: R: Reports*, 145 (2021) 100596. https://doi.org/10.1016/j.mser.2020.100596

[7] M. Gao, L. Li, Q. Wang, Z. Ma, X. Li, and Z. Liu, "Integration of Additive Manufacturing in Casting: Advances, Challenges, and Prospects," *Int. J. of Precis. Eng. and Manuf.-Green Tech.*, 9 (2022) 305-322. https://doi.org/10.1007/s40684-021-00323-w

[8] D. Chantzis et al., "Review on additive manufacturing of tooling for hot stamping," *Int J Adv Manuf Technol*, 109 (2020) 87-107. https://doi.org/10.1007/s00170-020-05622-1
[9] A. Huskic, B. Behrens, J. Giedenbacher, and A. Huskic, "Standzeituntersuchungen generativ hergestellter Schmiedewerkzeuge," *Schmiede J*, 92013 2013) 66-70.

[10] D. Junker, O. Hentschel, R. Schramme, M. Schmidt, and M. Merklein, "Performance of hot forging tools built by laser metal deposition of hot work tool steel X37CrMoV5-1," in *Proceedings of the laser in manufacturing conference*, 2017.

[11] R. Stache, "Untersuchungen zur Herstellung von Werkzeugelementen für das Presshärten mittels Laser Beam Melting," PhD, Technische Universität Chemnitz, Chemnitz, 2020.

[12] "DIN EN ISO/ASTM 52900:2022-03 Additive Fertigung - Grundlagen – Terminologie."

[13] S. L. Sing and W. Y. Yeong, "Laser powder bed fusion for metal additive manufacturing: perspectives on recent developments," *Virtual and Physical Prototyping*, 15 (2020) 359-370.
https://doi.org/10.1080/17452759.2020.1779999

[14] Ayed, A. et al. (2020). Effects of WAAM Process Parameters on Metallurgical and Mechanical Properties of Ti-6Al-4V Deposits. In: Chaari, F., et al. Advances in Materials, Mechanics and Manufacturing. Lecture Notes in Mechanical Engineering. Springer, Cham.
https://doi.org/10.1007/978-3-030-24247-3_4

[15] M. Benakis, D. Costanzo, and A. Patran, "Current mode effects on weld bead geometry and heat affected zone in pulsed wire arc additive manufacturing of Ti-6-4 and Inconel 718," *Journal of Manufacturing Processes*, 60 (2020) 61-74.
https://doi.org/10.1016/j.jmapro.2020.10.018

[16] A. Alimov, A. Sviridov, B. Sydow, F. Jensch, and S. Härtel, "Additive manufacturing of hot-forming dies using laser powder bed fusion and wire arc direct energy deposition technologies," *Metals*, 13 (2023) 1842.
https://doi.org/10.3390/met13111842

[17] A. A. Emamverdian, Y. Sun, C. Cao, C. Pruncu, and Y. Wang, "Current failure mechanisms and treatment methods of hot forging tools (dies) - a review," *Engineering Failure Analysis*, vol. 129 (2021) 105678.
https://doi.org/10.1016/j.engfailanal.2021.105678

[18] Z. Gronostajski, M. Kaszuba, S. Polak, M. Zwierzchowski, A. Niechajowicz, and M. Hawryluk, "The failure mechanisms of hot forging dies," *Materials Science and Engineering: A*, 657 (2016) 147-160.
https://doi.org/10.1016/j.msea.2016.01.030

[19] J. Radhakrishnan, P. Kumar, S. S. Gan, A. Bryl, J. McKinnell, and U. Ramamurty, "Fatigue resistance of the binder jet printed 17-4 precipitation hardened martensitic stainless steel," *Materials Science and Engineering: A*, 865 (2023) 144451.
https://doi.org/10.1016/j.msea.2022.144451

[20] F. Concli, L. Fraccaroli, F. Nalli, and L. Cortese, "High and low-cycle-fatigue properties of 17-4 PH manufactured via selective laser melting in as-built, machined and hipped conditions," *Prog Addit Manuf*, 7 (2022) 99-109.
https://doi.org/10.1007/s40964-021-00217-y

[21] L. Carneiro, B. Jalalahmadi, A. Ashtekar, and Y. Jiang, "Cyclic deformation and fatigue behavior of additively manufactured 17-4 PH stainless steel," *International Journal of Fatigue*, 123 (2019) 22-30.
https://doi.org/10.1016/j.ijfatigue.2019.02.006

[22] S. Suresh, *Fatigue of materials*, Cambridge University Press, 1998.
https://doi.org/10.1017/CBO9780511806575

[23] A. V. Vlasov, "Thermomechanical fatigue of dies for hot stamping," *Steel Transl.*, 46 (2016) 356-360.
https://doi.org/10.3103/S0967091216050156

[24] V. Velay, G. Bernhart, D. Delagnes, and L. Penazzi, "A continuum damage model applied to high-temperature fatigue lifetime prediction of a martensitic tool steel," *Fatigue Fract Eng Mat Struct*, 28 (2005) 1009-1023.
https://doi.org/10.1111/j.1460-2695.2005.00939.x
[9] A. Huskic, B. Behrens, J. Giedenbacher, and A. Huskic, "Standzeituntersuchungen generativ hergestellter Schmiedewerkzeuge," *Schmiede J*, 92013 2013) 66-70.

[10] D. Junker, O. Hentschel, R. Schramme, M. Schmidt, and M. Merklein, "Performance of hot forging tools built by laser metal deposition of hot work tool steel X37CrMoV5-1," in *Proceedings of the laser in manufacturing conference*, 2017.

[11] R. Stache, "Untersuchungen zur Herstellung von Werkzeugelementen für das Presshärten mittels Laser Beam Melting," PhD, Technische Universität Chemnitz, Chemnitz, 2020.

[12] "DIN EN ISO/ASTM 52900:2022-03 Additive Fertigung - Grundlagen – Terminologie."

[13] S. L. Sing and W. Y. Yeong, "Laser powder bed fusion for metal additive manufacturing: perspectives on recent developments," *Virtual and Physical Prototyping*, 15 (2020) 359-370.
https://doi.org/10.1080/17452759.2020.1779999

[14] Ayed, A. et al. (2020). Effects of WAAM Process Parameters on Metallurgical and Mechanical Properties of Ti-6Al-4V Deposits. In: Chaari, F., et al. Advances in Materials, Mechanics and Manufacturing. Lecture Notes in Mechanical Engineering. Springer, Cham.
https://doi.org/10.1007/978-3-030-24247-3_4

[15] M. Benakis, D. Costanzo, and A. Patran, "Current mode effects on weld bead geometry and heat affected zone in pulsed wire arc additive manufacturing of Ti-6-4 and Inconel 718," *Journal of Manufacturing Processes*, 60 (2020) 61-74.
https://doi.org/10.1016/j.jmapro.2020.10.018

[16] A. Alimov, A. Sviridov, B. Sydow, F. Jensch, and S. Härtel, "Additive manufacturing of hot-forming dies using laser powder bed fusion and wire arc direct energy deposition technologies," *Metals*, 13 (2023) 1842.
https://doi.org/10.3390/met13111842

[17] A. A. Emamverdian, Y. Sun, C. Cao, C. Pruncu, and Y. Wang, "Current failure mechanisms and treatment methods of hot forging tools (dies) - a review," *Engineering Failure Analysis*, vol. 129 (2021) 105678.
https://doi.org/10.1016/j.engfailanal.2021.105678

[18] Z. Gronostajski, M. Kaszuba, S. Polak, M. Zwierzchowski, A. Niechajowicz, and M. Hawryluk, "The failure mechanisms of hot forging dies," *Materials Science and Engineering: A*, 657 (2016) 147-160.
https://doi.org/10.1016/j.msea.2016.01.030

[19] J. Radhakrishnan, P. Kumar, S. S. Gan, A. Bryl, J. McKinnell, and U. Ramamurty, "Fatigue resistance of the binder jet printed 17-4 precipitation hardened martensitic stainless steel," *Materials Science and Engineering: A*, 865 (2023) 144451.
https://doi.org/10.1016/j.msea.2022.144451

[20] F. Concli, L. Fraccaroli, F. Nalli, and L. Cortese, "High and low-cycle-fatigue properties of 17-4 PH manufactured via selective laser melting in as-built, machined and hipped conditions," *Prog Addit Manuf*, 7 (2022) 99-109.
https://doi.org/10.1007/s40964-021-00217-y

[21] L. Carneiro, B. Jalalahmadi, A. Ashtekar, and Y. Jiang, "Cyclic deformation and fatigue behavior of additively manufactured 17-4 PH stainless steel," *International Journal of Fatigue*, 123 (2019) 22-30.
https://doi.org/10.1016/j.ijfatigue.2019.02.006

[22] S. Suresh, *Fatigue of materials*, Cambridge University Press, 1998.
https://doi.org/10.1017/CBO9780511806575

[23] A. V. Vlasov, "Thermomechanical fatigue of dies for hot stamping," *Steel Transl.*, 46 (2016) 356-360.
https://doi.org/10.3103/S0967091216050156

[24] V. Velay, G. Bernhart, D. Delagnes, and L. Penazzi, "A continuum damage model applied to high-temperature fatigue lifetime prediction of a martensitic tool steel," *Fatigue Fract Eng Mat Struct*, 28 (2005) 1009-1023.
https://doi.org/10.1111/j.1460-2695.2005.00939.x
# The prospects of implementation of artificial intelligence for modelling of microstructural parameters in metal forming processes

TRETYAKOV Denis¹,³,ᵃ, BYLYA Olga²,ᵇ, SHITIKOV Andrei⁴,ᶜ, GARTVIG Artur¹,ᵈ, STEBUNOV Sergey¹,ᵉ, BIBA Nikolay⁴,ᶜ*

¹QForm Group FZ LLC, Fujairah - Creative Tower P.O. Box 4422 Fujairah, UAE

²National Manufacturing Research Centre, University of Strathclyde, 3 Netherton Sq, Paisley, Renfrew PA3 2EF

³Moscow State University, Lomonosovskiy Prospekt, Moscow, Russia

⁴MICAS Simulation Ltd, 107 Oxford Road, Oxford, OX4 2ER, UK

ᵃtretyakov.denis.a@gmail.com, ᵇolga.bylya@strath.ac.uk, ᶜa.a.sh.ggml@gmail.com,

ᵈgartvig@qform3d.com, ᵇserg@qform3d.com, ᶠnick@qform3d.com

**Keywords:** Metals, Forging, Inconel 718, Microstructure Evolution, Recrystallisation, FEM, Deep Learning, AI Models

**Abstract.** The primary trend in modern metal forming can be characterised by the increase in the complexity of the technological processes and higher demand for the quality of the products. This naturally raises the requirements for the quality of modelling prediction of various aspects of metal forming process, such as tool wear, metal flow, fracture and defects formation, microstructure evolution and mechanical properties. However, various independent benchmarking studies [1] have shown that modelling predictions can be wrong even for well-calibrated models, and all the efforts with more detailed and metrologically better experiments didn't lead to any significant leap in the prediction quality. As an attempt to implement some alternative approach, this paper investigates the applicability of an Artificial Intelligence (AI) approach, in particular Deep Learning models. The example of a recurrent neural network model predicting recrystallisation during hot forging of Inconel 718 is presented. The model considers the entire thermo-mechanical history at every point and is trained and blind-tested using actual experimental data.

## Introduction

The author's experience shows that there exists a certain scepticism about AI-based models among people involved in microstructural modelling in the industrial and scientific fields. It is often said that analytical models describing various phenomena involved in metal forming reflect the “physics” of the process and can, therefore, be trusted, while AI models are a sort of “black box” somehow trained to guess few correct answers and can’t be reliable. Both parts of this assumption are not true. Almost all classical phenomenological models are the curve-fitting of experimental data accompanied by logical interpretations of the observed trends and dependencies. This nature is the origin of the major limitations of such models, i.e. their mathematical structure and long, costly and complicated calibration, i.e. the process of finding actual model parameters [2].

Taking recrystallisation as an example and assuming that it depends on only three major process parameters: strain, strain-rate and temperature, the relation for recrystallisation should be written as a differential to reflect history dependence, i.e. $dR/dt=f(T, \varepsilon, \dot{\varepsilon})$. For calibration of this model, we can change only one parameter at a time; others should be fixed, though it can be technically difficult. Thus, to get data for just three temperatures, strains and strain-rates, a minimum of 9 experiments are needed and a total of 36 for four strain values [3-5]. As a result, identification of the model parameters, in reality, is always based on a very limited number of points (which limits the mathematical complexity of the model), with limited strain value (generally not more than 0.7),

<figure><img src="image_2045.png" /></figure>
# The prospects of implementation of artificial intelligence for modelling of microstructural parameters in metal forming processes

TRETYAKOV Denis¹,³,a, BYLYA Olga²,b, SHITIKOV Andrei⁴,c,
GARTVIG Artur¹,d, STEBUNOV Sergey¹,e, BIBA Nikolay⁴,f*

¹QForm Group FZ LLC, Fujairah - Creative Tower P.O. Box 4422 Fujairah, UAE

²National Manufacturing Research Centre, University of Strathclyde, 3 Netherton Sq, Paisley, Renfrew PA3 2EF

³Moscow State University, Lomonosovskiy Prospekt, Moscow, Russia

⁴MICAS Simulation Ltd, 107 Oxford Road, Oxford, OX4 2ER, UK

ªtretyakov.denis.a@gmail.com, ᵇolga.bylya@strath.ac.uk, ᶜa.a.sh.ggml@gmail.com,

ᵈgartvig@qform3d.com, ᵈserg@qform3d.com, ᶠnick@qform3d.com

**Keywords:** Metals, Forging, Inconel 718, Microstructure Evolution, Recrystallisation, FEM, Deep Learning, AI Models

**Abstract.** The primary trend in modern metal forming can be characterised by the increase in the complexity of the technological processes and higher demand for the quality of the products. This naturally raises the requirements for the quality of modelling prediction of various aspects of metal forming process, such as tool wear, metal flow, fracture and defects formation, microstructure evolution and mechanical properties. However, various independent benchmarking studies [1] have shown that modelling predictions can be wrong even for well-calibrated models, and all the efforts with more detailed and metrologically better experiments didn't lead to any significant leap in the prediction quality. As an attempt to implement some alternative approach, this paper investigates the applicability of an Artificial Intelligence (AI) approach, in particular Deep Learning models. The example of a recurrent neural network model predicting recrystallisation during hot forging of Inconel 718 is presented. The model considers the entire thermo-mechanical history at every point and is trained and blind-tested using actual experimental data.

**Introduction**

The author's experience shows that there exists a certain scepticism about AI-based models among people involved in microstructural modelling in the industrial and scientific fields. It is often said that analytical models describing various phenomena involved in metal forming reflect the “physics” of the process and can, therefore, be trusted, while AI models are a sort of “black box” somehow trained to guess few correct answers and can’t be reliable. Both parts of this assumption are not true. Almost all classical phenomenological models are the curve-fitting of experimental data accompanied by logical interpretations of the observed trends and dependencies. This nature is the origin of the major limitations of such models, i.e. their mathematical structure and long, costly and complicated calibration, i.e. the process of finding actual model parameters [2].

Taking recrystallisation as an example and assuming that it depends on only three major process parameters: strain, strain-rate and temperature, the relation for recrystallisation should be written as a differential to reflect history dependence, i.e. $dR/dt=f(T, \varepsilon, \dot{\varepsilon})$. For calibration of this model, we can change only one parameter at a time; others should be fixed, though it can be technically difficult. Thus, to get data for just three temperatures, strains and strain-rates, a minimum of 9 experiments are needed and a total of 36 for four strain values [3-5]. As a result, identification of the model parameters, in reality, is always based on a very limited number of points (which limits the mathematical complexity of the model), with limited strain value (generally not more than 0.7),

<figure><img src="image_2037.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.
and an assumption that the reactions on individual parameters are additive and transients in parameter changes are negligible. This forces the model to use plenty of extrapolations when utilised in real industrial processes and often causes poor results.

Returning to the second part of the first statement regarding AI models. As it will be shown in this paper, the principle of their operation is not much different from phenomenological models. The recurrent cell used in the modelling of recrystallisation is analogous to the above-mentioned R, with a few additional state variables dR/dt=f(T, ε, έ, R, h1, h2, h3, ...), having a different numerical representation and a bigger number of model parameters. This substitute is based on the universal approximation theorem established by Cybenko [6], which states that a neural network with one hidden layer, a finite number of neurons and a sigmoid activation function can approximate any continuous n-dimensional function with any set precision.

The second peculiarity of the AI models is the technique of their calibration or “training”. Traditional analytical models could not use data directly from industrial forging processes because of complicated thermo-mechanical histories, as manual curve-fitting needs to change one parameter at a time. In contrast, machine learning utilises finding model parameters based on recognising patterns from data. In this case, the complexity of the data is compensated for by its amount.

Summarising everything mentioned above, AI-based models have quite a similar phenomenological nature but with a more flexible mathematical structure and advanced calibration techniques. Their implementation in process modelling can simplify the process of tuning models for metal forging industries. Industrial forging data can be used directly instead of “special” uniaxial lab tests. This will also improve the prediction quality because the model will work in the regime of interpolation. An example of one such model is presented in this paper.

**Methodology**

**Forging trials.** The quality of model calibration is always extremely dependent on the quality of the experimental data. For analytical models, it is critical to keep only a single process parameter changing and others fixed, limiting specimen geometry and loading parameters. For AI models, these restrictions are unnecessary, and forged parts of different geometry can be used. In this investigation, the relatively simple geometry of Double Truncated Cones (DTC) was used for model calibration, and more complicated profiled disk geometry for blind validation (Fig.1a). Forging trials were performed on an industrial 21 MN screw press (Fig.1b) at the temperatures 970°C and 1080°C. During the experiment, the ram velocity and energy were recorded on the press and temperature distribution was measured with three thermocouples embedded in the specimen (Fig.1c). All this data was used for the validation of FE simulations of the forging processes performed in the commercial metal forming simulation software QForm UK®. After forging, specimens were quenched in water, cut and analysed under optical and electron microscopes. The obtained distribution of the percentage of recrystallisation over the specimen's cross-section is shown in Fig.1d. More information on these experiments and experimental results can be found in the following papers [7, 8].
and an assumption that the reactions on individual parameters are additive and transients in parameter changes are negligible. This forces the model to use plenty of extrapolations when utilised in real industrial processes and often causes poor results.

Returning to the second part of the first statement regarding AI models. As it will be shown in this paper, the principle of their operation is not much different from phenomenological models. The recurrent cell used in the modelling of recrystallisation is analogous to the above-mentioned R, with a few additional state variables dR/dt=f(T, ε, έ, R, h1, h2, h3, ...), having a different numerical representation and a bigger number of model parameters. This substitute is based on the universal approximation theorem established by Cybenko [6], which states that a neural network with one hidden layer, a finite number of neurons and a sigmoid activation function can approximate any continuous n-dimensional function with any set precision.

The second peculiarity of the AI models is the technique of their calibration or “training”. Traditional analytical models could not use data directly from industrial forging processes because of complicated thermo-mechanical histories, as manual curve-fitting needs to change one parameter at a time. In contrast, machine learning utilises finding model parameters based on recognising patterns from data. In this case, the complexity of the data is compensated for by its amount.

Summarising everything mentioned above, AI-based models have quite a similar phenomenological nature but with a more flexible mathematical structure and advanced calibration techniques. Their implementation in process modelling can simplify the process of tuning models for metal forging industries. Industrial forging data can be used directly instead of “special” uniaxial lab tests. This will also improve the prediction quality because the model will work in the regime of interpolation. An example of one such model is presented in this paper.

**Methodology**

**Forging trials.** The quality of model calibration is always extremely dependent on the quality of the experimental data. For analytical models, it is critical to keep only a single process parameter changing and others fixed, limiting specimen geometry and loading parameters. For AI models, these restrictions are unnecessary, and forged parts of different geometry can be used. In this investigation, the relatively simple geometry of Double Truncated Cones (DTC) was used for model calibration, and more complicated profiled disk geometry for blind validation (Fig.1a). Forging trials were performed on an industrial 21 MN screw press (Fig.1b) at the temperatures 970°C and 1080°C. During the experiment, the ram velocity and energy were recorded on the press and temperature distribution was measured with three thermocouples embedded in the specimen (Fig.1c). All this data was used for the validation of FE simulations of the forging processes performed in the commercial metal forming simulation software QForm UK®. After forging, specimens were quenched in water, cut and analysed under optical and electron microscopes. The obtained distribution of the percentage of recrystallisation over the specimen's cross-section is shown in Fig.1d. More information on these experiments and experimental results can be found in the following papers [7, 8].
<figure><img src="image_2046.png" /><figcaption>Figure 1. Forging trials: a) geometry of the parts; b) screw press; c) location of thermocouples; d) experimental distribution of recrystallisation.</figcaption></figure>

**Extracting Data from FE models.** It's crucial that the FE models of these processes are calibrated correctly before any further work can be done, as the data acquired from these models will later be used for training the microstructural models in question. This is why data such as energy and movement of the press as well as specimen temperature over time at multiple points is absolutely necessary to validate the model [7]. More detailed information on the constitutive model and FE simulations used in this paper as well as the methods used to calibrate and validate them can be found in the following papers [7, 8]. After the model is validated, a large distribution of points is chosen over the specimen. Using modern FE modelling software such as QForm UK® we can extract full thermo-mechanical histories over the course of the simulation for each of the chosen points [T(t), ε(t), ɛ̇(t), ...].

Unlike in traditional lab experiments performed on small specimens in a presumed homogeneous state where these values are constant over the whole specimen and in time, we used more complicated industrial forgings where these histories are vastly different at different points in the specimen and are far from constant in time (Fig 2.), this can also be seen from analysing the drastically different temperature profiles from the three thermocouples [1]. This fact, combined with the new opportunity to extract complete thermo-mechanical histories from different points of a specimen, allows us to gather an unprecedented amount of data from just a few industrial forgings. Having analysed the cross-section of our specimens at the chosen points, we can now map the thermo-mechanical histories to their corresponding resulting microstructural parameters such as percentage of recrystallisation, grain size and others. For simplicity, only the percentage of recrystallisation was considered in this study. An additional bonus of this large body of data is
<figure><img src="image_2038.png" /><figcaption>Figure 1. Forging trials: a) geometry of the parts; b) screw press; c) location of thermocouples; d) experimental distribution of recrystallisation.</figcaption></figure>

**Extracting Data from FE models.** It's crucial that the FE models of these processes are calibrated correctly before any further work can be done, as the data acquired from these models will later be used for training the microstructural models in question. This is why data such as energy and movement of the press as well as specimen temperature over time at multiple points is absolutely necessary to validate the model [7]. More detailed information on the constitutive model and FE simulations used in this paper as well as the methods used to calibrate and validate them can be found in the following papers [7, 8]. After the model is validated, a large distribution of points is chosen over the specimen. Using modern FE modelling software such as QForm UK® we can extract full thermo-mechanical histories over the course of the simulation for each of the chosen points [T(t), ε(t), ɛ̇(t), ...].

Unlike in traditional lab experiments performed on small specimens in a presumed homogeneous state where these values are constant over the whole specimen and in time, we used more complicated industrial forgings where these histories are vastly different at different points in the specimen and are far from constant in time (Fig 2.), this can also be seen from analysing the drastically different temperature profiles from the three thermocouples [1]. This fact, combined with the new opportunity to extract complete thermo-mechanical histories from different points of a specimen, allows us to gather an unprecedented amount of data from just a few industrial forgings. Having analysed the cross-section of our specimens at the chosen points, we can now map the thermo-mechanical histories to their corresponding resulting microstructural parameters such as percentage of recrystallisation, grain size and others. For simplicity, only the percentage of recrystallisation was considered in this study. An additional bonus of this large body of data is
that it lends itself very well to a machine learning approach that, despite its benefits mentioned earlier, requires a lot of training data to be implemented successfully.

<figure><img src="image_2047.png" /><figcaption>Figure 2. Example of numerical data: thermo-mechanical histories at two different points of the specimen: a) locations of two different points in two configurations of the deformed part; b) temperature over time at chosen points; c) strain over time at chosen points; d) strain-rate over time at chosen points.</figcaption></figure>

**AI model.** The model consists of a recurrent neural network with a long short-term memory (LSTM) cell [9] with a hidden state vector of size 10 as the recurrent cell. The LSTM cell was modified so that one of the values of the hidden state vector would start at 0, wouldn't decline with each time step and wouldn't exceed 1; this value was used as the output of the model.

The model includes a total of 840 trainable parameters. As for the training data, exactly two forgings were used to calibrate the model: DTC at 970°C and 1080°C. For both of these forgings four FE simulations with different time step lengths were made in QForm UK®. This was necessary to ensure consistent behaviour of the trained model when applied to the same FE simulation with differing time steps, as this isn't inherently part of the models' architecture. From each of these eight simulations, around 1500 points in a uniform grid in finish configuration were chosen, as shown in (Fig 2a) for a total of 12000 training points. Thermo-mechanical histories used for training the model from each of these points consisted of the following 10 values at each time step that form the vector x:

$$x = [dt, T, \varepsilon, \dot{\varepsilon}, \sigma, d\varepsilon, d\sigma, dW, W, ZH]$$

$$dt - \text{Time Step [s]}$$

$$T - \text{Temperature [°C]}$$

$$\varepsilon - \text{Accumulated Plastic Strain}$$

$$\dot{\varepsilon} - \text{Strain-Rate [1/s]}$$

$$\sigma - \text{Effective Stress [MPa]}$$

$$d\varepsilon = \varepsilon_i - \varepsilon_{i-1}$$

$$d\sigma = \sigma_i - \sigma_{i-1}$$
that it lends itself very well to a machine learning approach that, despite its benefits mentioned earlier, requires a lot of training data to be implemented successfully.

<figure><img src="image_2039.png" /><figcaption>Figure 2. Example of numerical data: thermo-mechanical histories at two different points of the specimen: a) locations of two different points in two configurations of the deformed part; b) temperature over time at chosen points; c) strain over time at chosen points; d) strain-rate over time at chosen points.</figcaption></figure>

**AI model.** The model consists of a recurrent neural network with a long short-term memory (LSTM) cell [9] with a hidden state vector of size 10 as the recurrent cell. The LSTM cell was modified so that one of the values of the hidden state vector would start at 0, wouldn't decline with each time step and wouldn't exceed 1; this value was used as the output of the model.

The model includes a total of 840 trainable parameters. As for the training data, exactly two forgings were used to calibrate the model: DTC at 970°C and 1080°C. For both of these forgings four FE simulations with different time step lengths were made in QForm UK®. This was necessary to ensure consistent behaviour of the trained model when applied to the same FE simulation with differing time steps, as this isn't inherently part of the models' architecture. From each of these eight simulations, around 1500 points in a uniform grid in finish configuration were chosen, as shown in (Fig 2a) for a total of 12000 training points. Thermo-mechanical histories used for training the model from each of these points consisted of the following 10 values at each time step that form the vector x:

$$x = [dt, T, \varepsilon, \dot{\varepsilon}, \sigma, d\varepsilon, d\sigma, dW, W, ZH]$$

$$dt - \text{Time Step [s]}$$

$$T - \text{Temperature [°C]}$$

$$\varepsilon - \text{Accumulated Plastic Strain}$$

$$\dot{\varepsilon} - \text{Strain-Rate [1/s]}$$

$$\sigma - \text{Effective Stress [MPa]}$$

$$d\varepsilon = \varepsilon_i - \varepsilon_{i-1}$$

$$d\sigma = \sigma_i - \sigma_{i-1}$$
$$ 
\begin{aligned}
dW &= \sigma \cdot d\epsilon \\
W &= W_{i-1} + dW \\
ZH &= \dot{\epsilon} \cdot \exp\left(\frac{Q}{R \cdot T_{abs}}\right)
\end{aligned}
 $$

where Q - Activation Energy [J·mol⁻¹], R - Gas Constant [J·K⁻¹·mol⁻¹], Tₐbs - Temperature [K], i - index of time step.

The choice of process parameters included in x was based on preliminary data analysis and previous microstructural modelling experience.

The training data was normalised and split into 80 batches. The model was trained for 500 epochs with an ADAM optimiser, minimising the mean square error between experimental recrystallisation percentage values and the model's predictions at the chosen points.

A benefit of using such architecture is that the trained recurrent cell can be implemented in metal forming simulation software and run during FE simulations using data from the simulation at each step as input and feeding microstructural data back into the material model if needed. After the model was trained, it was implemented as a Lua user subroutine in QForm UK® for further testing and analysis. The model was blind tested on simulations of the profiled disk forgings.

<figure><img src="image_2048.png" /><figcaption>Figure 3. The results of the blind test of the proposed AI-based model: a) disk forged at 970°C; b) disk forged at 1080°C.</figcaption></figure>

## Results

A comparison of results from the FE simulations of the forging processes with the AI-based model used as a user subroutine against experimental data is shown in Fig. 3. Analysing obtained results, a few important points should be emphasised:

1. Despite the fact that the disks trials were used as a completely blind test for the model, the acquired results are physically consistent, and the field of the percentage of recrystallisation is smooth and continuous over the specimen and over time. Traditional analytical models are mathematical formulae describing the relations between different process parameters. They are generally expressed as smooth, continuous functions differentiable with respect to time and coordinates. On the other hand, such smoothness and continuity are not a generic property of an AI-based model's architecture. Thus, the continuity observed in the blind tests shows that the model succeeded in identifying some process features from the provided data during training.
$$ 
\begin{aligned}
dW &= \sigma \cdot d\epsilon \\
W &= W_{i-1} + dW \\
ZH &= \dot{\epsilon} \cdot \exp\left(\frac{Q}{R \cdot T_{abs}}\right)
\end{aligned}
 $$

where Q - Activation Energy [J·mol⁻¹], R - Gas Constant [J·K⁻¹·mol⁻¹], Tₐbs - Temperature [K], i - index of time step.

The choice of process parameters included in x was based on preliminary data analysis and previous microstructural modelling experience.

The training data was normalised and split into 80 batches. The model was trained for 500 epochs with an ADAM optimiser, minimising the mean square error between experimental recrystallisation percentage values and the model's predictions at the chosen points.

A benefit of using such architecture is that the trained recurrent cell can be implemented in metal forming simulation software and run during FE simulations using data from the simulation at each step as input and feeding microstructural data back into the material model if needed. After the model was trained, it was implemented as a Lua user subroutine in QForm UK® for further testing and analysis. The model was blind tested on simulations of the profiled disk forgings.

<figure><img src="image_2040.png" /><figcaption>Figure 3. The results of the blind test of the proposed AI-based model: a) disk forged at 970°C; b) disk forged at 1080°C.</figcaption></figure>

## Results

A comparison of results from the FE simulations of the forging processes with the AI-based model used as a user subroutine against experimental data is shown in Fig. 3. Analysing obtained results, a few important points should be emphasised:

1. Despite the fact that the disks trials were used as a completely blind test for the model, the acquired results are physically consistent, and the field of the percentage of recrystallisation is smooth and continuous over the specimen and over time. Traditional analytical models are mathematical formulae describing the relations between different process parameters. They are generally expressed as smooth, continuous functions differentiable with respect to time and coordinates. On the other hand, such smoothness and continuity are not a generic property of an AI-based model's architecture. Thus, the continuity observed in the blind tests shows that the model succeeded in identifying some process features from the provided data during training.
2. Quantitative results of the blind tests are surprisingly good. The profiled disks have more complicated geometries, and they were obtained in a “closed die” forging process, which leads to more complex process histories than the ones used to train the model, and this will be discussed in more detail later in the paper. However, out of 1500 traced points, the error in prediction was more than 30% only for 250 points in the 970°C trial and about 300 points for the 1080°C trial (Fig.3).

<figure><img src="image_2049.png" /><figcaption>Figure 4. Peculiarities of the AI-based model: a) some puzzling microstructural results; b) comparison of the AI-based model prediction with the experimental results in the questionable zone for disk at 970°C; c) the same for disk at 1080°C.</figcaption></figure>

3. Microstructural analysis of the profiled disks of the given geometry forged on a screw press yields some puzzling peculiarities of this process [2,7], which, in principle, can't be described by any JMAK-type model. As shown in Fig. 4a, at the centre of the disk, at the point of first contact with the upper die there is a zone marked as “1”, which, according to modelling and common sense, is characterised by very low strains as material there doesn't flow anywhere and relatively low temperatures due to contact with the 350°C die. However, this zone shows a very pronounced high level of recrystallisation, and this result is consistently reproducible. For comparison, in a zone marked as “2” with higher strains and temperatures, only a few sparse recrystallisation nuclei were observed. It was unexpected to see that the utilisation of an AI-based model helped predict recrystallisation in zone “1” in both blind tests (Fig. 4c). Most likely, this means that in the process of training the AI-based model was able to isolate some hidden correlations in the data, reflecting not very evident mechanical or microstructural mechanisms, which is a natural for deep learning algorithms.

Despite having trained the AI-based model on data from only two industrial forgings, the model outputs competitive results when compared to traditional JMAK based models. Fig. 5 shows the results of AI prediction against one of the most popular traditional JMAK models [10] used as default in multiple commercial metal forming simulation software programs, e.g. QForm UK® [11] and DEFORM® [12]. It can be observed that the AI model demonstrates a more correct qualitative distribution of recrystallisation levels. Comparing the figures of total recrystallisation level, it can be observed that the AI model has the capability to better describe intermediate zones between non-recrystallised and fully recrystallised zones (which is typically a weak point of all JMAK-type models). Quantitatively, the AI prediction is also better, however, this comparison is not completely fair as the AI model was calibrated for the same material as the one used in experiments with the profiled disks, while in the case of the JMAK model, standard default coefficients for Inconel 718 were used.
2. Quantitative results of the blind tests are surprisingly good. The profiled disks have more complicated geometries, and they were obtained in a “closed die” forging process, which leads to more complex process histories than the ones used to train the model, and this will be discussed in more detail later in the paper. However, out of 1500 traced points, the error in prediction was more than 30% only for 250 points in the 970°C trial and about 300 points for the 1080°C trial (Fig.3).

<figure><img src="image_2041.png" /><figcaption>Figure 4. Peculiarities of the AI-based model: a) some puzzling microstructural results; b) comparison of the AI-based model prediction with the experimental results in the questionable zone for disk at 970°C; c) the same for disk at 1080°C.</figcaption></figure>

3. Microstructural analysis of the profiled disks of the given geometry forged on a screw press yields some puzzling peculiarities of this process [2,7], which, in principle, can't be described by any JMAK-type model. As shown in Fig. 4a, at the centre of the disk, at the point of first contact with the upper die there is a zone marked as “1”, which, according to modelling and common sense, is characterised by very low strains as material there doesn't flow anywhere and relatively low temperatures due to contact with the 350°C die. However, this zone shows a very pronounced high level of recrystallisation, and this result is consistently reproducible. For comparison, in a zone marked as “2” with higher strains and temperatures, only a few sparse recrystallisation nuclei were observed. It was unexpected to see that the utilisation of an AI-based model helped predict recrystallisation in zone “1” in both blind tests (Fig. 4c). Most likely, this means that in the process of training the AI-based model was able to isolate some hidden correlations in the data, reflecting not very evident mechanical or microstructural mechanisms, which is a natural for deep learning algorithms.

Despite having trained the AI-based model on data from only two industrial forgings, the model outputs competitive results when compared to traditional JMAK based models. Fig. 5 shows the results of AI prediction against one of the most popular traditional JMAK models [10] used as default in multiple commercial metal forming simulation software programs, e.g. QForm UK® [11] and DEFORM® [12]. It can be observed that the AI model demonstrates a more correct qualitative distribution of recrystallisation levels. Comparing the figures of total recrystallisation level, it can be observed that the AI model has the capability to better describe intermediate zones between non-recrystallised and fully recrystallised zones (which is typically a weak point of all JMAK-type models). Quantitatively, the AI prediction is also better, however, this comparison is not completely fair as the AI model was calibrated for the same material as the one used in experiments with the profiled disks, while in the case of the JMAK model, standard default coefficients for Inconel 718 were used.
<figure><img src="image_2050.png" /><figcaption>Figure 5. Comparison of the predictions of AI-based model and standard JMAK model for the “blind” tests - disks forged at the screw press at: a) 970°C and b) 1080°C.</figcaption></figure>

Discussion

When it comes to implementing the described methodology and model in practice for modelling microstructure evolution in commercial forging processes, there are a few clear advantages and challenges compared to the methodology and models currently in use that we would like to discuss.

The potential predictive capabilities of an AI-based model, such as the one showcased in this paper, are far superior to a traditional analytical model, since the model itself has a much higher predictive capacity and can predict unknown dependencies found in the data without the need to describe them analytically. This was clearly shown with the “unexpected” prediction of recrystallisation shown in Fig. 4. It obviously could be argued that such results are nothing more than luck; meanwhile, only a large amount of further testing and data analysis with various other forgings can give a definitive answer regarding the robustness of the model. However, in our opinion, this is not random luck for the following reasons. Standard JMAK models use only three process parameters (temperature, strain and strain rate), assuming that the influence of other process parameters is negligible even though it was not proven by any investigation. As described in section 2.3., the model presented in this paper uses 10 process parameters, providing the artificial neural network the freedom to “decide” whether these parameters are essential for correct prediction of the results or whether they should be ignored. This may, for example, be important in the description of various transient, unsterilised processes where the role of gradients of the process parameters is not very clear but whose influence could have been picked up by the model during the training process.

Another example of a higher predictive capability can be seen in the comparison with the JMAK model (Fig.5). Screw press forging is a fast process where the total deformation takes just a few seconds. The time of the process is not sufficient for the material to absorb all the strain energy provided to it, and as such, the Dynamic Recrystallisation (DRX) initiated during the forging actively continues as Meta-Dynamic (MDRX) or Post-Dynamic Recrystallisation (PDRX) in the few seconds after deformation [13]. This is why JMAK-type models traditionally consist of three main parts: DRX, MDRX and SRX, which is Static Recrystallisation, which generally occurs during reheating or heat treatment [10-12]. Each part contains 7-10 material parameters, which are expected to be found analytically from pure uniaxial tests. Unfortunately, in practice, it's almost impossible to separate Meta-Dynamic from Dynamic recrystallisation; as such, the MDRX part of JMAK-type models often suffers from poor calibration (which can be observed in Fig. 5). Machine learning algorithms may help with this for two reasons. Firstly, as most of the information about the underlying mechanical and microstructural processes is acquired by an AI-based model and reflected in its parameters from the data, there is no need to actively separate these processes in the underlying architecture as long as they are both sufficiently represent in the experiments used
<figure><img src="image_2042.png" /><figcaption>Figure 5. Comparison of the predictions of AI-based model and standard JMAK model for the “blind” tests - disks forged at the screw press at: a) 970°C and b) 1080°C.</figcaption></figure>

Discussion

When it comes to implementing the described methodology and model in practice for modelling microstructure evolution in commercial forging processes, there are a few clear advantages and challenges compared to the methodology and models currently in use that we would like to discuss.

The potential predictive capabilities of an AI-based model, such as the one showcased in this paper, are far superior to a traditional analytical model, since the model itself has a much higher predictive capacity and can predict unknown dependencies found in the data without the need to describe them analytically. This was clearly shown with the “unexpected” prediction of recrystallisation shown in Fig. 4. It obviously could be argued that such results are nothing more than luck; meanwhile, only a large amount of further testing and data analysis with various other forgings can give a definitive answer regarding the robustness of the model. However, in our opinion, this is not random luck for the following reasons. Standard JMAK models use only three process parameters (temperature, strain and strain rate), assuming that the influence of other process parameters is negligible even though it was not proven by any investigation. As described in section 2.3., the model presented in this paper uses 10 process parameters, providing the artificial neural network the freedom to “decide” whether these parameters are essential for correct prediction of the results or whether they should be ignored. This may, for example, be important in the description of various transient, unsterilised processes where the role of gradients of the process parameters is not very clear but whose influence could have been picked up by the model during the training process.

Another example of a higher predictive capability can be seen in the comparison with the JMAK model (Fig.5). Screw press forging is a fast process where the total deformation takes just a few seconds. The time of the process is not sufficient for the material to absorb all the strain energy provided to it, and as such, the Dynamic Recrystallisation (DRX) initiated during the forging actively continues as Meta-Dynamic (MDRX) or Post-Dynamic Recrystallisation (PDRX) in the few seconds after deformation [13]. This is why JMAK-type models traditionally consist of three main parts: DRX, MDRX and SRX, which is Static Recrystallisation, which generally occurs during reheating or heat treatment [10-12]. Each part contains 7-10 material parameters, which are expected to be found analytically from pure uniaxial tests. Unfortunately, in practice, it's almost impossible to separate Meta-Dynamic from Dynamic recrystallisation; as such, the MDRX part of JMAK-type models often suffers from poor calibration (which can be observed in Fig. 5). Machine learning algorithms may help with this for two reasons. Firstly, as most of the information about the underlying mechanical and microstructural processes is acquired by an AI-based model and reflected in its parameters from the data, there is no need to actively separate these processes in the underlying architecture as long as they are both sufficiently represent in the experiments used
for calibration. Secondly, the fixed mathematical structure of a traditional analytical model with only 20 parameters may simply be too “stiff” for successful optimisation.

Speaking about other prerequisites of higher predictive quality, it should be noted that the proposed methodology for data collection and calibration from industrial forgings results in the model generally working in an interpolation regime. Models based and calibrated on uniaxial lab tests, on the other hand, are often forced to be used in an extrapolation regime as they attempt to process data from actual industrial forging processes that are far more complicated.

Nevertheless, the existing challenges must also be addressed when discussing the potential advantages of AI-based models. The main advantage of traditional analytical models comes from the capability to analyse their behaviour, though practically, this isn't easy to do for complicated models. Analytical models can give wrong predictions in extrapolation, but the worst scenario can be accessed at least theoretically. This is not the case for deep learning models. As mentioned earlier, the quality of AI-based models significantly depends on how well they recognise important data patterns during training. Developing the methodology for guaranteed successful training is a serious challenge in AI research and is beyond the scope of this paper. However, some steps can be proposed to increase the chance of successful training.

The key prerequisite of successful training is appropriate data. As mentioned in the introduction, training of an AI-based model is similar to curve fitting and consistency and homogeneity of training data may be more important than the total amount of data.

<figure><img src="image_2051.png" /><figcaption>Figure 6. Projection onto the [Temperature, Strain-rate] plane of the data used: a) for training the AI model, b) for blind validation of the AI model. Orange dots represent process data from standard uniaxial lab tests generally used to calibrate traditional JMAK-type models.</figcaption></figure>

Fig. 6a shows the data obtained from FE simulations of the two forgings of double truncated cones and the typical set of results that can be obtained from uniaxial lab tests. For each point of the DTCs, curved trajectories show thermo-mechanical histories traced at those points and a coloured circle – the final state. Uniaxial tests have constant temperature and constant strain rate, so they don't have a path in this space and are sufficiently denoted by a single point. The first thing to note in this figure is how far-reaching the data from the forging trials is compared to uniaxial tests. One single forging trial can provide hundreds of trajectories. It can also be observed that there can be vastly different histories resulting in the same or similar final states (the same can be said about points with similar experimental final recrystallisation values).

In the example showcased in this paper around 3000 trajectories were taken from just two forging trials and were successfully used to train 840 parameters. Is this dataset sufficient? Is the
for calibration. Secondly, the fixed mathematical structure of a traditional analytical model with only 20 parameters may simply be too “stiff” for successful optimisation.

Speaking about other prerequisites of higher predictive quality, it should be noted that the proposed methodology for data collection and calibration from industrial forgings results in the model generally working in an interpolation regime. Models based and calibrated on uniaxial lab tests, on the other hand, are often forced to be used in an extrapolation regime as they attempt to process data from actual industrial forging processes that are far more complicated.

Nevertheless, the existing challenges must also be addressed when discussing the potential advantages of AI-based models. The main advantage of traditional analytical models comes from the capability to analyse their behaviour, though practically, this isn't easy to do for complicated models. Analytical models can give wrong predictions in extrapolation, but the worst scenario can be accessed at least theoretically. This is not the case for deep learning models. As mentioned earlier, the quality of AI-based models significantly depends on how well they recognise important data patterns during training. Developing the methodology for guaranteed successful training is a serious challenge in AI research and is beyond the scope of this paper. However, some steps can be proposed to increase the chance of successful training.

The key prerequisite of successful training is appropriate data. As mentioned in the introduction, training of an AI-based model is similar to curve fitting and consistency and homogeneity of training data may be more important than the total amount of data.

<figure><img src="image_2043.png" /><figcaption>Figure 6. Projection onto the [Temperature, Strain-rate] plane of the data used: a) for training the AI model, b) for blind validation of the AI model. Orange dots represent process data from standard uniaxial lab tests generally used to calibrate traditional JMAK-type models.</figcaption></figure>

Fig. 6a shows the data obtained from FE simulations of the two forgings of double truncated cones and the typical set of results that can be obtained from uniaxial lab tests. For each point of the DTCs, curved trajectories show thermo-mechanical histories traced at those points and a coloured circle – the final state. Uniaxial tests have constant temperature and constant strain rate, so they don't have a path in this space and are sufficiently denoted by a single point. The first thing to note in this figure is how far-reaching the data from the forging trials is compared to uniaxial tests. One single forging trial can provide hundreds of trajectories. It can also be observed that there can be vastly different histories resulting in the same or similar final states (the same can be said about points with similar experimental final recrystallisation values).

In the example showcased in this paper around 3000 trajectories were taken from just two forging trials and were successfully used to train 840 parameters. Is this dataset sufficient? Is the
quality of this dataset sufficient? What additional forging trials can improve model training? The answers to these questions are not evident. But in the first approximation, it can be concluded that good results in blind prediction show that an AI-based model was able to recognise important patterns that previously were not recognisable with a traditional approach. However, Fig. 6b shows that a significant amount of thermo-mechanical histories belongs to a part of the temperature-strain rate space, in which the model either was not trained or was trained on a small amount of homogeneously distributed data. Over 30% error in about 20% of points is likely related to this circumstance, however this requires more detailed analysis. The gaps in available data suggest what extra forging trials can be beneficial and how they should be designed. Fortunately, they are much cheaper and easier to conduct for the forging industry than lab tests.

The last thing that should be mentioned when talking about challenges relates to the fact that all the trajectories shown in Fig. 6 and used for training the model are obtained from FE simulations. Thus, the quality of this data and in turn the quality of the trained model is significantly dependent on the quality of the FE simulation.

## Conclusion

This paper proposed a new approach to microstructure evolution modelling in hot forging processes using an AI-based model paired with a new methodology for data collection and model calibration using full thermo-mechanical histories acquired directly from industrial forgings with the use of FE simulation in QForm UK®.

An example of such a model has been made and calibrated using industrial hot forgings of two Double Truncated Cones made of Inconel 718. The model has been implemented into existing commercial metal forming simulation software as a user subroutine and blind tested on two complex disk forgings, showing the viability and high potential of the proposed approach.

Further development and implementation of AI-based models into industrial metal forming practice has the potential to increase the quality and reliability of microstructure evolution predictions compared to currently used models while making the process of calibrating such models drastically faster and cheaper.

## References

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[3] Guimaraes, A.A. and Jonas, J.J., 1981. Recrystallization and aging effects associated with the high temperature deformation of Waspaloy and Inconel 718. Metallurgical Transactions A, 12, pp.1655-1666. https://doi.org/10.1007/BF02643571

[4] Sui, F.L., Xu, L.X., Chen, L.Q. and Liu, X.H., 2011. Processing map for hot working of Inconel 718 alloy. Journal of Materials Processing Technology, 211(3), pp.433-440. https://doi.org/10.1016/j.jmatprotec.2010.10.015

[5] Nicolaÿ, A., Fiorucci, G., Franchet, J.M., Cormier, J. and Bozzolo, N., 2019. Influence of strain rate on subsolvus dynamic and post-dynamic recrystallization kinetics of Inconel 718. Acta Materialia, 174, pp.406-417. https://doi.org/10.1016/j.actamat.2019.05.061

[6] Cybenko, G., 1989. Approximation by superpositions of a sigmoidal function. Math. Control. Signals, Syst. 2, 303-314. https://doi.org/10.1007/BF02551274
quality of this dataset sufficient? What additional forging trials can improve model training? The answers to these questions are not evident. But in the first approximation, it can be concluded that good results in blind prediction show that an AI-based model was able to recognise important patterns that previously were not recognisable with a traditional approach. However, Fig. 6b shows that a significant amount of thermo-mechanical histories belongs to a part of the temperature-strain rate space, in which the model either was not trained or was trained on a small amount of homogeneously distributed data. Over 30% error in about 20% of points is likely related to this circumstance, however this requires more detailed analysis. The gaps in available data suggest what extra forging trials can be beneficial and how they should be designed. Fortunately, they are much cheaper and easier to conduct for the forging industry than lab tests.

The last thing that should be mentioned when talking about challenges relates to the fact that all the trajectories shown in Fig. 6 and used for training the model are obtained from FE simulations. Thus, the quality of this data and in turn the quality of the trained model is significantly dependent on the quality of the FE simulation.

## Conclusion

This paper proposed a new approach to microstructure evolution modelling in hot forging processes using an AI-based model paired with a new methodology for data collection and model calibration using full thermo-mechanical histories acquired directly from industrial forgings with the use of FE simulation in QForm UK®.

An example of such a model has been made and calibrated using industrial hot forgings of two Double Truncated Cones made of Inconel 718. The model has been implemented into existing commercial metal forming simulation software as a user subroutine and blind tested on two complex disk forgings, showing the viability and high potential of the proposed approach.

Further development and implementation of AI-based models into industrial metal forming practice has the potential to increase the quality and reliability of microstructure evolution predictions compared to currently used models while making the process of calibrating such models drastically faster and cheaper.

## References

[1] Reshetov, A. and Bylya, O., 2023, April. Benchmarking mean-field models available in commercial FE software in application to two-blow forging of IN718 alloy. Materials Research Forum LLC. https://doi.org/10.21741/9781644902479-74

[2] Bylya, O., Reshetov, A., Stefani, N., Rosochowska, M. and Blackwell, P., 2017. Applicability of JMAK-type model for predicting microstructural evolution in nickel-based superalloys. Procedia engineering, 207, pp.1105-1110. https://doi.org/10.1016/j.proeng.2017.10.1067

[3] Guimaraes, A.A. and Jonas, J.J., 1981. Recrystallization and aging effects associated with the high temperature deformation of Waspaloy and Inconel 718. Metallurgical Transactions A, 12, pp.1655-1666. https://doi.org/10.1007/BF02643571

[4] Sui, F.L., Xu, L.X., Chen, L.Q. and Liu, X.H., 2011. Processing map for hot working of Inconel 718 alloy. Journal of Materials Processing Technology, 211(3), pp.433-440. https://doi.org/10.1016/j.jmatprotec.2010.10.015

[5] Nicolaÿ, A., Fiorucci, G., Franchet, J.M., Cormier, J. and Bozzolo, N., 2019. Influence of strain rate on subsolvus dynamic and post-dynamic recrystallization kinetics of Inconel 718. Acta Materialia, 174, pp.406-417. https://doi.org/10.1016/j.actamat.2019.05.061

[6] Cybenko, G., 1989. Approximation by superpositions of a sigmoidal function. Math. Control. Signals, Syst. 2, 303-314. https://doi.org/10.1007/BF02551274
[7] Reshetov, A., Stefani, N., Bylya, O., Krishnamurthy, B. and Blackwell, P., 2020. Aspects of high strain rate industrial forging of Inconel 718. In Superalloys 2020: Proceedings of the 14th International Symposium on Superalloys (pp. 461-470). Springer International Publishing. https://doi.org/10.1007/978-3-030-51834-9_45

[8] Stefani, N., 2020. Novel experimental methodology for the investigation of recrystallisation during industrial hot forging of Inconel 718, Engineering Doctoral Thesis, pp243

[9] Gers, F., 2001. Long short-term memory in recurrent neural networks, Doctoral dissertation.

[10] Huang, D., Wu, W.T., Lambert, D. and Semiatin, S.L., 2001. Computer simulation of microstructure evolution during hot forging of Waspaloy and nickel alloy 718. Minerals, Metals and Materials Society/AIME, Microstructure Modeling and Prediction During Thermomechanical Processing (USA), pp.137-146.

[11] Micas Simulation LTD, "QForm UK version 11, The software documentation: Description of Microstructure evolution model", 2023.

[12] “DEFORMTM V11.0.1 System Documentation (Reed Road Columbus, Ohio: Scientific Forming Technologies Corporation).” pp. 175–180, 2014.

[13] Meriem Zouari, Roland E, Logé, Nathalie Bozzolo In Situ Characterization of Inconel 718 Post-Dynamic Recrystallization within a Scanning Electron Microscope, Metals 2017, 7, 476. https://doi.org/10.3390/met7110476
[7] Reshetov, A., Stefani, N., Bylya, O., Krishnamurthy, B. and Blackwell, P., 2020. Aspects of high strain rate industrial forging of Inconel 718. In Superalloys 2020: Proceedings of the 14th International Symposium on Superalloys (pp. 461-470). Springer International Publishing. https://doi.org/10.1007/978-3-030-51834-9_45

[8] Stefani, N., 2020. Novel experimental methodology for the investigation of recrystallisation during industrial hot forging of Inconel 718, Engineering Doctoral Thesis, pp243

[9] Gers, F., 2001. Long short-term memory in recurrent neural networks, Doctoral dissertation.

[10] Huang, D., Wu, W.T., Lambert, D. and Semiatin, S.L., 2001. Computer simulation of microstructure evolution during hot forging of Waspaloy and nickel alloy 718. Minerals, Metals and Materials Society/AIME, Microstructure Modeling and Prediction During Thermomechanical Processing (USA), pp.137-146.

[11] Micas Simulation LTD, "QForm UK version 11, The software documentation: Description of Microstructure evolution model", 2023.

[12] “DEFORMTM V11.0.1 System Documentation (Reed Road Columbus, Ohio: Scientific Forming Technologies Corporation).” pp. 175–180, 2014.

[13] Meriem Zouari, Roland E, Logé, Nathalie Bozzolo In Situ Characterization of Inconel 718 Post-Dynamic Recrystallization within a Scanning Electron Microscope, Metals 2017, 7, 476. https://doi.org/10.3390/met7110476
# Investigation of temperature-dependent mechanical behaviours of polycarbonate with an innovative fractional order model

SUN Lin<fnref n="1" />,ᵃ*, CHENG Gang<fnref n="2" />,ᵇ* and BARRIERE Thierry<fnref n="1" />,ᶜ

**Keywords:** Polycarbonate, Glass Transition Temperature, Variable Order Fractional Model, Mechanical Behaviour

**Abstract.** Amorphous thermoplastic polymers are considered as essential engineering materials. However, their strongly temperature-dependent elastic-viscoplastic behaviour is still not well understood. The large deformation response of these materials in the temperature range spanning the glass transition temperature is highly required. In this paper, the uniaxial compression tests are performed with large deformation on a polycarbonate amorphous polymer, within a forming temperature range from 60 °C to 180 °C. The variable order fractional constitutive model is improved to describe the evolution of mechanical behaviour of polycarbonate at different temperatures. The fractional order in the model is proposed as a function of time. The simulation results are in good agreement with the experimental data. It is concluded that the variable order fractional model is an efficient tool to predict the large deformation of amorphous polymers under different temperatures.

**Introduction**

Polycarbonate (PC) exhibits excellent mechanical, chemical and physical properties such as transparency, toughness, ductility and elasticity, making it widely used as a structural material [1]. However, significant influences of temperature are observed on its physical properties, such as color changes, surface cracking and mass loss [2]. It is necessary to propose an effective constitutive behaviour law to accurately describe its mechanical properties due to the strong dependence on temperature.

The mechanical behaviour of PC is greatly changed above its glass transition temperature (Tg). It behaves as a rubbery phase with a viscoelastic response. Below Tg, strain softening occurs before strain hardening, and it behaves as a glassy phase [3]. Over the past few decades, many constitutive models have been developed to describe the complex behaviour of polymers considering the glass transition [4-7]. Alves et al. [8] proposed a constitutive model for the deformation behaviour of PC below Tg. These constitutive models usually require a considerable number of parameters to identify the characteristics of polymers due to their complex temperature-dependent behaviours [9,10]. Temperature has a great impact on rheological property of PC during the forming process [11,12]. In order to simulate these manufacturing processes with fewer material parameters, an accurate and efficient constitutive model is still strongly required in a wide temperature range.

A variable order fractional (VOF) derivative operator is introduced into the viscoelastic model to improve its accuracy and reduce the number of identified parameters. The order of fractional derivative is supposed to be a function of the independent variable (time or space) [13]. It provides a new way to describe the time dependence and historical memory of viscoelastic materials. Based on the molecular chain conflict theory, Meng et al. [14] verified that the corresponding mechanical

<figure><img src="image_2052.png" /></figure>

<footnote n="1">¹Université de Franche-Comté, CNRS, institut FEMTO-ST, 25000, Besançon, France</footnote>

<footnote n="2">²INSA Centre Val de Loire, Univ. Tours, Univ. Orléans, LaMé, 41034, Blois, France</footnote>

<footnote>ᵃlin.sun@femto-st.fr, ᵇgang.cheng@insa-cvl.fr, ᶜthierry.barriere@univ-fcomte.fr</footnote>
# Investigation of temperature-dependent mechanical behaviours of polycarbonate with an innovative fractional order model

SUN Lin<fnref n="1" />,ᵃ*, CHENG Gang<fnref n="2" />,ᵇ* and BARRIERE Thierry<fnref n="1" />,ᶜ

**Keywords:** Polycarbonate, Glass Transition Temperature, Variable Order Fractional Model, Mechanical Behaviour

**Abstract.** Amorphous thermoplastic polymers are considered as essential engineering materials. However, their strongly temperature-dependent elastic-viscoplastic behaviour is still not well understood. The large deformation response of these materials in the temperature range spanning the glass transition temperature is highly required. In this paper, the uniaxial compression tests are performed with large deformation on a polycarbonate amorphous polymer, within a forming temperature range from 60 °C to 180 °C. The variable order fractional constitutive model is improved to describe the evolution of mechanical behaviour of polycarbonate at different temperatures. The fractional order in the model is proposed as a function of time. The simulation results are in good agreement with the experimental data. It is concluded that the variable order fractional model is an efficient tool to predict the large deformation of amorphous polymers under different temperatures.

**Introduction**

Polycarbonate (PC) exhibits excellent mechanical, chemical and physical properties such as transparency, toughness, ductility and elasticity, making it widely used as a structural material [1]. However, significant influences of temperature are observed on its physical properties, such as color changes, surface cracking and mass loss [2]. It is necessary to propose an effective constitutive behaviour law to accurately describe its mechanical properties due to the strong dependence on temperature.

The mechanical behaviour of PC is greatly changed above its glass transition temperature (Tg). It behaves as a rubbery phase with a viscoelastic response. Below Tg, strain softening occurs before strain hardening, and it behaves as a glassy phase [3]. Over the past few decades, many constitutive models have been developed to describe the complex behaviour of polymers considering the glass transition [4-7]. Alves et al. [8] proposed a constitutive model for the deformation behaviour of PC below Tg. These constitutive models usually require a considerable number of parameters to identify the characteristics of polymers due to their complex temperature-dependent behaviours [9,10]. Temperature has a great impact on rheological property of PC during the forming process [11,12]. In order to simulate these manufacturing processes with fewer material parameters, an accurate and efficient constitutive model is still strongly required in a wide temperature range.

A variable order fractional (VOF) derivative operator is introduced into the viscoelastic model to improve its accuracy and reduce the number of identified parameters. The order of fractional derivative is supposed to be a function of the independent variable (time or space) [13]. It provides a new way to describe the time dependence and historical memory of viscoelastic materials. Based on the molecular chain conflict theory, Meng et al. [14] verified that the corresponding mechanical

<figure><img src="image_2044.png" /></figure>

<footnote n="1">¹Université de Franche-Comté, CNRS, institut FEMTO-ST, 25000, Besançon, France</footnote>

<footnote n="2">²INSA Centre Val de Loire, Univ. Tours, Univ. Orléans, LaMé, 41034, Blois, France</footnote>

<footnote>ᵃlin.sun@femto-st.fr, ᵇgang.cheng@insa-cvl.fr, ᶜthierry.barriere@univ-fcomte.fr</footnote>
property evolution revealed by fractional order was reasonable. They demonstrated that the VOF model could effectively describe the compression deformation of amorphous polymers.

In this study, the mechanical behaviours of PC below and above Tg are analyzed numerically and experimentally. A linear function is proposed for the fractional order to simulate the viscoelastic behaviour above Tg. A power function is applied to the fractional order to describe the large deformation of polymer below Tg. The material parameters of the models are identified by inverse method according to the experimental data.

## Fractional viscoelastic constitutive model

The VOF model of viscoelasticity is expressed as follows [15]:

$$ \sigma(t) = E\theta^{\alpha(t)}D_{t}^{\alpha(t)}\varepsilon(t), 0 < \alpha(t) < 1. \quad (1) $$

where $\sigma(t)$ is stress, $\varepsilon(t)$ is strain, $E$ is elastic modulus, $\theta = \eta/E$ is relaxation time, $\eta$ is viscosity coefficient. $D_{t}^{\alpha(t)}\varepsilon(t)$ represents VOF derivative of $\varepsilon(t)$, where $\alpha(t)$ is the variable fractional order. When $\alpha(t) = 0$, the model becomes pure elastic behaviour law, represented by a Hookean spring, while $\alpha(t) = 1$, it becomes Newton's law of viscosity, represented by a dashpot. The fractional order is defined between 0 and 1, represents the viscoelastic behaviour of the material. Three parameters $E$, $\theta$ and $\alpha(t)$, need to be identified in VOF, shown in Fig. 1.

<figure><img src="image_2053.png" /><figcaption>Fig. 1. Schematic of VOF model.</figcaption></figure>

In this study, the VOF differential operator proposed by Ramirez and Coimbra [15] is adopted:

$$ D^{\alpha(t)}f(t) = \frac{1}{\Gamma[1-\alpha(t)]} \int_{0}^{t} (t-s)^{-\alpha(t)} f(s)^{(1)}ds + \frac{(f(0+)-f(0-))t^{-\alpha(t)}}{\Gamma[1-\alpha(t)]}. \quad (2) $$

where $\Gamma(*) = \int_{0}^{*} e^{-s}s^{*-1} ds$ is Gamma function, $f(s)^{(1)}ds$ represents the first derivative of $f(s)$ with respect to variable $s$.

When the deformation rate of the material is constant, its strain can be expressed as follows:

$$ \varepsilon(t) = \dot{\varepsilon}t. \quad (3) $$

where $\dot{\varepsilon}$ is the constant strain rate.

Substituting of the strain rate into Eq. (1) and using the definition of Eq. (2) gives:

$$ \sigma(t) = E\theta^{\alpha(t)} \frac{\dot{\varepsilon}t^{1-\alpha(t)}}{\Gamma(2-\alpha(t))}. \quad (4) $$

The equation of VOF stress-strain relationship is obtained:

$$ \sigma(t) = E(\dot{\varepsilon}\theta)^{\alpha(t)} \frac{\varepsilon(t)^{1-\alpha(t)}}{\Gamma(2-\alpha(t))}. \quad (5) $$

The fractional order in the VOF model exhibits a linear relationship with strain [16], which is mathematically formulated as $\alpha(\varepsilon) = a\varepsilon + b$, where $a$ and $b$ are constants. Other functions are proposed for the fractional order to describe the complex behaviours with large deformations. Strain-dependent power function $\alpha(\varepsilon) = c\varepsilon^d + e$, where $c$, $d$ and $e$ are constants, has recently
property evolution revealed by fractional order was reasonable. They demonstrated that the VOF model could effectively describe the compression deformation of amorphous polymers.

In this study, the mechanical behaviours of PC below and above Tg are analyzed numerically and experimentally. A linear function is proposed for the fractional order to simulate the viscoelastic behaviour above Tg. A power function is applied to the fractional order to describe the large deformation of polymer below Tg. The material parameters of the models are identified by inverse method according to the experimental data.

## Fractional viscoelastic constitutive model

The VOF model of viscoelasticity is expressed as follows [15]:

$$ \sigma(t) = E\theta^{\alpha(t)}D_{t}^{\alpha(t)}\varepsilon(t), 0 < \alpha(t) < 1. \quad (1) $$

where $\sigma(t)$ is stress, $\varepsilon(t)$ is strain, $E$ is elastic modulus, $\theta = \eta/E$ is relaxation time, $\eta$ is viscosity coefficient. $D_{t}^{\alpha(t)}\varepsilon(t)$ represents VOF derivative of $\varepsilon(t)$, where $\alpha(t)$ is the variable fractional order. When $\alpha(t) = 0$, the model becomes pure elastic behaviour law, represented by a Hookean spring, while $\alpha(t) = 1$, it becomes Newton's law of viscosity, represented by a dashpot. The fractional order is defined between 0 and 1, represents the viscoelastic behaviour of the material. Three parameters $E$, $\theta$ and $\alpha(t)$, need to be identified in VOF, shown in Fig. 1.

<figure><img src="image_2045.png" /><figcaption>Fig. 1. Schematic of VOF model.</figcaption></figure>

In this study, the VOF differential operator proposed by Ramirez and Coimbra [15] is adopted:

$$ D^{\alpha(t)} f(t) = \frac{1}{\Gamma[1-\alpha(t)]} \int_{0}^{t} (t-s)^{-\alpha(t)} f(s)^{(1)} ds + \frac{(f(0+)-f(0-))t^{-\alpha(t)}}{\Gamma[1-\alpha(t)]}. \quad (2) $$

where $\Gamma(*) = \int_{0}^{*} e^{-s}s^{*-1} ds$ is Gamma function, $f(s)^{(1)} ds$ represents the first derivative of $f(s)$ with respect to variable $s$.

When the deformation rate of the material is constant, its strain can be expressed as follows:

$$ \varepsilon(t) = \dot{\varepsilon}t. \quad (3) $$

where $\dot{\varepsilon}$ is the constant strain rate.

Substituting of the strain rate into Eq. (1) and using the definition of Eq. (2) gives:

$$ \sigma(t) = E\theta^{\alpha(t)} \frac{\dot{\varepsilon}t^{1-\alpha(t)}}{\Gamma(2-\alpha(t))}. \quad (4) $$

The equation of VOF stress-strain relationship is obtained:

$$ \sigma(t) = E(\dot{\varepsilon}\theta)^{\alpha(t)} \frac{\varepsilon(t)^{1-\alpha(t)}}{\Gamma(2-\alpha(t))}. \quad (5) $$

The fractional order in the VOF model exhibits a linear relationship with strain [16], which is mathematically formulated as $\alpha(\varepsilon) = a\varepsilon + b$, where $a$ and $b$ are constants. Other functions are proposed for the fractional order to describe the complex behaviours with large deformations. Strain-dependent power function $\alpha(\varepsilon) = c\varepsilon^d + e$, where $c$, $d$ and $e$ are constants, has recently
been proposed to describe complex non-linear stress-strain responses [17]. The model needs to be validated with amorphous thermoplastic polymers. In this study, it is applied to describe the mechanical behaviours of an amorphous polymer and the simulation accuracy is confirmed with experimental data.

## Experimental tests and numerical approach

Thermoplastic polymer PC was selected in this study. Injection molding technology was applied to fabricate the cylindrical samples. Uniaxial compression tests were carried on to obtain the stress-strain relationships. Both ends of the PC sample were polished to reduce friction between the polymer and mould. The sample was inserted between two metal plates in an oven. Uniaxial compression tests were repeated several times at different temperatures. Based on the standard ISO 604, a constant strain rate 0.003 s⁻¹ was applied during the test and the true strain of the sample was more than 50% [18]. The material parameters in VOF model were identified based on the experimental data with the least squares method, via MATLAB. The effectiveness of the identification was assessed by the values of root mean square error (RMSE) and coefficient of determination (R²). A flowchart describing the numerical approach is shown in Fig. 2.

<figure><img src="image_2054.png" /><figcaption>Fig. 2. Flowchart of the general methodology for the identification of material parameters in VOF model.</figcaption></figure>

## Results

The compression tests were conducted on PC in the temperature range of 60-180°C, with a Tg approximately 150 °C. Fig. 3 shows the evolution of true stress in function of true strain of PC under different temperatures. For the same true strain, the true stress decreases with temperature. At temperature below Tg (Fig. 3 (a)), the stress-strain curve exhibits an obvious glassy phase, with a strain softening stage after yielding and a strain hardening stage at large strains. At temperature above Tg (Fig. 3 (b)), the yield point and strain softening disappear, but strain hardening still exists. The true stress of PC is significantly reduced and a rubbery phase appears. The true stress tends to increase with true strain at temperature above Tg. PC exhibits different true stress-strain curve at 140 °C and 160 °C, which confirms the glass transition stage in this temperature range.
been proposed to describe complex non-linear stress-strain responses [17]. The model needs to be validated with amorphous thermoplastic polymers. In this study, it is applied to describe the mechanical behaviours of an amorphous polymer and the simulation accuracy is confirmed with experimental data.

## Experimental tests and numerical approach

Thermoplastic polymer PC was selected in this study. Injection molding technology was applied to fabricate the cylindrical samples. Uniaxial compression tests were carried on to obtain the stress-strain relationships. Both ends of the PC sample were polished to reduce friction between the polymer and mould. The sample was inserted between two metal plates in an oven. Uniaxial compression tests were repeated several times at different temperatures. Based on the standard ISO 604, a constant strain rate 0.003 s⁻¹ was applied during the test and the true strain of the sample was more than 50% [18]. The material parameters in VOF model were identified based on the experimental data with the least squares method, via MATLAB. The effectiveness of the identification was assessed by the values of root mean square error (RMSE) and coefficient of determination (R²). A flowchart describing the numerical approach is shown in Fig. 2.

<figure><img src="image_2046.png" /><figcaption>Fig. 2. Flowchart of the general methodology for the identification of material parameters in VOF model.</figcaption></figure>

## Results

The compression tests were conducted on PC in the temperature range of 60-180°C, with a Tg approximately 150 °C. Fig. 3 shows the evolution of true stress in function of true strain of PC under different temperatures. For the same true strain, the true stress decreases with temperature. At temperature below Tg (Fig. 3 (a)), the stress-strain curve exhibits an obvious glassy phase, with a strain softening stage after yielding and a strain hardening stage at large strains. At temperature above Tg (Fig. 3 (b)), the yield point and strain softening disappear, but strain hardening still exists. The true stress of PC is significantly reduced and a rubbery phase appears. The true stress tends to increase with true strain at temperature above Tg. PC exhibits different true stress-strain curve at 140 °C and 160 °C, which confirms the glass transition stage in this temperature range.
<figure><img src="image_2055.png" /></figure>

<figure><img src="image_2056.png" /><figcaption>Fig. 3. True stress-strain curves obtained in uniaxial compression tests of PC at constant strain rate 0.003 s⁻¹ across the glass transition (a) below Tg (b) above Tg.</figcaption></figure>

From a physical perspective, the mechanical responses of polymers in glassy and rubbery phase are dominated by intermolecular and intramolecular chain resistance respectively [19]. The glass transition is the critical state where these two mechanisms work together. The evolution of true stress of polymer is greatly affected by temperature. It is difficult to apply the same material model to effectively describe its mechanical behaviour over the entire glass transition range.

<figure><img src="image_2057.png" /><figcaption>Fig. 4. Comparison of true stress-strain curves between experiment data and model predictions above Tg.</figcaption></figure>

VOF model was proposed to characterize the mechanical behaviour above Tg. A linear function $α(ε) = aε + b$ was proposed for the fractional order, because no obvious yield phenomenon was observed in the true stress-strain curve, shown in Fig. 3(b). The comparisons between the numerical model predictions and experimental data are shown in Fig. 4. It can be observed that the model predictions are in good agreement with the experimental data, which confirms that the VOF model can well fit the stress-strain response of PC at temperatures above Tg. The parameters in the model were identified by using the least squares method. The model parameters are shown in Table 1. The elastic modulus *E* decreases with temperature and the relaxation time θ increases with temperature. The value of the fractional order increases and is always less than 1. The decrease of fractional order reflects the strain hardening of PC under large deformation. The fractional order possesses a larger value at higher temperature, which indicates that the mechanical behaviour of PC is close to Newton's law.
<figure><img src="image_2047.png" /></figure>

<figure><img src="image_2048.png" /><figcaption>Fig. 3. True stress-strain curves obtained in uniaxial compression tests of PC at constant strain rate 0.003 s⁻¹ across the glass transition (a) below Tg (b) above Tg.</figcaption></figure>

From a physical perspective, the mechanical responses of polymers in glassy and rubbery phase are dominated by intermolecular and intramolecular chain resistance respectively [19]. The glass transition is the critical state where these two mechanisms work together. The evolution of true stress of polymer is greatly affected by temperature. It is difficult to apply the same material model to effectively describe its mechanical behaviour over the entire glass transition range.

<figure><img src="image_2049.png" /><figcaption>Fig. 4. Comparison of true stress-strain curves between experiment data and model predictions above Tg.</figcaption></figure>

VOF model was proposed to characterize the mechanical behaviour above Tg. A linear function $α(ε) = aε + b$ was proposed for the fractional order, because no obvious yield phenomenon was observed in the true stress-strain curve, shown in Fig. 3(b). The comparisons between the numerical model predictions and experimental data are shown in Fig. 4. It can be observed that the model predictions are in good agreement with the experimental data, which confirms that the VOF model can well fit the stress-strain response of PC at temperatures above Tg. The parameters in the model were identified by using the least squares method. The model parameters are shown in Table 1. The elastic modulus *E* decreases with temperature and the relaxation time θ increases with temperature. The value of the fractional order increases and is always less than 1. The decrease of fractional order reflects the strain hardening of PC under large deformation. The fractional order possesses a larger value at higher temperature, which indicates that the mechanical behaviour of PC is close to Newton's law.
<table><caption>Table 1. Identified parameters of VOF model for PC above Tg.</caption><thead><tr><td>T (°C)</td><td>E (MPa)</td><td>θ (s)</td><td>a</td><td>b</td><td>RMSE</td><td>R<sup>2</sup></td></tr></thead><tbody><tr><td>160</td><td>1.54</td><td>68.20</td><td>0.14</td><td>0.44</td><td>0.0016</td><td>0.9999</td></tr><tr><td>180</td><td>0.07</td><td>137.11</td><td>0.07</td><td>0.85</td><td>0.000096</td><td>0.9989</td></tr></tbody></table>

Fig. 5 (a) shows the PC true stress-strain curve, which is divided into two regions by the yield point. The value of the yield point is obtained according to the experimental data. The VOF model with linear fractional order function is applied to describe the relationship of true stress and strain below Tg. The results are shown in Fig. 5 (a), the experimental data cannot be well fitted by this model. Because the mechanical properties of PC change significantly before and after the yield point. Therefore, it is necessary to propose different fractional order function before and after yield point in this temperature range. A VOF model with linear fractional order function is applied to characterize the mechanical behaviour of PC before yielding, with a power function selected after yielding.

<figure><img src="image_2058.png" /><figcaption>Fig. 5. Identification of true stress-strain curves below Tg by using VOF model with (a) linear fractional order function and (b) linear and power fractional order function.</figcaption></figure>

The two regions of the stress-strain curves are characterised separately. The first region represents viscoelastic deformation, and is described by VOF model with a linear fractional order function $α(ε) = aε + b$. The second region represents non-linear viscoplastic deformation and is described by using a power fractional order function $α(ε) = cε^2 + e$. The two functions were obtained by fitting the experimental data before and after PC yielding with the least squares method. In Fig. 5(b), the proposed model accurately fits the experimental data at different temperatures below Tg. The evolution of fractional order in function of true strain at different temperatures below Tg is shown in Fig. 6. In the viscoelastic and viscoplastic deformation stage, the fractional order increases with temperature. VOF models with various fractional order functions are quite efficient to characterize the mechanical properties below Tg.
<table><caption>Table 1. Identified parameters of VOF model for PC above Tg.</caption><thead><tr><td>T (°C)</td><td>E (MPa)</td><td>θ (s)</td><td>a</td><td>b</td><td>RMSE</td><td>R<sup>2</sup></td></tr></thead><tbody><tr><td>160</td><td>1.54</td><td>68.20</td><td>0.14</td><td>0.44</td><td>0.0016</td><td>0.9999</td></tr><tr><td>180</td><td>0.07</td><td>137.11</td><td>0.07</td><td>0.85</td><td>0.000096</td><td>0.9989</td></tr></tbody></table>

Fig. 5 (a) shows the PC true stress-strain curve, which is divided into two regions by the yield point. The value of the yield point is obtained according to the experimental data. The VOF model with linear fractional order function is applied to describe the relationship of true stress and strain below Tg. The results are shown in Fig. 5 (a), the experimental data cannot be well fitted by this model. Because the mechanical properties of PC change significantly before and after the yield point. Therefore, it is necessary to propose different fractional order function before and after yield point in this temperature range. A VOF model with linear fractional order function is applied to characterize the mechanical behaviour of PC before yielding, with a power function selected after yielding.

<figure><img src="image_2050.png" /><figcaption>Fig. 5. Identification of true stress-strain curves below Tg by using VOF model with (a) linear fractional order function and (b) linear and power fractional order function.</figcaption></figure>

The two regions of the stress-strain curves are characterised separately. The first region represents viscoelastic deformation, and is described by VOF model with a linear fractional order function $α(ε) = aε + b$. The second region represents non-linear viscoplastic deformation and is described by using a power fractional order function $α(ε) = cε^2 + e$. The two functions were obtained by fitting the experimental data before and after PC yielding with the least squares method. In Fig. 5(b), the proposed model accurately fits the experimental data at different temperatures below Tg. The evolution of fractional order in function of true strain at different temperatures below Tg is shown in Fig. 6. In the viscoelastic and viscoplastic deformation stage, the fractional order increases with temperature. VOF models with various fractional order functions are quite efficient to characterize the mechanical properties below Tg.
<figure><img src="image_2059.png" /><figcaption>Fig. 6. Evolution of fractional order vs. true strain at different temperatures below Tg.</figcaption></figure>

Based on Fig. 6, the linear and power fractional order functions are continuous at the yield point for each temperature. The linear function increases with strain before the yield point, followed by the power function that decreases after the yield point. The fractional order varies between 0 and 1 during the viscoelastic and viscoplastic deformation stage, satisfying the definition of variable fractional order. The evolution of mechanical property is accurately described by using a fractional constitutive behaviour law. The variable fractional order is expressed by a piecewise function, consisted of a linear and power fractional order function. The numerical simulation results are consistent with the existing experimental research [20], the viscoplastic response of materials is strongly affected by the yield stress, strain rate, and temperature.

## Conclusions

In this study, the VOF model was proposed to describe the mechanical behaviours of amorphous polymer across a wide temperature range (below and above Tg). Compression experimental data were used to identify the material parameters and validate the accuracy of the proposed constitutive model. The main conclusions are as follows:

1. The VOF model effectively captures the evolution of mechanical properties of amorphous polymers across Tg.

2. Above Tg, VOF model with a linear fractional order function is suitable to characterize the mechanical behaviours of PC. While below Tg, VOF model with linear and power fractional order function is required.

3. The fractional order increases gradually with temperature, which confirms the efficiency of VOF model in describing the glass transition of amorphous thermoplastic polymer.

## References

[1] S.M. Mirkhalaf, F.M. Andrade Pires, R. Simões, An elasto-viscoplastic constitutive model for polymers at finite strains: formulation and computational aspects, Comput. Struct. 166 (2016) 60-74. https://doi.org/10.1016/j.compstruc.2016.01.002

[2] N. Bouida, A.S. Bouchikhi, A. Megueni, S. Gouasmi, A finite element analysis for evaluation of J-integral in plates made of functionally graded materials with a semicircular notch, J. Fail. Anal. Prev. 18 (6) (2018) 1573-1586. https://doi.org/10.1007/s11668-018-0558-6

[3] R. Xiao, H.G. Sun, W. Chen, A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers, Int. J. Nonlin. Mech. 93 (2017) 7-14. https://doi.org/10.1016/j.ijnonlinmec.2017.04.019
<figure><img src="image_2051.png" /><figcaption>Fig. 6. Evolution of fractional order vs. true strain at different temperatures below Tg.</figcaption></figure>

Based on Fig. 6, the linear and power fractional order functions are continuous at the yield point for each temperature. The linear function increases with strain before the yield point, followed by the power function that decreases after the yield point. The fractional order varies between 0 and 1 during the viscoelastic and viscoplastic deformation stage, satisfying the definition of variable fractional order. The evolution of mechanical property is accurately described by using a fractional constitutive behaviour law. The variable fractional order is expressed by a piecewise function, consisted of a linear and power fractional order function. The numerical simulation results are consistent with the existing experimental research [20], the viscoplastic response of materials is strongly affected by the yield stress, strain rate, and temperature.

## Conclusions

In this study, the VOF model was proposed to describe the mechanical behaviours of amorphous polymer across a wide temperature range (below and above Tg). Compression experimental data were used to identify the material parameters and validate the accuracy of the proposed constitutive model. The main conclusions are as follows:

1. The VOF model effectively captures the evolution of mechanical properties of amorphous polymers across Tg.

2. Above Tg, VOF model with a linear fractional order function is suitable to characterize the mechanical behaviours of PC. While below Tg, VOF model with linear and power fractional order function is required.

3. The fractional order increases gradually with temperature, which confirms the efficiency of VOF model in describing the glass transition of amorphous thermoplastic polymer.

## References

[1] S.M. Mirkhalaf, F.M. Andrade Pires, R. Simões, An elasto-viscoplastic constitutive model for polymers at finite strains: formulation and computational aspects, Comput. Struct. 166 (2016) 60-74. https://doi.org/10.1016/j.compstruc.2016.01.002

[2] N. Bouida, A.S. Bouchikhi, A. Megueni, S. Gouasmi, A finite element analysis for evaluation of J-integral in plates made of functionally graded materials with a semicircular notch, J. Fail. Anal. Prev. 18 (6) (2018) 1573-1586. https://doi.org/10.1007/s11668-018-0558-6

[3] R. Xiao, H.G. Sun, W. Chen, A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers, Int. J. Nonlin. Mech. 93 (2017) 7-14. https://doi.org/10.1016/j.ijnonlinmec.2017.04.019
[4] C. Yu, G.Z. Kang, K.J. Chen, A hygro-thermo-mechanical coupled cyclic constitutive model for polymers with considering glass transition, Int. J. Plast 89 (2017) 29-65.
https://doi.org/10.1016/j.ijplas.2016.11.001

[5] R.A. Loretz, T.J. Loretz, Corrections to theoretical glass transition temperature models and interpretations with application examples to chalcogenide glass, J Non Cryst Solids 628 (2024) 122845.
https://doi.org/10.1016/j.jnoncrysol.2024.122845

[6] P. Giuseppe, G. Jean-Claude, G. Marco, V. Alain, Assessment and simulation of the thermomechanical tensile and creep behavior of C/PEKK composites for aircraft applications above the glass transition temperature, Compos. Struct. 318 (2023) 117069.
https://doi.org/10.1016/j.compstruct.2023.117069

[7] Z.C. Li, M. Lei, Q.Q. Kou, X.L. Ma, L.H. Wen, J.Y. Xiao, X. Hou, A multiscale viscoelastic constitutive model of unidirectional carbon fiber reinforced PEEK over a wide temperature range, Compos. Struct. 321 (2023) 117258.
https://doi.org/10.1016/j.compstruct.2023.117258

[8] A. F.C. Alves, B. P. Ferreira, F.M. A. Pires, A constitutive model for amorphous thermoplastics from low to high strain rates: Formulation and computational aspects, Int. J. Plast 169 (2023) 103712.
https://doi.org/10.1016/j.ijplas.2023.103712

[9] T.X. Lan, Y.D. Jiang, P.D. Wu, A constitutive model for poly-methyl-methacrylate over a wide range of strain rates and temperatures, Polymer 240 (2022) 124524.
https://doi.org/10.1016/j.polymer.2022.124524

[10] A. Francisca Carvalho Alves, Bernardo P. Ferreira, F.M. Andrade Pires, Constitutive modeling of amorphous thermoplastics from low to high strain rates: Formulation and critical comparison employing an optimization-based parameter identification, Int J Solids Struct 273 (2023) 112258.
https://doi.org/10.1016/j.ijsolstr.2023.112258

[11] Y. Kusaka a, T. Kawamura, M. Nakagawa, K. Okamoto, K. Tanaka, N. Fukuda, Fabrication of extremely conductive high-aspect silver traces buried in hot-embossed polycarbonate films via the direct gravure doctoring method, Adv Powder Technol 32 (2021)764-770.
https://doi.org/10.1016/j.apt.2021.01.027

[12] C.C. Yang, X.Y. Tian, D.C. Li, Y. Cao, F. Zhao, C.Q. Shi, Influence of thermal processing conditions in 3D printing on the crystallinity and mechanical properties of PEEK material, J Mater Process Technol 248 (2017) 1-7.
http://dx.doi.org/10.1016/j.jmatprotec.2017.04.027

[13] T. A. Assiri, Time-space variable-order fractional nonlinear system of thermoelasticity: numerical treatment, Adv. Differ. Equ. 2020 (2020) 288.
https://doi.org/10.1186/s13662-020-02740-8

[14] R.F. Meng, D.S. Yin, S.Y. Lu, G.J. Xiang, Variable-order fractional constitutive model for the time-dependent mechanical behavior of polymers across the glass Transition, Eur. Phys. J. Plus 134 (2019) 376.
https://doi.org/10.1140/epjp/i2019-12767-x

[15] L.E.S. Ramirez, C.F.M. Coimbra, Variable order constitutive relation for viscoelasticity. Ann. Phys. 16 (2007) 543-552.
https://doi.org/10.1002/andp.200710246

[16] R. Meng, D. Yin, C.S. Drapaca. Variable-order fractional description of compression deformation of amorphous glassy polymers. Comput. Mech. 64 (2019) 163-171.
https://doi.org/10.1007/s00466-018-1663-9
[4] C. Yu, G.Z. Kang, K.J. Chen, A hygro-thermo-mechanical coupled cyclic constitutive model for polymers with considering glass transition, Int. J. Plast 89 (2017) 29-65.
https://doi.org/10.1016/j.ijplas.2016.11.001

[5] R.A. Loretz, T.J. Loretz, Corrections to theoretical glass transition temperature models and interpretations with application examples to chalcogenide glass, J Non Cryst Solids 628 (2024) 122845.
https://doi.org/10.1016/j.jnoncrysol.2024.122845

[6] P. Giuseppe, G. Jean-Claude, G. Marco, V. Alain, Assessment and simulation of the thermomechanical tensile and creep behavior of C/PEKK composites for aircraft applications above the glass transition temperature, Compos. Struct. 318 (2023) 117069.
https://doi.org/10.1016/j.compstruct.2023.117069

[7] Z.C. Li, M. Lei, Q.Q. Kou, X.L. Ma, L.H. Wen, J.Y. Xiao, X. Hou, A multiscale viscoelastic constitutive model of unidirectional carbon fiber reinforced PEEK over a wide temperature range, Compos. Struct. 321 (2023) 117258.
https://doi.org/10.1016/j.compstruct.2023.117258

[8] A. F.C. Alves, B. P. Ferreira, F.M. A. Pires, A constitutive model for amorphous thermoplastics from low to high strain rates: Formulation and computational aspects, Int. J. Plast 169 (2023) 103712.
https://doi.org/10.1016/j.ijplas.2023.103712

[9] T.X. Lan, Y.D. Jiang, P.D. Wu, A constitutive model for poly-methyl-methacrylate over a wide range of strain rates and temperatures, Polymer 240 (2022) 124524.
https://doi.org/10.1016/j.polymer.2022.124524

[10] A. Francisca Carvalho Alves, Bernardo P. Ferreira, F.M. Andrade Pires, Constitutive modeling of amorphous thermoplastics from low to high strain rates: Formulation and critical comparison employing an optimization-based parameter identification, Int J Solids Struct 273 (2023) 112258.
https://doi.org/10.1016/j.ijsolstr.2023.112258

[11] Y. Kusaka a, T. Kawamura, M. Nakagawa, K. Okamoto, K. Tanaka, N. Fukuda, Fabrication of extremely conductive high-aspect silver traces buried in hot-embossed polycarbonate films via the direct gravure doctoring method, Adv Powder Technol 32 (2021)764-770.
https://doi.org/10.1016/j.apt.2021.01.027

[12] C.C. Yang, X.Y. Tian, D.C. Li, Y. Cao, F. Zhao, C.Q. Shi, Influence of thermal processing conditions in 3D printing on the crystallinity and mechanical properties of PEEK material, J Mater Process Technol 248 (2017) 1-7.
http://dx.doi.org/10.1016/j.jmatprotec.2017.04.027

[13] T. A. Assiri, Time-space variable-order fractional nonlinear system of thermoelasticity: numerical treatment, Adv. Differ. Equ. 2020 (2020) 288.
https://doi.org/10.1186/s13662-020-02740-8

[14] R.F. Meng, D.S. Yin, S.Y. Lu, G.J. Xiang, Variable-order fractional constitutive model for the time-dependent mechanical behavior of polymers across the glass Transition, Eur. Phys. J. Plus 134 (2019) 376.
https://doi.org/10.1140/epjp/i2019-12767-x

[15] L.E.S. Ramirez, C.F.M. Coimbra, Variable order constitutive relation for viscoelasticity. Ann. Phys. 16 (2007) 543-552.
https://doi.org/10.1002/andp.200710246

[16] R. Meng, D. Yin, C.S. Drapaca. Variable-order fractional description of compression deformation of amorphous glassy polymers. Comput. Mech. 64 (2019) 163-171.
https://doi.org/10.1007/s00466-018-1663-9
[17] W. Cai, P. Wang. A nonlinear strain-dependent variable-order fractional model with applications to aluminum foams. Fractals 30 (2021) 2250021. https://doi.org/10.1142/S0218348X22500219

[18] ISO 604: 2002. Plastics-Determination of compressive properties, International Organization for Standardization: Geneva, Switzerland, (2002).

[19] G.Z. Voyiadjis, A. Samadi-Dooki. Constitutive modeling of large inelastic deformation of amorphous polymers: free volume and shear transformation zone dynamics. J. Appl. Phys. 119 (2016) 225104. https://doi.org/10.1063/1.4953355

[20] N. Girgis, B. Li, S. Akhtar, B. Courcelles, Experimental study of rate-dependent uniaxial compressive behaviors of two artificial frozen sandy clay soils. Cold Reg. Sci. Technol. 180 (2020) 103166. https://doi.org/10.1016/j.coldregions.2020.103166
[17] W. Cai, P. Wang. A nonlinear strain-dependent variable-order fractional model with applications to aluminum foams. Fractals 30 (2021) 2250021. https://doi.org/10.1142/S0218348X22500219

[18] ISO 604: 2002. Plastics-Determination of compressive properties, International Organization for Standardization: Geneva, Switzerland, (2002).

[19] G.Z. Voyiadjis, A. Samadi-Dooki. Constitutive modeling of large inelastic deformation of amorphous polymers: free volume and shear transformation zone dynamics. J. Appl. Phys. 119 (2016) 225104. https://doi.org/10.1063/1.4953355

[20] N. Girgis, B. Li, S. Akhtar, B. Courcelles, Experimental study of rate-dependent uniaxial compressive behaviors of two artificial frozen sandy clay soils. Cold Reg. Sci. Technol. 180 (2020) 103166. https://doi.org/10.1016/j.coldregions.2020.103166
# Influence of manufacturing defaults on the behavior of 3D printed lattice structures with a multiscale data-driven approach

COURT Clément¹,ᵃ*, MARENIC Eduard¹,ᵇ and PASSIEUX Jean-Charles¹,ᶜ

¹Institut Clément Ader (ICA), Université de Toulouse, INSA-ISAE-Mines Albi-UPS-CNRS, Toulouse, France

ᵃccourt@insa-toulouse.fr, ᵇmarenic@insa-toulouse.fr, ᶜpassieux@insa-toulouse.fr

**Keywords:** Lattice, Additive Manufacturing, Data-Driven, Multiscale, Digital Twins

**Abstract.** The inherent multiscale (MS) complexity of lattice materials necessitates specialized modeling techniques to predict their mechanical behavior accurately and efficiently. A combination of the multiscale finite element method (e.g. FE²), with a Data-Driven paradigm, leads to Multi Scale Data-Driven (MSDD) modeling procedure to simulate the behavior of these materials. On top of the inherent complexity of these materials, limitations of the Laser Powder Bed Fusion (LPBF) manufacturing process result in geometrical imperfections of the manufactured materials giving rise to so called *as-manufactured* configuration. Furthermore, theses imperfections have significant influence on the mechanical behavior and introduce biases and variance from expected behavior based on the modeling with initially designed lattice material, so called *as-designed* configuration. The proposed research is the first step in the reduction of the cost of MSDD approach by aligning the model's accuracy with the manufacturing capabilities. The aim is to exploit the inherent variance in *as-manufactured* lattice material and to minimizing the material database to the essential material state points. To that end, in this work, using X-ray tomography, we create digital twins of various *as-manufactured* aluminum lattice unit cells and perform a number of simulations. The goal is to get further understanding of the variance in mechanical behavior resulting from geometrical variations and defects. These results will be further used to enhance the efficiency of the MSDD approach with no compromise on predictive capabilities.

**Introduction**

**Lattice materials.** The rapid development of additive manufacturing in particular Laser Powder Bed Fusion (LPBF), enabled the design at the material scale in order to create, so called, lattice materials. These materials are a subset of cellular materials [1] and exhibit a high stiffness-over-mass ratio, making them particularly attractive for aerospace applications [2]. Lattice samples shown in this work were manufactured with Selective Laser Melting (SLM) [3] process on the machine SLM®125.

Manufacturing process depends on several parameters, namely laser displacement velocity, laser power, laser spot size, the powder constitutive material or its grain size. There is obviously a number of influential factors, some of them are random and their interaction and effects are still not completely understood. This further leads to differences between the *as-designed* and *as-manufactured* configurations in terms of relative volume and shape.

In [4], the authors enlighten different types of defects (Fig. 1): local cross-section variations of struts, waviness, porosity and even notches. In their work, the authors study the influence of these

<figure><img src="image_2060.png" /></figure>
# Influence of manufacturing defaults on the behavior of 3D printed lattice structures with a multiscale data-driven approach

COURT Clément¹,ᵃ*, MARENIC Eduard¹,ᵇ and PASSIEUX Jean-Charles¹,ᶜ

¹Institut Clément Ader (ICA), Université de Toulouse, INSA-ISAE-Mines Albi-UPS-CNRS, Toulouse, France

ᵃccourt@insa-toulouse.fr, ᵇmarenic@insa-toulouse.fr, ᶜpassieux@insa-toulouse.fr

**Keywords:** Lattice, Additive Manufacturing, Data-Driven, Multiscale, Digital Twins

**Abstract.** The inherent multiscale (MS) complexity of lattice materials necessitates specialized modeling techniques to predict their mechanical behavior accurately and efficiently. A combination of the multiscale finite element method (e.g. FE²), with a Data-Driven paradigm, leads to Multi Scale Data-Driven (MSDD) modeling procedure to simulate the behavior of these materials. On top of the inherent complexity of these materials, limitations of the Laser Powder Bed Fusion (LPBF) manufacturing process result in geometrical imperfections of the manufactured materials giving rise to so called *as-manufactured* configuration. Furthermore, theses imperfections have significant influence on the mechanical behavior and introduce biases and variance from expected behavior based on the modeling with initially designed lattice material, so called *as-designed* configuration. The proposed research is the first step in the reduction of the cost of MSDD approach by aligning the model's accuracy with the manufacturing capabilities. The aim is to exploit the inherent variance in *as-manufactured* lattice material and to minimizing the material database to the essential material state points. To that end, in this work, using X-ray tomography, we create digital twins of various *as-manufactured* aluminum lattice unit cells and perform a number of simulations. The goal is to get further understanding of the variance in mechanical behavior resulting from geometrical variations and defects. These results will be further used to enhance the efficiency of the MSDD approach with no compromise on predictive capabilities.

**Introduction**

**Lattice materials.** The rapid development of additive manufacturing in particular Laser Powder Bed Fusion (LPBF), enabled the design at the material scale in order to create, so called, lattice materials. These materials are a subset of cellular materials [1] and exhibit a high stiffness-over-mass ratio, making them particularly attractive for aerospace applications [2]. Lattice samples shown in this work were manufactured with Selective Laser Melting (SLM) [3] process on the machine SLM®125.

Manufacturing process depends on several parameters, namely laser displacement velocity, laser power, laser spot size, the powder constitutive material or its grain size. There is obviously a number of influential factors, some of them are random and their interaction and effects are still not completely understood. This further leads to differences between the *as-designed* and *as-manufactured* configurations in terms of relative volume and shape.

In [4], the authors enlighten different types of defects (Fig. 1): local cross-section variations of struts, waviness, porosity and even notches. In their work, the authors study the influence of these

<figure><img src="image_2052.png" /></figure>
<figure><img src="image_2061.png" /><figcaption>Fig. 1: Scanning Electron Microscope images of rhombicuboctahedron (A) and regular octet (B) lattices enlightening manufacturing defects [4].</figcaption></figure>

defects on the mechanical behavior of lattice material samples. They classify the defects and introduce a statistical distribution in a beam-based finite element model.

On the contrary, in this work we avoid the defects classification and we focus on creating the three-dimensional (3D) finite element (FE) models. These models are built from the 3D images obtained from the X-ray tomography of the real manufactured lattice samples.

**Multiscale methods for lattice materials.** Lattice materials represent complex cellular structures composed of repeating unit cells. If we add on top of this rather complex structure the details of the *as-manufactured* configuration (geometry variations described above), prohibitive computational costs render the direct numerical simulation virtually impossible. This type of problems where two distinguished length scales exist – local unit cells at the mesoscopic scale and global structure at the macroscopic scale – are often modeled using MS approach, e.g. FE² [5]. This method solves the macroscopic boundary value problem of the global structure and the local problem of the unit cell considered as a Representative Volume Element (RVE), as illustrated in Fig. 2.

<figure><img src="image_2062.png" /><figcaption>Fig. 2: Multiscale mechanical problem illustration</figcaption></figure>

Contrary to classical FE methods, the macroscopic problem is not driven by a material constitutive law, but by the homogenized results of mesoscopic scales problems of the RVE. Typically, the effective material behavior at the macroscopic scale is obtained by homogenizing the stress field of the RVE loaded with a given strain state, for each integration point on the macroscale. Although this kind of procedure can be parallelized due to independent mesoscopic problems, one needs to run a generally non-linear simulation for each integration points and each load increment of the macroscopic problems. Moreover, linear perturbation evaluations on the RVE are needed to obtain the tangent operator for the macroscopic problem. To reduce the number of simulations on RVE it is possible to reuse the previously calculated solutions using model reduction techniques or machine learning [6]. In the following work we will use an alternative approach based on Data-Driven simulations.
<figure><img src="image_2053.png" /><figcaption>Fig. 1: Scanning Electron Microscope images of rhombicuboctahedron (A) and regular octet (B) lattices enlightening manufacturing defects [4].</figcaption></figure>

defects on the mechanical behavior of lattice material samples. They classify the defects and introduce a statistical distribution in a beam-based finite element model.

On the contrary, in this work we avoid the defects classification and we focus on creating the three-dimensional (3D) finite element (FE) models. These models are built from the 3D images obtained from the X-ray tomography of the real manufactured lattice samples.

**Multiscale methods for lattice materials.** Lattice materials represent complex cellular structures composed of repeating unit cells. If we add on top of this rather complex structure the details of the *as-manufactured* configuration (geometry variations described above), prohibitive computational costs render the direct numerical simulation virtually impossible. This type of problems where two distinguished length scales exist – local unit cells at the mesoscopic scale and global structure at the macroscopic scale – are often modeled using MS approach, e.g. FE² [5]. This method solves the macroscopic boundary value problem of the global structure and the local problem of the unit cell considered as a Representative Volume Element (RVE), as illustrated in Fig. 2.

<figure><img src="image_2054.png" /><figcaption>Fig. 2: Multiscale mechanical problem illustration</figcaption></figure>

Contrary to classical FE methods, the macroscopic problem is not driven by a material constitutive law, but by the homogenized results of mesoscopic scales problems of the RVE. Typically, the effective material behavior at the macroscopic scale is obtained by homogenizing the stress field of the RVE loaded with a given strain state, for each integration point on the macroscale. Although this kind of procedure can be parallelized due to independent mesoscopic problems, one needs to run a generally non-linear simulation for each integration points and each load increment of the macroscopic problems. Moreover, linear perturbation evaluations on the RVE are needed to obtain the tangent operator for the macroscopic problem. To reduce the number of simulations on RVE it is possible to reuse the previously calculated solutions using model reduction techniques or machine learning [6]. In the following work we will use an alternative approach based on Data-Driven simulations.
**Data-Driven Computational Mechanics.** In 2015, Kirchdoerfer and Ortiz [7] introduced a formulation of the boundary value problem where the material constitutive law model is replaced by data. In this context one still needs to verify the kinematic compatibility condition (Eq. 1) and the static equilibrium condition (Eq. 2).

$$ \varepsilon = 1/2 (\nabla \mathbf{q} + (\nabla \mathbf{q})^T) \qquad (1) $$

$$ \mathrm{div}(\boldsymbol{\sigma}) + \mathbf{f}_v = \mathbf{0} \qquad (2) $$

where $\mathbf{q}$, $\varepsilon$, $\boldsymbol{\sigma}$, $\mathbf{f}_v$ are displacements field, strain field, stress field and external forces, respectively. However, the material behavior is now encoded as a cloud of points given as couples of strain and stress tensors ($\varepsilon$, $\boldsymbol{\sigma}$). The space where the data points are defined is called the *phase space* and is usually noted $Z$, it has 12 dimensions (6 for the strain components and 6 for the stress components, both in Voigt notation). Solution to the boundary value problem is obtained by minimizing a distance $d$ in $Z$, between the admissible mechanical states from the set called $C$ – which verify Eq. 1 and Eq. 2 – and the states from the material database called $D$. To that end, we use an energy norm, that is, the squared distance between two points of the phase space ($\varepsilon_c$, $\boldsymbol{\sigma}_c$) $\in C$ and ($\varepsilon_d$, $\boldsymbol{\sigma}_d$) $\in D$ is given by

$$ d^2 = 1/2 ((\varepsilon_c - \varepsilon_d) : \mathcal{C} : (\varepsilon_c - \varepsilon_d)) + 1/2 ((\boldsymbol{\sigma}_c - \boldsymbol{\sigma}_d) : \mathcal{C}^{-1} : (\boldsymbol{\sigma}_c - \boldsymbol{\sigma}_d)) \qquad (3) $$

where $\mathcal{C}$ is a purely numerical Hooke-like operator.

In this work we turn to the combination of FE² MS strategy with the Data-Driven paradigm [8], leading to MSDD method. The goal is to substitute the systematic online simulations on the RVE by the nearest neighbor search in terms of the distance defined in Eq. 3 and within the previously created material database. This makes the MSDD approach a good candidate to reduce the costs of the FE² method. We note that the material database is in the MSDD context usually created before in the offline phase, however it is still possible to add new data points in the material database by simulation of the RVE if necessary [9]. The principle obstacle within MSDD solver is the nearest neighbor search step which can be expensive if the database is large. Some of the standard efficient searching methods are readily available to accelerate this step, such as tree searching (kd-trees, k-means trees, etc.), see also some other techniques proposed in [10].

In our work we would like to explore the possibility of taking into account the variance inherent to the manufacturing process in order to reduce the size of the material database and avoid expensive and over-sampled material databases. More precisely, the overall goal is to propose a method to use the knowledge of the manufacturing uncertainties to filter an over-sampled database while keeping the same predictive capabilities at the macroscopic scale. In this work the first step considers exploiting X-ray tomography images and creation of digital twins of various as-manufactured aluminum lattice unit cells. With the accurate 3D description in hands, we tend to perform a number of RVE simulations with as-manufactured configuration to get further understanding of the variance in mechanical behavior resulting from geometrical variations and defects. These results will be used in our further work to enhance the efficiency of the MSDD approach.
**Data-Driven Computational Mechanics.** In 2015, Kirchdoerfer and Ortiz [7] introduced a formulation of the boundary value problem where the material constitutive law model is replaced by data. In this context one still needs to verify the kinematic compatibility condition (Eq. 1) and the static equilibrium condition (Eq. 2).

$$ \varepsilon = 1/2 (\nabla \mathbf{q} + (\nabla \mathbf{q})^T) \qquad (1) $$

$$ \mathrm{div}(\boldsymbol{\sigma}) + \mathbf{f}_v = \mathbf{0} \qquad (2) $$

where $\mathbf{q}$, $\varepsilon$, $\boldsymbol{\sigma}$, $\mathbf{f}_v$ are displacements field, strain field, stress field and external forces, respectively. However, the material behavior is now encoded as a cloud of points given as couples of strain and stress tensors ($\varepsilon$, $\boldsymbol{\sigma}$). The space where the data points are defined is called the *phase space* and is usually noted $Z$, it has 12 dimensions (6 for the strain components and 6 for the stress components, both in Voigt notation). Solution to the boundary value problem is obtained by minimizing a distance $d$ in $Z$, between the admissible mechanical states from the set called $C$ – which verify Eq. 1 and Eq. 2 – and the states from the material database called $D$. To that end, we use an energy norm, that is, the squared distance between two points of the phase space ($\varepsilon_c$, $\boldsymbol{\sigma}_c$) $\in C$ and ($\varepsilon_d$, $\boldsymbol{\sigma}_d$) $\in D$ is given by

$$ d^2 = 1/2 ((\varepsilon_c - \varepsilon_d) : \mathcal{C} : (\varepsilon_c - \varepsilon_d)) + 1/2 ((\boldsymbol{\sigma}_c - \boldsymbol{\sigma}_d) : \mathcal{C}^{-1} : (\boldsymbol{\sigma}_c - \boldsymbol{\sigma}_d)) \qquad (3) $$

where $\mathcal{C}$ is a purely numerical Hooke-like operator.

In this work we turn to the combination of FE² MS strategy with the Data-Driven paradigm [8], leading to MSDD method. The goal is to substitute the systematic online simulations on the RVE by the nearest neighbor search in terms of the distance defined in Eq. 3 and within the previously created material database. This makes the MSDD approach a good candidate to reduce the costs of the FE² method. We note that the material database is in the MSDD context usually created before in the offline phase, however it is still possible to add new data points in the material database by simulation of the RVE if necessary [9]. The principle obstacle within MSDD solver is the nearest neighbor search step which can be expensive if the database is large. Some of the standard efficient searching methods are readily available to accelerate this step, such as tree searching (kd-trees, k-means trees, etc.), see also some other techniques proposed in [10].

In our work we would like to explore the possibility of taking into account the variance inherent to the manufacturing process in order to reduce the size of the material database and avoid expensive and over-sampled material databases. More precisely, the overall goal is to propose a method to use the knowledge of the manufacturing uncertainties to filter an over-sampled database while keeping the same predictive capabilities at the macroscopic scale. In this work the first step considers exploiting X-ray tomography images and creation of digital twins of various as-manufactured aluminum lattice unit cells. With the accurate 3D description in hands, we tend to perform a number of RVE simulations with as-manufactured configuration to get further understanding of the variance in mechanical behavior resulting from geometrical variations and defects. These results will be used in our further work to enhance the efficiency of the MSDD approach.
## Methods

**Manufacturing and 3D imagery.** For this work, lattice sample (Fig. 3b) was manufactured with Selective Laser Melting (SLM) process on the machine SLM®125 and using an aluminum alloy powder AlSi7Mg0,6 according to the norm DIN EN 1706 / EN AC-42200. A unit cell in this sample is CFCC type, hybrid cell composed of the Face Centered Cubic and Center cross-like reinforcement (Fig. 3a). The *as-designed* unit cells have a side length of 4.069 mm and a relative volume of 16.81% (Fig. 3a). The use of X-ray tomography and 3D imagery with *18µm voxel size enabled* to reconstruct the surfaces of the *as-manufactured* unit cells (Fig. 3c).

<figure><img src="image_2063.png" /><figcaption>Fig. 3: Sample and unit cells illustrations.</figcaption></figure>

**Creation of RVEs from digital twins.** Digital twins are numerical models based on real structures. Firstly, the unit cells are extracted from the whole sample 3D image and treated separately. As can be seen from Fig. 3c reconstructed surfaces of the manufactured unit cell with 18µm voxel size are rough and too rich for intended FE modeling. To be able to create a 3D finite element mesh, it was necessary to apply a filter on the images to artificially smoothen the surfaces. This filtering must be as low as possible to minimize the modification of the extracted cell's geometry. After the images are smoothed, a *marching cubes* algorithm was used to reconstruct the outer surfaces. Then another meshing algorithm is used to fill the volume with tetrahedrons. The process of digital twin generation is illustrated in Fig. 4.

<figure><img src="image_2064.png" /><figcaption>Fig. 4: Digital twin RVE creation process illustration.</figcaption></figure>
## Methods

**Manufacturing and 3D imagery.** For this work, lattice sample (Fig. 3b) was manufactured with Selective Laser Melting (SLM) process on the machine SLM®125 and using an aluminum alloy powder AlSi7Mg0,6 according to the norm DIN EN 1706 / EN AC-42200. A unit cell in this sample is CFCC type, hybrid cell composed of the Face Centered Cubic and Center cross-like reinforcement (Fig. 3a). The *as-designed* unit cells have a side length of 4.069 mm and a relative volume of 16.81% (Fig. 3a). The use of X-ray tomography and 3D imagery with *18µm voxel size enabled* to reconstruct the surfaces of the *as-manufactured* unit cells (Fig. 3c).

<figure><img src="image_2055.png" /><figcaption>Fig. 3: Sample and unit cells illustrations.</figcaption></figure>

**Creation of RVEs from digital twins.** Digital twins are numerical models based on real structures. Firstly, the unit cells are extracted from the whole sample 3D image and treated separately. As can be seen from Fig. 3c reconstructed surfaces of the manufactured unit cell with 18µm voxel size are rough and too rich for intended FE modeling. To be able to create a 3D finite element mesh, it was necessary to apply a filter on the images to artificially smoothen the surfaces. This filtering must be as low as possible to minimize the modification of the extracted cell's geometry. After the images are smoothed, a *marching cubes* algorithm was used to reconstruct the outer surfaces. Then another meshing algorithm is used to fill the volume with tetrahedrons. The process of digital twin generation is illustrated in Fig. 4.

<figure><img src="image_2056.png" /><figcaption>Fig. 4: Digital twin RVE creation process illustration.</figcaption></figure>
In this work, six different cells were extracted from the same lattice sample. Using the previously described process, digital twins of our RVE are created, three of them are shown in Fig. 5. In addition, one *as-designed* RVE has been created for comparison purpose. The relative volumes of the RVEs are shown in Table 1. The RVEs FE models were created using DS Abaqus software using linear tetrahedral elements (C3D4). Each RVE is composed of approximately 1.6 * 10⁵ elements. The material behavior for the RVE simulation was modeled using standard elastic-plastic law for aluminum with isotropic hardening.

<figure><img src="image_2065.png" /><figcaption>Fig. 5: Three different as-manufactured RVEs from the same lattice sample.</figcaption></figure>

<table><caption>Table 1: RVEs relative volumes.</caption><thead><tr><td>RVE</td><td>Relative volume (%)</td></tr></thead><tbody><tr><td>As-designed</td><td>16.58</td></tr><tr><td>As-manufactured #1</td><td>19.39</td></tr><tr><td>As-manufactured #2</td><td>19.46</td></tr><tr><td>As-manufactured #3</td><td>19.25</td></tr><tr><td>As-manufactured #4</td><td>19.56</td></tr><tr><td>As-manufactured #5</td><td>19.54</td></tr><tr><td>As-manufactured #6</td><td>18.12</td></tr><tr><td>As-manufactured mean</td><td>19.22</td></tr></tbody></table>

In this work the finite element method is used for the simulation the RVE problem. We note however that other methods exist, such as, Fast Fourier Transform, see [11]. These methods can be based directly on images without the necessity of building a mesh making them particularly efficient to model RVEs with large voids. Exploring this alternative is out of scope of this work but will be explored in future study.

**Computational Homogenization.** Localization of the macroscopic strain state on the RVE is performed by applying boundary conditions on its external surfaces $\Gamma_D$. The displacements field $q_\Gamma$ of the nodes belonging to a face of the RVE's enveloping cube is constrained and driven to ensure a mean value of the RVE strain $\varepsilon$. In the literature, three different types of boundary conditions (ensuring Hill-Mandel energy condition and allowing to impose an average deformation are used, namely, i) kinematic uniform, ii) uniform traction and iii) periodic boundary conditions.

The kinematic uniform boundary conditions are easy to implement and are used in this work. Here the displacements of the boundary nodes are driven by their position (Eq. 4). To that end, the displacements of the boundary nodes are driven by $\varepsilon$ and by their position $X_\Gamma$ as

$$q_{\Gamma} = \varepsilon \cdot X_{\Gamma} \qquad (4)$$
In this work, six different cells were extracted from the same lattice sample. Using the previously described process, digital twins of our RVE are created, three of them are shown in Fig. 5. In addition, one *as-designed* RVE has been created for comparison purpose. The relative volumes of the RVEs are shown in Table 1. The RVEs FE models were created using DS Abaqus software using linear tetrahedral elements (C3D4). Each RVE is composed of approximately 1.6 * 10⁵ elements. The material behavior for the RVE simulation was modeled using standard elastic-plastic law for aluminum with isotropic hardening.

<figure><img src="image_2057.png" /><figcaption>Fig. 5: Three different as-manufactured RVEs from the same lattice sample.</figcaption></figure>

<table><caption>Table 1: RVEs relative volumes.</caption><thead><tr><td>RVE</td><td>Relative volume (%)</td></tr></thead><tbody><tr><td>As-designed</td><td>16.58</td></tr><tr><td>As-manufactured #1</td><td>19.39</td></tr><tr><td>As-manufactured #2</td><td>19.46</td></tr><tr><td>As-manufactured #3</td><td>19.25</td></tr><tr><td>As-manufactured #4</td><td>19.56</td></tr><tr><td>As-manufactured #5</td><td>19.54</td></tr><tr><td>As-manufactured #6</td><td>18.12</td></tr><tr><td>As-manufactured mean</td><td>19.22</td></tr></tbody></table>

In this work the finite element method is used for the simulation the RVE problem. We note however that other methods exist, such as, Fast Fourier Transform, see [11]. These methods can be based directly on images without the necessity of building a mesh making them particularly efficient to model RVEs with large voids. Exploring this alternative is out of scope of this work but will be explored in future study.

**Computational Homogenization.** Localization of the macroscopic strain state on the RVE is performed by applying boundary conditions on its external surfaces $\Gamma_D$. The displacements field $q_\Gamma$ of the nodes belonging to a face of the RVE's enveloping cube is constrained and driven to ensure a mean value of the RVE strain $\varepsilon$. In the literature, three different types of boundary conditions (ensuring Hill-Mandel energy condition and allowing to impose an average deformation are used, namely, i) kinematic uniform, ii) uniform traction and iii) periodic boundary conditions.

The kinematic uniform boundary conditions are easy to implement and are used in this work. Here the displacements of the boundary nodes are driven by their position (Eq. 4). To that end, the displacements of the boundary nodes are driven by $\varepsilon$ and by their position $X_\Gamma$ as

$$q_{\Gamma} = \varepsilon \cdot X_{\Gamma} \qquad (4)$$
In order to describe the effective response of our RVEs and use it in MSDD approach we need to perform so-called meso-to-macro transition (Fig. 2). In our context, this simply boils down to the stress homogenization, that is, the macroscopic stress state σ becomes the volume average of the stress field σm [12] over the RVE volume (Eq. 5).

$$ \sigma = \langle \sigma_m \rangle \quad \text{with} \quad \langle \cdot \rangle = \frac{1}{v_m} \int_{\Omega_m} \cdot dv \qquad (5) $$

For each strain state ε applied to the RVE, homogenized stresses are computed at 20 equally spaced increments. We note here that the input strain state ε is defined in 6-dimensional strain space. In order to limit the number of evaluation points exploring at the same time the strain space as much as possible, careful design of experiments is needed [13]. Thus, we exploit here the hypercube sampling and we choose to sample the strain space in three dimensions associated to the principal components of the strain tensor εI, εII, εIII.

<figure><img src="image_2066.png" /><figcaption>Fig. 6: Overview of used procedure to sample the 6D strain space.</figcaption></figure>

The proposed sampling for this first attempt is made of 13 423 points of the strain space.

## Results

As this work is still in progress, the performed simulations created a database that is currently composed of 5517 points related to 7 different RVEs: 6 related to as-manufactured and 1 to as-designed configuration. We recall that each of the material points is defined in 12-dimensional phase space Z with couples of strain and stress tensors (ε, σ). The evaluations and homogenization procedures are performed sequentially on a laptop Intel® Core™ i7-10510U CPU @ 1.80GHz with 32Gb memory. Projections on two of the 12 dimensions of Z enlighten the differences in mechanical response between the RVEs. Points resulting from unidimensional loading cases (only one component of the loading strain is non-zero) are presented in Fig. 7.
In order to describe the effective response of our RVEs and use it in MSDD approach we need to perform so-called meso-to-macro transition (Fig. 2). In our context, this simply boils down to the stress homogenization, that is, the macroscopic stress state $\sigma$ becomes the volume average of the stress field $\sigma_m$ [12] over the RVE volume (Eq. 5).

$$ \sigma = \langle \sigma_m \rangle \quad \text{with} \quad \langle \cdot \rangle = \frac{1}{v_m} \int_{\Omega_m} \cdot dv \qquad (5) $$

For each strain state $\varepsilon$ applied to the RVE, homogenized stresses are computed at 20 equally spaced increments. We note here that the input strain state $\varepsilon$ is defined in 6-dimensional strain space. In order to limit the number of evaluation points exploring at the same time the strain space as much as possible, careful design of experiments is needed [13]. Thus, we exploit here the hypercube sampling and we choose to sample the strain space in three dimensions associated to the principal components of the strain tensor $\varepsilon_I, \varepsilon_{II}, \varepsilon_{III}$.

<figure><img src="image_2058.png" /><figcaption>Fig. 6: Overview of used procedure to sample the 6D strain space.</figcaption></figure>

The proposed sampling for this first attempt is made of 13 423 points of the strain space.

## Results

As this work is still in progress, the performed simulations created a database that is currently composed of 5517 points related to 7 different RVEs: 6 related to as-manufactured and 1 to as-designed configuration. We recall that each of the material points is defined in 12-dimensional phase space Z with couples of strain and stress tensors ($\varepsilon, \sigma$). The evaluations and homogenization procedures are performed sequentially on a laptop Intel® Core™ i7-10510U CPU @ 1.80GHz with 32Gb memory. Projections on two of the 12 dimensions of Z enlighten the differences in mechanical response between the RVEs. Points resulting from unidimensional loading cases (only one component of the loading strain is non-zero) are presented in Fig. 7.
<figure><img src="image_2067.png" /><figcaption>Fig. 7: Extract of data created by simulations on the 7 RVEs, showing the projected stress and strain values on the dimension of the phase space that corresponds to the loaded strain component. The results for the as-manufactured configuration are displayed with error bars showing the variability (the mean value and the standard deviation) of generated data from different RVE realizations.</figcaption></figure>

As can be clearly seen from the presented examples, different geometries of the RVEs lead to variation in homogenized mechanical responses. The first noticeable difference is between the *as-designed* configuration (blue points in Fig. 7) and the average of the *as-manufactured* configurations. This systematic bias is associated both with the manufacturing process and the creation of the digital twins. Not surprisingly, different cells within the same lattice sample exhibit different defects, and therefore, do not demonstrate the same behavior. Consequently, for the *as-manufactured* RVEs, there is a random variability in the homogenized stress states. In Fig. 7, error bar type of plot is used for the graphical representation of the variability of the data. We note in passing that observed variability increases with the magnitude of the loading as shown in Fig. 7.

## Discussion

The systematic bias in the stress response can be partly explained by the difference in relative volume. As presented in Table 1, the average relative volume of the *as-manufactured* configurations is increased by 15.9% compared to that of the *as-designed* configuration.

As a correlation between relative volume and mechanical response is expected, in forthcoming work, we will delve deeper into its influence and the origin of its deviation from the *as-designed* value. However, this difference can come either from a systematic bias related to the manufacturing process or from the digital twin creation procedure. The relative volume of the digital twins is influenced by the parameters of their creation process namely the cell extraction, filtering, surface reconstruction and meshing. One of the examples of significant parameter is the threshold of the marching cubes algorithm used to reconstruct the outer surfaces which impacts the relative density and the overall homogenized response. As mentioned previously, our perspective is to profit from the variability of responses from the *as-manufactured* configurations and to propose a database ‘filtering’ method. This method will consider the fact that variability is not identical in all loading directions and for different amplitudes of the loading.
<figure><img src="image_2059.png" /><figcaption>Fig. 7: Extract of data created by simulations on the 7 RVEs, showing the projected stress and strain values on the dimension of the phase space that corresponds to the loaded strain component. The results for the as-manufactured configuration are displayed with error bars showing the variability (the mean value and the standard deviation) of generated data from different RVE realizations.</figcaption></figure>

As can be clearly seen from the presented examples, different geometries of the RVEs lead to variation in homogenized mechanical responses. The first noticeable difference is between the *as-designed* configuration (blue points in Fig. 7) and the average of the *as-manufactured* configurations. This systematic bias is associated both with the manufacturing process and the creation of the digital twins. Not surprisingly, different cells within the same lattice sample exhibit different defects, and therefore, do not demonstrate the same behavior. Consequently, for the *as-manufactured* RVEs, there is a random variability in the homogenized stress states. In Fig. 7, error bar type of plot is used for the graphical representation of the variability of the data. We note in passing that observed variability increases with the magnitude of the loading as shown in Fig. 7.

## Discussion

The systematic bias in the stress response can be partly explained by the difference in relative volume. As presented in Table 1, the average relative volume of the *as-manufactured* configurations is increased by 15.9% compared to that of the *as-designed* configuration.

As a correlation between relative volume and mechanical response is expected, in forthcoming work, we will delve deeper into its influence and the origin of its deviation from the *as-designed* value. However, this difference can come either from a systematic bias related to the manufacturing process or from the digital twin creation procedure. The relative volume of the digital twins is influenced by the parameters of their creation process namely the cell extraction, filtering, surface reconstruction and meshing. One of the examples of significant parameter is the threshold of the marching cubes algorithm used to reconstruct the outer surfaces which impacts the relative density and the overall homogenized response. As mentioned previously, our perspective is to profit from the variability of responses from the *as-manufactured* configurations and to propose a database ‘filtering’ method. This method will consider the fact that variability is not identical in all loading directions and for different amplitudes of the loading.
References

[1] L. J. Gibson et M. F. Ashby, *Cellular Solids: Structure and Properties*. Cambridge University Press, 1997.

[2] B. Blakey-Milner et al., Metal additive manufacturing in aerospace: A review, Materials & Design, 209 (2021) 110008. https://doi.org/10.1016/j.matdes.2021.110008

[3] L. Riva, P. S. Ginestra, E. Ceretti, Mechanical characterization and properties of laser-based powder bed-fused lattice structures: a review, Int J Adv Manuf Technol, 113 (2021) 649-671, doi: 10.1007/s00170-021-06631-4

[4] L. Liu, P. Kamm, F. García-Moreno, J. Banhart, D. Pasini, Elastic and failure response of imperfect three-dimensional metallic lattices: the role of geometric defects induced by Selective Laser Melting, Journal of the Mechanics and Physics of Solids, 107 (2017) 160-184. https://doi.org/10.1016/j.jmps.2017.07.003

[5] F. Feyel, A multilevel finite element method (FE2) to describe the response of highly nonlinear structures using generalized continua, Computer Methods in Applied Mechanics and Engineering, 192 (2003) 3233-3244. https://doi.org/10.1016/S0045-7825(03)00348-7

[6] B. A. Le, J. Yvonnet, et Q.-C. He, Computational homogenization of nonlinear elastic materials using neural networks, International Journal for Numerical Methods in Engineering, 104 (2015) 1061-1084, doi: 10.1002/nme.1586

[7] T. Kirchdoerfer et M. Ortiz, Data-driven computational mechanics, Computer Methods in Applied Mechanics and Engineering, 304 (2016) 81-101. https://doi.org/10.1016/j.cma.2016.02.001

[8] R. Xu et al., Data-driven multiscale finite element method: From concurrence to separation, Computer Methods in Applied Mechanics and Engineering, 363 (2020), 112893. https://doi.org/10.1016/j.cma.2020.112893

[9] A. Platzer, A. Leygue, L. Stainier, Stratégie adaptative de calcul multiéchelle piloté par les données, 15ème Colloque National en Calcul des Structures, mai 2022.

[10] R. Eggersmann, L. Stainier, M. Ortiz, et S. Reese, Model-free data-driven computational mechanics enhanced by tensor voting, Computer Methods in Applied Mechanics and Engineering, 373 (2021) 113499. https://doi.org/10.1016/j.cma.2020.113499

[11] H. Moulinec, P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, 157 (1998) 69-94. https://doi.org/10.1016/S0045-7825(97)00218-1

[12] E. Sanchez-Palencia, Homogenization in mechanics: A survey of solved and open problems, Rend. Sem. Mat. Univers. Politecn. Torino, 1986.

[13] M. A. Bessa et al., A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality, Computer Methods in Applied Mechanics and Engineering, 320 (2017) 633-667. https://doi.org/10.1016/j.cma.2017.03.037
## References

[1] L. J. Gibson et M. F. Ashby, *Cellular Solids: Structure and Properties*. Cambridge University Press, 1997.

[2] B. Blakey-Milner et al., Metal additive manufacturing in aerospace: A review, Materials & Design, 209 (2021) 110008. https://doi.org/10.1016/j.matdes.2021.110008

[3] L. Riva, P. S. Ginestra, E. Ceretti, Mechanical characterization and properties of laser-based powder bed-fused lattice structures: a review, Int J Adv Manuf Technol, 113 (2021) 649-671, doi: 10.1007/s00170-021-06631-4

[4] L. Liu, P. Kamm, F. García-Moreno, J. Banhart, D. Pasini, Elastic and failure response of imperfect three-dimensional metallic lattices: the role of geometric defects induced by Selective Laser Melting, Journal of the Mechanics and Physics of Solids, 107 (2017) 160-184. https://doi.org/10.1016/j.jmps.2017.07.003

[5] F. Feyel, A multilevel finite element method (FE2) to describe the response of highly nonlinear structures using generalized continua, Computer Methods in Applied Mechanics and Engineering, 192 (2003) 3233-3244. https://doi.org/10.1016/S0045-7825(03)00348-7

[6] B. A. Le, J. Yvonnet, et Q.-C. He, Computational homogenization of nonlinear elastic materials using neural networks, International Journal for Numerical Methods in Engineering, 104 (2015) 1061-1084, doi: 10.1002/nme.1586

[7] T. Kirchdoerfer et M. Ortiz, Data-driven computational mechanics, Computer Methods in Applied Mechanics and Engineering, 304 (2016) 81-101. https://doi.org/10.1016/j.cma.2016.02.001

[8] R. Xu et al., Data-driven multiscale finite element method: From concurrence to separation, Computer Methods in Applied Mechanics and Engineering, 363 (2020), 112893. https://doi.org/10.1016/j.cma.2020.112893

[9] A. Platzer, A. Leygue, L. Stainier, Stratégie adaptative de calcul multiéchelle piloté par les données, 15ème Colloque National en Calcul des Structures, mai 2022.

[10] R. Eggersmann, L. Stainier, M. Ortiz, et S. Reese, Model-free data-driven computational mechanics enhanced by tensor voting, Computer Methods in Applied Mechanics and Engineering, 373 (2021) 113499. https://doi.org/10.1016/j.cma.2020.113499

[11] H. Moulinec, P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, 157 (1998) 69-94. https://doi.org/10.1016/S0045-7825(97)00218-1

[12] E. Sanchez-Palencia, Homogenization in mechanics: A survey of solved and open problems, Rend. Sem. Mat. Univers. Politecn. Torino, 1986.

[13] M. A. Bessa et al., A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality, Computer Methods in Applied Mechanics and Engineering, 320 (2017) 633-667. https://doi.org/10.1016/j.cma.2017.03.037
# Modelling ductile fracture in an Al alloy with crystal plasticity models

KHADYKO Mikhail¹,²,*, FRODAL Bjørn Håkon³ and HOPPERSTAD Odd Sture²,⁴

¹UiT The Arctic University of Norway, Campus Narvik, Department of Building, Energy and Material Technology, Lodve Langesgate 2, 8514 Narvik, Norway

²Centre for Advanced Structural Analysis (CASA), NTNU, Trondheim, Norway

³Waves AS, Slettaveien 21, 1553 Son, Norway

⁴Structural Impact Laboratory (SIMLab), Department of Structural Engineering, NTNU –
Norwegian University of Science and Technology, Trondheim, Norway

ªmikhail.khadyko@uit.no

**Keywords:** Ductile Fracture, Crystal Plasticity, Finite Element Method, Constituent Particles, Al Alloys

**Abstract.** Crystal plasticity models enhanced with coupled or uncoupled damage and fracture criteria give an opportunity to account for the role of microstructure in ductile fracture, most directly representing the local variations of stress and strain fields inside and between the grains, voids and particles. Some computationally efficient crystal plasticity, damage and fracture models have recently been developed and applied to some cases of polycrystalline fracture. Such models allow to investigate in a direct way the effects of, e.g., shear bands, larger voids, particles, free surfaces and load direction on the development of damage and fracture. The cast and homogenized Al1.2Mn alloy investigated previously is used here as a basis for simulations. The alloy has an equiaxed grain structure with no texture and contains a population of larger particles and a population of dispersoids. The grain structure and the large particles are modelled directly in the finite element model, while the effect of dispersoids is represented by the damage and fracture part of the single crystal plasticity model. The study investigates the effect of different model parameters and features on the global and local behaviour of the material during localization and fracture, in light of available experimental data.

**Introduction**

One of the important directions of research in material science is establishing the connection between the macroscopic properties of metal alloys and their microstructure. The multiscale modelling approach considers different length scales from the atomic lattice to crystalline grains, macroscopic structures and components, and aims to predict the behaviour at the larger scale based on the properties at the smaller scale. While the prediction of structural properties based on the fundamental physical laws acting on the atomic lattice level is still out of reach, some more limited studies of the underlying microscopic mechanisms of the macroscopic phenomena may be useful for both predicting and explaining the behaviour of existing materials and for designing new materials.

The present study investigates some aspects of the ductile fracture process in crystalline materials on the scale of the grains and constituent particles, using a combination of well-established methods and theories. The first component to this investigation is the crystal plasticity (CP) theory. The crystal plasticity theory describes a type of plastic deformation in metals as a combination of slips on a set of slip systems [1]. The plastic deformation in polycrystals is heterogeneous on the grain level. This heterogeneity has been experimentally observed, using the digital image correlation (DIC) method [2, 3]. While the slip bands observed in DIC studies [2] cannot yet be reproduced by the CP model, the strain concentrations caused by variation in

<figure><img src="image_2068.png" /></figure>
# Modelling ductile fracture in an Al alloy with crystal plasticity models

KHADYKO Mikhail¹,²,*, FRODAL Bjørn Håkon³ and HOPPERSTAD Odd Sture²,⁴

¹UiT The Arctic University of Norway, Campus Narvik, Department of Building, Energy and Material Technology, Lodve Langesgate 2, 8514 Narvik, Norway

²Centre for Advanced Structural Analysis (CASA), NTNU, Trondheim, Norway

³Waves AS, Slettaveien 21, 1553 Son, Norway

⁴Structural Impact Laboratory (SIMLab), Department of Structural Engineering, NTNU –
Norwegian University of Science and Technology, Trondheim, Norway

ªmikhail.khadyko@uit.no

**Keywords:** Ductile Fracture, Crystal Plasticity, Finite Element Method, Constituent Particles, Al Alloys

**Abstract.** Crystal plasticity models enhanced with coupled or uncoupled damage and fracture criteria give an opportunity to account for the role of microstructure in ductile fracture, most directly representing the local variations of stress and strain fields inside and between the grains, voids and particles. Some computationally efficient crystal plasticity, damage and fracture models have recently been developed and applied to some cases of polycrystalline fracture. Such models allow to investigate in a direct way the effects of, e.g., shear bands, larger voids, particles, free surfaces and load direction on the development of damage and fracture. The cast and homogenized Al1.2Mn alloy investigated previously is used here as a basis for simulations. The alloy has an equiaxed grain structure with no texture and contains a population of larger particles and a population of dispersoids. The grain structure and the large particles are modelled directly in the finite element model, while the effect of dispersoids is represented by the damage and fracture part of the single crystal plasticity model. The study investigates the effect of different model parameters and features on the global and local behaviour of the material during localization and fracture, in light of available experimental data.

**Introduction**

One of the important directions of research in material science is establishing the connection between the macroscopic properties of metal alloys and their microstructure. The multiscale modelling approach considers different length scales from the atomic lattice to crystalline grains, macroscopic structures and components, and aims to predict the behaviour at the larger scale based on the properties at the smaller scale. While the prediction of structural properties based on the fundamental physical laws acting on the atomic lattice level is still out of reach, some more limited studies of the underlying microscopic mechanisms of the macroscopic phenomena may be useful for both predicting and explaining the behaviour of existing materials and for designing new materials.

The present study investigates some aspects of the ductile fracture process in crystalline materials on the scale of the grains and constituent particles, using a combination of well-established methods and theories. The first component to this investigation is the crystal plasticity (CP) theory. The crystal plasticity theory describes a type of plastic deformation in metals as a combination of slips on a set of slip systems [1]. The plastic deformation in polycrystals is heterogeneous on the grain level. This heterogeneity has been experimentally observed, using the digital image correlation (DIC) method [2, 3]. While the slip bands observed in DIC studies [2] cannot yet be reproduced by the CP model, the strain concentrations caused by variation in

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crystallographic orientations between the grains can be relatively well predicted by the CP simulations. This ability has been demonstrated in [4] for tantalum oligocrystals and in [5] for copper oligocrystals. The slip activity, i.e., which slip systems are active in which grain, has also been predicted in [5]. Another important aspect of polycrystal plasticity is the appearance of shear bands. Shear band is a type of localization, arising often due to crystallographic softening and can span across multiple grains. The CP models can predict the initiation point and intensity of shear bands in the polycrystals [6, 7].

The second component of the present study is the damage and fracture model. Ductile fracture in metals is known to occur due to nucleation, growth and coalescence of microscopic voids. The plasticity of a material containing voids and their evolution were first modelled in [8, 9]. The Gurson model [8] described the yielding of an isotropic perfect plastic material containing spherical voids. It was later modified to include work-hardening, void coalescence [10], plastic anisotropy [11] and numerous other features. A similar variation of CP model for a crystal containing voids was developed in [12] and implemented and applied in finite element (FE) analyses in [13]. Another less rigorous but more computationally efficient CP model with porosity induced damage was formulated and implemented in [14].

Fracture in metallic materials is an inherently microscopic process. The void size at coalescence and fracture is usually on the micrometre scale and in many cases comparable to the grain size. The voids nucleate at certain parts of the grains due to local stress concentrations and grow in the crystalline medium, which often deforms by plastic slip. Therefore, CP models can be useful in investigating the effects of microstructure features on fracture. Some studies have already been carried out in this direction. In [14, 15] the tensile tests of Al alloys were modelled using a CP model with coupled damage. In [16] the effect of microstructure on ductile fracture in bending was reproduced with the same material model. Other studies include [17], where a CP model with damage was used to study tensile fracture of an olygocrystal, [18], where the CP model was combined with a cohesive zone model, and [19] which implemented a strain-gradient regularization of a porous CP model. Void formation around second phase particles was modelled in [20]. The Fast Fourier Transform solver was used to investigate a polycrystal with non-metallic particles, using a CP model with coupled damage in [21]. The high computational cost of CP models limits either the size of the modelled structure to an oligocrystal or the resolution of the mesh to a few elements per grain, and often 2D plane strain models are used. In addition, the uncertainty in the microscopic material parameters make the results mostly valid in the qualitative sense. Nevertheless, these studies may become a valuable supplement to the other experimental and numerical methods of investigating fracture.

In the present work some aspects of modelling the ductile fracture in polycrystalline materials is investigated with CP models with coupled damage. An experimental study of the cast and homogenized Al1.2Mn alloy [22] is used as the basis for modelling. The microstructure of the alloy including the grain morphology, the crystallographic texture and the constituent particles, is reproduced in a 2D FE model. The constituent particles are explicitly modelled as elastic inclusions, which fracture at a given level of the von Mises stress, creating voids, whereas previous studies usually only introduced the already existing voids [23]. The coupled damage CP model is used to account for the dispersoids in the Al1.2Mn alloy and their contribution to the ductile fracture. The study investigates the onset and development of localization and ductile fracture for a microstructure containing constituent particles with different strength. The simulations show a complex interaction between the grain boundaries, the voids, the shear bands and the ductile crack and the influence of the two different particle populations on ductile fracture.