<figure><img src="image_1642.png" /><figcaption>Fig. 1: Experimental setup of the punch indentation test</figcaption></figure>

## Local ductility testing

Tool concept. To characterize the ductility of the untreated and conditioned aluminum die casting material, a punctual punch indentation test is used as described. Specific characteristic values such as the maximum punch force and the associated punch displacement are used to identify correlations regarding the ductility of the casting material. The experimental setup of the test is shown in Fig. 2 a). A stamping tool is used that consists of a punch and a spring-loaded blank holder. Thus, it is possible to vary the punch and die geometry as required.

The tool is implemented in a Zwick Z1486 tensile-compression testing machine, which can be seen in Fig. 2 b). It has an extra-wide testing area that allows complete die-cast components to be tested. It is also characterized by precise displacement control. The punch displacement is measured locally using optical measurement systems and digital image correlation.

<figure><img src="image_1643.png" /><figcaption>Fig. 2: Experimental setup of the punch indentation test</figcaption></figure>

Numerical modeling. In order to identify an optimum tool configuration, the ductility test is first analyzed numerically. For the numerical analysis of the ductility test and to optimize the tool geometries, a 2D model was created in Simufact Forming, which can be seen in Fig. 3. It is a rotationally symmetrical setup with rigid tools. For the validation and parameter study, a material model of a wrought aluminum alloy EN AW-6014 T4 has been implemented in order to avoid the influence factor of varying material properties as it is common for cast materials. The implemented flow curve was determined on the basis of layer compression tests and the extrapolation using the Hockett-Sherby hardening law. A combined friction model consisting of the Coulomb friction model and the friction factor model was used to model the frictional behavior. The Coulomb friction coefficient $\mu$ was set to $\mu = 0.2$ and the friction factor $m = 0.4$. The macro-mechanical
damage criterion according to Cockcroft-Latham is used for damage modeling [10]. In this damage model, the first principal normal stress is used to identify critical material states and the plastic work that must be induced in the material to cause damage is calculated. It is possible to specify a damage threshold value so that elements in the simulation are deleted and a crack occurs if this value is exceeded. The damage threshold was set to 0.7. The friction parameters $\mu$ and $m$ as well as the damage threshold were adjusted iteratively so that the calculated contour and the force-displacement curve match the experimental data (see Fig. 4).

<figure><img src="image_1643.png" /><figcaption>Fig. 3: Numerical modeling of the punch indentation test</figcaption></figure>

Validation. Experimental stage tests were carried out to validate the model using the test setup shown in Fig. 2. The micrographs from the stage tests are shown in Fig. 4 a). In the simulations, the punch displacement was adjusted according to the experimentally measured displacement and the calculated contour in red was compared with the micrograph. In all stages, the calculated contour corresponds well with the geometry from the micrograph. In addition, the numerically calculated force-displacement curves were compared with the experimental data (Fig. 4 b)). The experimental maximum force is 6.0 kN and the numerically determined maximum force is 6.1 kN. The failure displacement also agrees with each other. While the experiment fails at 1.4 mm, the numerical failure displacement is 1.41 mm. As the percentage deviation of the two characteristic values between experiment and simulation is less than 1 % in each case, the model is therefore considered validated.

<figure><img src="image_1644.png" /><figcaption>Fig. 4: Validation of the numerical model using geometric contours and force-displacement-curves</figcaption></figure>

The validated model for the punch indentation test was used to analyze influencing factors of the tool geometries on the force-displacement curve behavior. The primary objective is to identify a setup that achieves the longest possible run-out of the force-displacement curve after reaching
damage criterion according to Cockcroft-Latham is used for damage modeling [10]. In this damage model, the first principal normal stress is used to identify critical material states and the plastic work that must be induced in the material to cause damage is calculated. It is possible to specify a damage threshold value so that elements in the simulation are deleted and a crack occurs if this value is exceeded. The damage threshold was set to 0.7. The friction parameters μ and m as well as the damage threshold were adjusted iteratively so that the calculated contour and the force-displacement curve match the experimental data (see Fig. 4).

<figure><img src="image_1644.png" /><figcaption>Fig. 3: Numerical modeling of the punch indentation test</figcaption></figure>

Validation. Experimental stage tests were carried out to validate the model using the test setup shown in Fig. 2. The micrographs from the stage tests are shown in Fig. 4 a). In the simulations, the punch displacement was adjusted according to the experimentally measured displacement and the calculated contour in red was compared with the micrograph. In all stages, the calculated contour corresponds well with the geometry from the micrograph. In addition, the numerically calculated force-displacement curves were compared with the experimental data (Fig. 4 b)). The experimental maximum force is 6.0 kN and the numerically determined maximum force is 6.1 kN. The failure displacement also agrees with each other. While the experiment fails at 1.4 mm, the numerical failure displacement is 1.41 mm. As the percentage deviation of the two characteristic values between experiment and simulation is less than 1 % in each case, the model is therefore considered validated.

<figure><img src="image_1645.png" /><figcaption>Fig. 4: Validation of the numerical model using geometric contours and force-displacement-curves</figcaption></figure>

The validated model for the punch indentation test was used to analyze influencing factors of the tool geometries on the force-displacement curve behavior. The primary objective is to identify a setup that achieves the longest possible run-out of the force-displacement curve after reaching
the maximum force. This should allow enough time to stop the test in a production-accompanying application of this ductility test without the material failing or causing cracks in the component.

As shown in Fig. 5 a), the punch diameter, die diameter and punch radius were varied and various resulting combinations were investigated numerically. Selected force-displacement curves are shown in Fig. 5 a). It can be seen that the maximum force is dependent on the punch diameter due to the squared relationship between the punch diameter and the specimen surface on which the punch impacts. A dependency between maximum force and cutting gap is not recognizable. The failure displacement and the run-out of the force after reaching the maximum is dependent on the punch radius. This is due to a reduced damage input due to reduced shear cutting processes. The larger the punch radius, the longer the run-out until the material finally fails, see Fig. 5 b).

<figure><img src="image_1645.png" /><figcaption>Fig. 5: Influence of tool geometry on the force-displacement curve of the punch indentation test</figcaption></figure>

For the final setup, a punch diameter of 4.0 mm and a die diameter of 8.0 mm were specified. With a smaller punch diameter, it is possible that local material defects such as shrinkage voids near the punch could have a strong influence on the result and this influence is reduced by a larger punch diameter. A larger punch diameter also increases the maximum force and, with regard to the application of the test with mobile tongs, the forces should be kept low. The punch radius was set at 0.5 mm, as this enabled the force-displacement curve to be as flat as possible.

## Experimental setup for local heat treatment

Inductive heating. The first investigated method for conditioning the die casting material is inductive heating. A test setup was implemented for this purpose, which is shown in Fig. 6.
the maximum force. This should allow enough time to stop the test in a production-accompanying application of this ductility test without the material failing or causing cracks in the component.

As shown in Fig. 5 a), the punch diameter, die diameter and punch radius were varied and various resulting combinations were investigated numerically. Selected force-displacement curves are shown in Fig. 5 a). It can be seen that the maximum force is dependent on the punch diameter due to the squared relationship between the punch diameter and the specimen surface on which the punch impacts. A dependency between maximum force and cutting gap is not recognizable. The failure displacement and the run-out of the force after reaching the maximum is dependent on the punch radius. This is due to a reduced damage input due to reduced shear cutting processes. The larger the punch radius, the longer the run-out until the material finally fails, see Fig. 5 b).

<figure><img src="image_1646.png" /><figcaption>Fig. 5: Influence of tool geometry on the force-displacement curve of the punch indentation test</figcaption></figure>

For the final setup, a punch diameter of 4.0 mm and a die diameter of 8.0 mm were specified. With a smaller punch diameter, it is possible that local material defects such as shrinkage voids near the punch could have a strong influence on the result and this influence is reduced by a larger punch diameter. A larger punch diameter also increases the maximum force and, with regard to the application of the test with mobile tongs, the forces should be kept low. The punch radius was set at 0.5 mm, as this enabled the force-displacement curve to be as flat as possible.

## Experimental setup for local heat treatment

Inductive heating. The first investigated method for conditioning the die casting material is inductive heating. A test setup was implemented for this purpose, which is shown in Fig. 6.
<figure><img src="image_1646.png" /><figcaption>Fig. 6: Experimental setup of the casting materials conditioning via induction heating</figcaption></figure>

It consists of an induction coil with two windings and a diameter of 13 mm, which is connected to a power controller. The coil current is controlled according to the temperature setting. A pyrometer and thermocouples were initially used to measure the temperature. However, the pyrometer had problems with the fluctuating surface properties and associated emissivity of the die-cast material, so the measurement was carried out exclusively with type K thermocouples.

The temperature curve of the sample is shown as an example on the right in Fig. 6. The specimen requires about 25 seconds to be heated to 500 °C and is then held at the specified target temperature for the specified time.

Contact heating. The second method for heating the joining parts is contact heating (Fig. 7). The tool used is an anvil with a heated base, which is heated by heating cartridges. The temperature is controlled using a PID controller and a type K thermocouple is positioned near the surface between the contact area of the joining parts and the underlying heating cartridges.

<figure><img src="image_1647.png" /><figcaption>Fig. 7: Experimental setup of the casting materials conditioning via contact heating</figcaption></figure>

This setup offers simplicity and cost-effectiveness compared to induction heating in industrial applications, but it also presents drawbacks in terms of temperature control. The tool temperature decreases upon contact with the joining part, and the temperature drop is subsequently compensated by the controller. It takes time until an equilibrium state is reached where the heat
<figure><img src="image_1647.png" /><figcaption>Fig. 6: Experimental setup of the casting materials conditioning via induction heating</figcaption></figure>

It consists of an induction coil with two windings and a diameter of 13 mm, which is connected to a power controller. The coil current is controlled according to the temperature setting. A pyrometer and thermocouples were initially used to measure the temperature. However, the pyrometer had problems with the fluctuating surface properties and associated emissivity of the die-cast material, so the measurement was carried out exclusively with type K thermocouples.

The temperature curve of the sample is shown as an example on the right in Fig. 6. The specimen requires about 25 seconds to be heated to 500 °C and is then held at the specified target temperature for the specified time.

Contact heating. The second method for heating the joining parts is contact heating (Fig. 7). The tool used is an anvil with a heated base, which is heated by heating cartridges. The temperature is controlled using a PID controller and a type K thermocouple is positioned near the surface between the contact area of the joining parts and the underlying heating cartridges.

<figure><img src="image_1648.png" /><figcaption>Fig. 7: Experimental setup of the casting materials conditioning via contact heating</figcaption></figure>

This setup offers simplicity and cost-effectiveness compared to induction heating in industrial applications, but it also presents drawbacks in terms of temperature control. The tool temperature decreases upon contact with the joining part, and the temperature drop is subsequently compensated by the controller. It takes time until an equilibrium state is reached where the heat
losses to the surroundings are balanced by the contact heating, resulting in a constant temperature in the conditioning zone of the joining part.

The curve shown in Fig. 7 depicts the temperature profile in the casting in the contact zone with the heated tool. For this purpose, a hole was drilled into a sample and a type K thermocouple was integrated for temperature measurement.

The measurement shows that it takes approximately 100 s to reach a nearly constant temperature. The temperature reached at that point is between 475 °C and 480 °C at a tool temperature of 500 °C. Tool temperatures above were not investigated because previous experiments showed that the aluminum material tends to stick to the tool at temperatures above this level. The process duration for contact heating includes the time for temperature ramp-up.

## Results

### Hardness Measurement and Microstructure.
In preliminary investigations, an assessment was made of the temperature range that could lead to a significant alteration in the material properties. The effect of conditioning was examined by measuring Vickers hardness on heat-treated samples, represented in Fig. 8 a).

The analysis of the sample treated by induction heating shows a lower minimum hardness level compared to contact heating with values below 60 HV1. The minimum hardness achieved with contact heating is approximately 65 HV1. The specimens treated by contact heating have become less soft throughout the measured range. In the edge area, they show hardness values close to the untreated state. In the center of the treated area, there is an increase in hardness with induction heating. Partial melting and solidification of the aluminum could be the cause for this effect. This phenomenon can be visible on the surface of the samples as well as occur between the grain boundaries, without being externally detectable [11].

The analysis of a micrograph shows the microstructure before (Fig. 8 b) and after heat treatment (Fig. 8 c) using contact heating as an example. It can be observed that the microstructural state of alpha dendrites of aluminum and the Al-Si eutectic hardly changes due to the heat treatment. A slight homogenization can be noticed, which could be attributed to the diffusion ability of Si caused by the increase in temperature. However, it is assumed that the effect of softening is due to the healing of lattice defects, such as dislocations.

The effect of heat treatment using contact heating appears to be limited to the contact area with the tool, compared to induction heating. This suggests different temperature distributions and disparities in energy input due to heating. Further investigations are needed to understand the specific mechanisms and parameters that lead to this observation.

<figure><img src="image_1648.png" /><figcaption>Fig. 8 a) Hardness measurement after heat treatment (T = 500 °C, t = 120 s); Micrograph before (b) and after (c) heat treatment by contact heating (T = 500 °C, t = 150 s), Material: AlSi10MnMg</figcaption></figure>

### Ductility test.
Measurements were carried out with the above-mentioned final setup of the punch indentation test on casting components in three conditions. The untreated condition serves
losses to the surroundings are balanced by the contact heating, resulting in a constant temperature in the conditioning zone of the joining part.

The curve shown in Fig. 7 depicts the temperature profile in the casting in the contact zone with the heated tool. For this purpose, a hole was drilled into a sample and a type K thermocouple was integrated for temperature measurement.

The measurement shows that it takes approximately 100 s to reach a nearly constant temperature. The temperature reached at that point is between 475 °C and 480 °C at a tool temperature of 500 °C. Tool temperatures above were not investigated because previous experiments showed that the aluminum material tends to stick to the tool at temperatures above this level. The process duration for contact heating includes the time for temperature ramp-up.

## Results

### Hardness Measurement and Microstructure.
In preliminary investigations, an assessment was made of the temperature range that could lead to a significant alteration in the material properties. The effect of conditioning was examined by measuring Vickers hardness on heat-treated samples, represented in Fig. 8 a).

The analysis of the sample treated by induction heating shows a lower minimum hardness level compared to contact heating with values below 60 HV1. The minimum hardness achieved with contact heating is approximately 65 HV1. The specimens treated by contact heating have become less soft throughout the measured range. In the edge area, they show hardness values close to the untreated state. In the center of the treated area, there is an increase in hardness with induction heating. Partial melting and solidification of the aluminum could be the cause for this effect. This phenomenon can be visible on the surface of the samples as well as occur between the grain boundaries, without being externally detectable [11].

The analysis of a micrograph shows the microstructure before (Fig. 8 b) and after heat treatment (Fig. 8 c) using contact heating as an example. It can be observed that the microstructural state of alpha dendrites of aluminum and the Al-Si eutectic hardly changes due to the heat treatment. A slight homogenization can be noticed, which could be attributed to the diffusion ability of Si caused by the increase in temperature. However, it is assumed that the effect of softening is due to the healing of lattice defects, such as dislocations.

The effect of heat treatment using contact heating appears to be limited to the contact area with the tool, compared to induction heating. This suggests different temperature distributions and disparities in energy input due to heating. Further investigations are needed to understand the specific mechanisms and parameters that lead to this observation.

<figure><img src="image_1649.png" /><figcaption>Fig. 8 a) Hardness measurement after heat treatment (T = 500 °C, t = 120 s); Micrograph before (b) and after (c) heat treatment by contact heating (T = 500 °C, t = 150 s), Material: AlSi10MnMg</figcaption></figure>

### Ductility test.
Measurements were carried out with the above-mentioned final setup of the punch indentation test on casting components in three conditions. The untreated condition serves
as the reference, with the input parameter for the heat treatment temperature assumed to be room temperature (RT = 20 °C) in this case. Additionally, two heat treatment conditions are investigated, generated using induction heating. The target temperature in the component was set to 450 °C and 500 °C, with a hold time of 120 s. At this point it should be mentioned that the heat-treated die-cast material is tested in the cold state (RT) with the ductility test and the joining tests are also carried out in the cold state. In Fig. 9, the distribution of individual values for all parameters is shown. It can be observed that $F_{max}$ decreases significantly in the heat-treated state at $T = 450$ °C. With further increase in temperature to 500 °C, there is a slight increase in $F_{max}$. This increase could also be related to the local hardening, as indicated by the Vickers hardness values in Fig. 8 a). The maximum punch displacement $s_{Fmax}$ and the forming energy $W_{Fmax}$ until reaching $F_{max}$ also tend to increase with higher heat treatment temperatures up to 500 °C.

<figure><img src="image_1649.png" /><figcaption>Fig. 9: Parameter distribution of heat treatment temperature and ductility test parameters</figcaption></figure>

Joining experiments. Joining experiments are being conducted using self-pierce riveting. The untreated state of the aluminum casting alloy is used as reference. The findings of the ductility test show that a significant increase in $W_{Fmax}$ can be achieved at a temperature of $T = 450$ °C and $T = 500$ °C, see Fig. 9. At a temperature of $T = 450$ °C, a significant reduction in cracks after joining compared to the reference can be observed. However, the best results are achieved at a setting temperature of $T = 500$ °C. Fig. 10 shows the SPR joint of a press-hardened steel material with AlSi10MnMg (F) aluminum casting alloy in the initial state of the casting material. Both in the cross-section and in the bottom view, cracks can be observed in the aluminum. The ductility of the material is not sufficient to achieve crack-free joining.

<table><thead><tr><th colspan="3">Joining parameters</th><th rowspan="2">Cross section</th><th rowspan="2">Closing head (bottom view of joint)</th></tr><tr><th>Parts to be joined</th><th>Material</th><th>Thickness</th></tr></thead><tbody><tr><td>Part 1</td><td>22MnB5</td><td>1.5 mm</td><td rowspan="5" colspan="2" style="vertical-align: middle; text-align: center; padding-bottom: 5px;"><i>[Image of cross section]</i></td></tr><tr><td>Part 2</td><td>AlSi10MnMg F</td><td>3.0 mm</td></tr><tr><td>Rivet</td><td colspan="2">HDZ Ø5.5 x 5.0 H6</td></tr><tr><td>Die</td><td colspan="2">Ø11 mm; die depth 1.5 mm</td></tr><tr><td>Joining Force</td><td colspan="2">68 kN</td></tr></tbody></table>

<figure><img src="image_1650.png" /></figure>

<figure><img src="image_1651.png" /><figcaption>Fig. 10: Self-pierce riveting without heat treatment of the die cast material</figcaption></figure>

Fig. 11 and Fig. 12 show the result of the joining process after the two heat treatment strategies, applied for $T = 500$ °C over a time period of 120 s for inductive heating (Fig. 11) and 150 s overall hold time for contact heating at tool temperature of 500 °C (Fig. 12). In both cases, it can be observed that the susceptibility to cracking significantly decreases as a result of the conditioning. The analysis in the cross-section reveals that cracks occur near the rivet foot in the sample treated by contact heating. This is because the temperature-time profile in contact heating is lower, resulting in a reduced effect of heat treatment on the cast material. The presence of such cracks is partially tolerable, as observed in studies on the joining of brittle materials [6]. However, cracks
as the reference, with the input parameter for the heat treatment temperature assumed to be room temperature (RT = 20 °C) in this case. Additionally, two heat treatment conditions are investigated, generated using induction heating. The target temperature in the component was set to 450 °C and 500 °C, with a hold time of 120 s. At this point it should be mentioned that the heat-treated die-cast material is tested in the cold state (RT) with the ductility test and the joining tests are also carried out in the cold state. In Fig. 9, the distribution of individual values for all parameters is shown. It can be observed that $F_{max}$ decreases significantly in the heat-treated state at $T = 450$ °C. With further increase in temperature to 500 °C, there is a slight increase in $F_{max}$. This increase could also be related to the local hardening, as indicated by the Vickers hardness values in Fig. 8 a). The maximum punch displacement $s_{Fmax}$ and the forming energy $W_{Fmax}$ until reaching $F_{max}$ also tend to increase with higher heat treatment temperatures up to 500 °C.

<figure><img src="image_1650.png" /><figcaption>Fig. 9: Parameter distribution of heat treatment temperature and ductility test parameters</figcaption></figure>

Joining experiments. Joining experiments are being conducted using self-pierce riveting. The untreated state of the aluminum casting alloy is used as reference. The findings of the ductility test show that a significant increase in $W_{Fmax}$ can be achieved at a temperature of $T = 450$ °C and $T = 500$ °C, see Fig. 9. At a temperature of $T = 450$ °C, a significant reduction in cracks after joining compared to the reference can be observed. However, the best results are achieved at a setting temperature of $T = 500$ °C. Fig. 10 shows the SPR joint of a press-hardened steel material with AlSi10MnMg (F) aluminum casting alloy in the initial state of the casting material. Both in the cross-section and in the bottom view, cracks can be observed in the aluminum. The ductility of the material is not sufficient to achieve crack-free joining.

<table><caption>Fig. 10: Self-pierce riveting without heat treatment of the die cast material</caption><thead><tr><th colspan="3">Joining parameters</th><th rowspan="2">Cross section</th><th rowspan="2">Closing head (bottom view of joint)</th></tr><tr><th>Parts to be joined</th><th>Material</th><th>Thickness</th></tr></thead><tbody><tr><td>Part 1</td><td>22MnB5</td><td>1.5 mm</td><td rowspan="5" colspan="2" style="vertical-align: middle; text-align: center;"><div style="font-family: monospace; white-space: pre; text-align: left;">1 mm</div><div style="border: 1px solid black; padding: 2px 5px; display: inline-block;">Image of a cross section of a die cast material</div></td></tr><tr><td>Part 2</td><td>AlSi10MnMg F</td><td>3.0 mm</td></tr><tr><td>Rivet</td><td colspan="2">HDZ Ø5.5 x 5.0 H6</td></tr><tr><td>Die</td><td colspan="2">Ø11 mm; die depth 1.5 mm</td></tr><tr><td>Joining Force</td><td colspan="2">68 kN</td></tr></tbody></table>

Fig. 11 and Fig. 12 show the result of the joining process after the two heat treatment strategies, applied for $T = 500$ °C over a time period of 120 s for inductive heating (Fig. 11) and 150 s overall hold time for contact heating at tool temperature of 500 °C (Fig. 12). In both cases, it can be observed that the susceptibility to cracking significantly decreases as a result of the conditioning. The analysis in the cross-section reveals that cracks occur near the rivet foot in the sample treated by contact heating. This is because the temperature-time profile in contact heating is lower, resulting in a reduced effect of heat treatment on the cast material. The presence of such cracks is partially tolerable, as observed in studies on the joining of brittle materials [6]. However, cracks
that manifest in larger quantities on the material surface and show a tendency to open up, as depicted in Fig. 10, are considered critical. Additionally, cracks extending from the rivet to the outer side of the joint in the micrograph are also deemed critical and are challenging to detect in a single micrograph.

<table><thead><tr><th colspan="3">Joining parameters</th><th rowspan="2">Cross section</th><th rowspan="2">Closing head (bottom view of joint)</th></tr><tr><th>Parts to be joined</th><th>Material</th><th>Thickness</th></tr></thead><tbody><tr><td>Part 1</td><td>22MnB5</td><td>1.5 mm</td><td rowspan="5"></td><td rowspan="5"></td></tr><tr><td>Part 2</td><td>AlSi10MnMg F</td><td>3.0 mm</td></tr><tr><td>Rivet</td><td colspan="2">HDZ Ø5.5 x 5.0 H6</td></tr><tr><td>Die</td><td colspan="2">Ø11 mm; die depth 1.5 mm</td></tr><tr><td>Joining Force</td><td colspan="2">63 kN</td></tr><tr><td>Heat treatment</td><td colspan="2">Inductive heating</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_1652.png" /></figure>

<figure><img src="image_1653.png" /><figcaption>Fig. 11: Self-pierce riveting after inductive heating of die cast material at local temperature $T = 500$ °C</figcaption></figure>

<table><thead><tr><th colspan="3">Joining parameters</th><th rowspan="2">Cross section</th><th rowspan="2">Closing head (bottom view of joint)</th></tr><tr><th>Parts to be joined</th><th>Material</th><th>Thickness</th></tr></thead><tbody><tr><td>Part 1</td><td>22MnB5</td><td>1.5 mm</td><td rowspan="5"></td><td rowspan="5"></td></tr><tr><td>Part 2</td><td>AlSi10MnMg F</td><td>3.0 mm</td></tr><tr><td>Rivet</td><td colspan="2">HDZ Ø5.5 x 5.0 H6</td></tr><tr><td>Die</td><td colspan="2">Ø11 mm; die depth 1.5 mm</td></tr><tr><td>Joining Force</td><td colspan="2">64 kN</td></tr><tr><td>Heat treatment</td><td colspan="2">Contact heating</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_1654.png" /></figure>

<figure><img src="image_1655.png" /><figcaption>Fig. 12: Self-pierce riveting after contact heating of die cast material at a tool temperature of $T = 500$ °C</figcaption></figure>

## Summary

In this study, two approaches for the robust mechanical joining of aluminum die-cast components were investigated. Firstly, a ductility test in the form of a punch indentation test was investigated experimentally as well as numerically. It was found that the maximum force is dependent on the punch diameter and independent of the cutting gap. Furthermore, the failure displacement and the run-out of the force after reaching the maximum is dependent on the punch radius. Based on these findings, a final setup for the ductility test was determined and tests were carried out on heat-treated samples.

Two different technologies were investigated for the heat treatment of the cast aluminum material. Heating by means of contact heating and induction heating. The comparison of the two strategies showed that both concepts had advantages and disadvantages. Induction heating enabled a fast process with efficient control, but required complex equipment such as fixtures, coils, generators, and control systems, especially for a larger number of joining points. Contact heating using heating cartridges offer a more cost-effective alternative, but the actual temperature achieved in the component could be lower than the set temperature due to energy losses. Therefore, individual adjustment of the setting parameters is necessary to enable crack-free joining of the casting material.

The correlation analysis of the values obtained from the ductility test demonstrated the relationship between the heat treatment and the characteristic output parameters of the ductility test. For example, $W_{Fmax}$ increased when the samples underwent heat treatment. This indicated that
that manifest in larger quantities on the material surface and show a tendency to open up, as depicted in Fig. 10, are considered critical. Additionally, cracks extending from the rivet to the outer side of the joint in the micrograph are also deemed critical and are challenging to detect in a single micrograph.

<table><thead><tr><th colspan="3">Joining parameters</th><th rowspan="2">Cross section</th><th rowspan="2">Closing head (bottom view of joint)</th></tr><tr><th>Parts to be joined</th><th>Material</th><th>Thickness</th></tr></thead><tbody><tr><td>Part 1</td><td>22MnB5</td><td>1.5 mm</td><td rowspan="5"></td><td rowspan="5"></td></tr><tr><td>Part 2</td><td>AlSi10MnMg F</td><td>3.0 mm</td></tr><tr><td>Rivet</td><td colspan="2">HDZ Ø5.5 x 5.0 H6</td></tr><tr><td>Die</td><td colspan="2">Ø11 mm; die depth 1.5 mm</td></tr><tr><td>Joining Force</td><td colspan="2">63 kN</td></tr><tr><td>Heat treatment</td><td colspan="2">Inductive heating</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_1651.png" /></figure>

<figure><img src="image_1652.png" /><figcaption>Fig. 11: Self-pierce riveting after inductive heating of die cast material at local temperature $T = 500$ °C</figcaption></figure>

<table><thead><tr><th colspan="3">Joining parameters</th><th rowspan="2">Cross section</th><th rowspan="2">Closing head (bottom view of joint)</th></tr><tr><th>Parts to be joined</th><th>Material</th><th>Thickness</th></tr></thead><tbody><tr><td>Part 1</td><td>22MnB5</td><td>1.5 mm</td><td rowspan="5"></td><td rowspan="5"></td></tr><tr><td>Part 2</td><td>AlSi10MnMg F</td><td>3.0 mm</td></tr><tr><td>Rivet</td><td colspan="2">HDZ Ø5.5 x 5.0 H6</td></tr><tr><td>Die</td><td colspan="2">Ø11 mm; die depth 1.5 mm</td></tr><tr><td>Joining Force</td><td colspan="2">64 kN</td></tr><tr><td>Heat treatment</td><td colspan="2">Contact heating</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_1653.png" /></figure>

<figure><img src="image_1654.png" /><figcaption>Fig. 12: Self-pierce riveting after contact heating of die cast material at a tool temperature of $T = 500$ °C</figcaption></figure>

## Summary

In this study, two approaches for the robust mechanical joining of aluminum die-cast components were investigated. Firstly, a ductility test in the form of a punch indentation test was investigated experimentally as well as numerically. It was found that the maximum force is dependent on the punch diameter and independent of the cutting gap. Furthermore, the failure displacement and the run-out of the force after reaching the maximum is dependent on the punch radius. Based on these findings, a final setup for the ductility test was determined and tests were carried out on heat-treated samples.

Two different technologies were investigated for the heat treatment of the cast aluminum material. Heating by means of contact heating and induction heating. The comparison of the two strategies showed that both concepts had advantages and disadvantages. Induction heating enabled a fast process with efficient control, but required complex equipment such as fixtures, coils, generators, and control systems, especially for a larger number of joining points. Contact heating using heating cartridges offer a more cost-effective alternative, but the actual temperature achieved in the component could be lower than the set temperature due to energy losses. Therefore, individual adjustment of the setting parameters is necessary to enable crack-free joining of the casting material.

The correlation analysis of the values obtained from the ductility test demonstrated the relationship between the heat treatment and the characteristic output parameters of the ductility test. For example, $W_{Fmax}$ increased when the samples underwent heat treatment. This indicated that
heat treatment had an influence on the formability of the material and thus affected the required joining properties.

The ductility test responses to the temperature difference in the range between 450 °C and 500 °C in the parameter $F_{max}$. In future research, a more comprehensive analysis of the ductility test will be carried out to facilitate quantitative correlations between the occurrence of cracks during the joining process and the output parameters of the ductility test. However, quantifying cracks in the joint poses a challenge as factors such as crack depth and the corrosion relevance of a crack are difficult to determine.

Current work is now investigating the application of local heat treatment to large components. The focus is on a robust process control and demonstrating that local heat treatment does not compromise the dimensional accuracy of the components.

## References

[1] Kaufman JG and Rooy EL. Aluminum alloy castings: Properties, processes, and applications. Materials Park, OH: ASM International, 2004.
https://doi.org/10.31399/asm.tb.aacppa.9781627083355

[2] Schuh, Günther, Georg Bergweiler, Leonhard Dworog and Falko Fiedler. 2022., Opportunities and Risks of Mega-Casting in Automotive Production – The Aluminum Die-Casted Body in White". 9. Wt Werkstattstechnik Online, 112, 580–585. https://doi.org/10.37544/1436-4980-2022-09-52

[3] Sun X, Khaleel M. Performance optimization of self-piercing rivets through analytical rivet strength estimation. J Manuf Process 2005;7:83e93. https://doi.org/10.1016/S1526-6125(05)70085-2

[4] Jäckel M, Grimm T, Niegsch R, Drossel WG. Overview of Current Challenges in Self-Pierce Riveting of Lightweight Materials. MDPI Proceedings 2018, 2, 384.
https://doi.org/10.3390/ICEM18-05206

[5] Ang HQ. An overview of self-piercing riveting process with focus on joint failures, corrosion issues and optimisation techniques. Chin J Mech Eng-En 2021;34:1e25.
https://doi.org/10.1186/s10033-020-00526-3

[6] Jäckel M, Coppieters S, Hofmann M, Vandermeiren N, Landgrebe D. Mechanical joining of materials with limited ductility: analysis of process-induced defects. AIP Conf Proc 2017;1896:110009. https://doi.org/10.1063/1.5008136

[7] Zhao X, Meng D, Zhang JY, Han QY. The effect of heat treatment on die casting aluminum to apply self-pierce riveting. Int J Adv Manuf Technol 2020;109:2409e19.
https://doi.org/10.1007/s00170-020-05833-6

[8] Ming Lou, YongBing Li, Yuan Wang, Bin Wang, Xinmin Lai, Influence of resistance heating on self-piercing riveted dissimilar joints of AA6061-T6 and galvanized DP590, Journal of Materials Processing Technology, Volume 214, Issue 10, 2014, Pages 2119-2126,
https://doi.org/10.1016/j.jmatprotec.2014.03.006

[9] Bingöl, M.; Böhnlein, C.; Dengler, A.; Seitz, I.; Storsberg, L.: Method for determining at least one characteristic value for the formability of a material and device for carrying out such a method. 2015. publication. Daimler AG. Publication no.: DE 10 2015 006 274 A1

[10] Cockcroft, M. G.; Latham, D. J.: Ductility and the Workability of Metals, Journal Institute of Metals, Vol. 96, 1968, pp. 33-39

[11] Ostermann, F.: Anwendungstechnologie Aluminium (2., neu bearb. u. aktualisierte ed.). Berlin: Springer. 2007
heat treatment had an influence on the formability of the material and thus affected the required joining properties.

The ductility test responses to the temperature difference in the range between 450 °C and 500 °C in the parameter $F_{max}$. In future research, a more comprehensive analysis of the ductility test will be carried out to facilitate quantitative correlations between the occurrence of cracks during the joining process and the output parameters of the ductility test. However, quantifying cracks in the joint poses a challenge as factors such as crack depth and the corrosion relevance of a crack are difficult to determine.

Current work is now investigating the application of local heat treatment to large components. The focus is on a robust process control and demonstrating that local heat treatment does not compromise the dimensional accuracy of the components.

## References

[1] Kaufman JG and Rooy EL. Aluminum alloy castings: Properties, processes, and applications. Materials Park, OH: ASM International, 2004.
https://doi.org/10.31399/asm.tb.aacppa.9781627083355

[2] Schuh, Günther, Georg Bergweiler, Leonhard Dworog and Falko Fiedler. 2022., Opportunities and Risks of Mega-Casting in Automotive Production – The Aluminum Die-Casted Body in White". 9. Wt Werkstattstechnik Online, 112, 580–585. https://doi.org/10.37544/1436-4980-2022-09-52

[3] Sun X, Khaleel M. Performance optimization of self-piercing rivets through analytical rivet strength estimation. J Manuf Process 2005;7:83e93. https://doi.org/10.1016/S1526-6125(05)70085-2

[4] Jäckel M, Grimm T, Niegsch R, Drossel WG. Overview of Current Challenges in Self-Pierce Riveting of Lightweight Materials. MDPI Proceedings 2018, 2, 384.
https://doi.org/10.3390/ICEM18-05206

[5] Ang HQ. An overview of self-piercing riveting process with focus on joint failures, corrosion issues and optimisation techniques. Chin J Mech Eng-En 2021;34:1e25.
https://doi.org/10.1186/s10033-020-00526-3

[6] Jäckel M, Coppieters S, Hofmann M, Vandermeiren N, Landgrebe D. Mechanical joining of materials with limited ductility: analysis of process-induced defects. AIP Conf Proc 2017;1896:110009. https://doi.org/10.1063/1.5008136

[7] Zhao X, Meng D, Zhang JY, Han QY. The effect of heat treatment on die casting aluminum to apply self-pierce riveting. Int J Adv Manuf Technol 2020;109:2409e19.
https://doi.org/10.1007/s00170-020-05833-6

[8] Ming Lou, YongBing Li, Yuan Wang, Bin Wang, Xinmin Lai, Influence of resistance heating on self-piercing riveted dissimilar joints of AA6061-T6 and galvanized DP590, Journal of Materials Processing Technology, Volume 214, Issue 10, 2014, Pages 2119-2126, https://doi.org/10.1016/j.jmatprotec.2014.03.006

[9] Bingöl, M.; Böhnlein, C.; Dengler, A.; Seitz, I.; Storsberg, L.: Method for determining at least one characteristic value for the formability of a material and device for carrying out such a method. 2015. publication. Daimler AG. Publication no.: DE 10 2015 006 274 A1

[10] Cockcroft, M. G.; Latham, D. J.: Ductility and the Workability of Metals, Journal Institute of Metals, Vol. 96, 1968, pp. 33-39

[11] Ostermann, F.: Anwendungstechnologie Aluminium (2., neu bearb. u. aktualisierte ed.). Berlin: Springer. 2007
# Heterogeneity of strain and texture inside roll-bonded multilaminates

HANON Guillaume¹,ª*, MALET Loïc²,ᵇ and DELANNAY Laurent¹,ᶜ

¹iMMC, UCLouvain, 1348 Louvain-la-Neuve, Belgium

²4MAT, Université Libre de Bruxelles, 1050 Bruxelles, Belgium

ªguillaume.hanon@uclouvain.be, ᵇloic.malet@ulb.be, ᶜlaurent.delannay@uclouvain.be

**Keywords:** Roll-Bonding, Anisotropic Plasticity, Finite Element

**Abstract.** This study demonstrates the usefulness of crystal plasticity modeling and crystallographic texture analysis when aiming to understand through-thickness strain heterogeneity after roll bonding of dissimilar materials. FE modeling was used at two length scales to study the deformation and texture heterogeneities inside a 9-layer multilaminate made of aluminum and steel, produced by roll-bonding. Microstructure and crystallographic texture were probed using EBSD. Numerical predictions indicated that plane strain compression was accompanied by _significant_ shear parallel to the rolling plane and inclined shear banding in aluminum. Predictions of the Texture development were more accurate in the bcc phase than the fcc phase.

## Introduction

Roll bonding enables solid state welding of dissimilar metals in a single rolling pass, on the condition that thickness reduction is above 50% [1]. Due to the mismatch of strength, achieving co-deformation of the assembly is a challenge. The harder phase tends to fragment with the emergence of inclined shear bands inside the softer phase [2,3]. This undesirable result should be avoided or limited as it hinders achieving composite mechanical properties and is hence likely to reduce ductility, strength and toughness.

Previous investigations of the texture gradient across the thickness of rolled sheets have demonstrated that the plastic strain is heterogeneous even when the sheets are made of a single metal [4]. The amplitude of shear parallel to the rolling plane varies as a function of the depth across the sheet and also depending on the rolling draught, i.e. on the ratio between the thickness of the sheet and the contact length with the rolls [5,6]. To the knowledge of the authors, such studies have never been extended to the case of roll bonded composite multilayers.

The present study demonstrates that “macroscopic” Finite Element (FE) modeling of the through-thickness strain heterogeneity during rolling can efficiently be combined with “microscopic” crystal plasticity based finite element modeling (CPFEM) [7] of the texture development. The confrontation of numerical predictions and experimental observations is aimed to draw the link between deformation heterogeneity and through-thickness texture gradient.

## Experimental samples

A 9-layer multilaminate was successfully roll-bonded at ambient temperature by stacking layers of 1050 aluminum alloy ($\sigma_{yield} = 75$MPa and $\sigma_{UTS} = 120$MPa) and Armco pure iron ($\sigma_{yield} = 250$MPa and $\sigma_{UTS} = 300$MPa). Before each rolling pass, all surfaces were degreased and brushed using a 0.3 mm diameter wire-brush, which removed oxide layers. Folding of the sheets impeded any relative motion between the layers [9]. The diameter of the rolls was 112 mm and no lubricant was used.

Two rolling passes were required to obtain the 9-layer stack. To start, a 3-layer was obtained by rolling a 0.8 mm-thick Iron sheet wrapped inside a 0.4 aluminum sheet folded in two. The first rolling pass involved 61% thickness reduction to ensure bonding. Two such 3-layers, stacked

<figure><img src="image_1656.png" /></figure>
# Heterogeneity of strain and texture inside roll-bonded multilaminates

HANON Guillaume¹,ª*, MALET Loïc²,ᵇ and DELANNAY Laurent¹,ᶜ

¹iMMC, UCLouvain, 1348 Louvain-la-Neuve, Belgium

²4MAT, Université Libre de Bruxelles, 1050 Bruxelles, Belgium

ªguillaume.hanon@uclouvain.be, ᵇloic.malet@ulb.be, ᶜlaurent.delannay@uclouvain.be

**Keywords:** Roll-Bonding, Anisotropic Plasticity, Finite Element

**Abstract.** This study demonstrates the usefulness of crystal plasticity modeling and crystallographic texture analysis when aiming to understand through-thickness strain heterogeneity after roll bonding of dissimilar materials. FE modeling was used at two length scales to study the deformation and texture heterogeneities inside a 9-layer multilaminate made of aluminum and steel, produced by roll-bonding. Microstructure and crystallographic texture were probed using EBSD. Numerical predictions indicated that plane strain compression was accompanied by _significant_ shear parallel to the rolling plane and inclined shear banding in aluminum. Predictions of the Texture development were more accurate in the bcc phase than the fcc phase.

## Introduction

Roll bonding enables solid state welding of dissimilar metals in a single rolling pass, on the condition that thickness reduction is above 50% [1]. Due to the mismatch of strength, achieving co-deformation of the assembly is a challenge. The harder phase tends to fragment with the emergence of inclined shear bands inside the softer phase [2,3]. This undesirable result should be avoided or limited as it hinders achieving composite mechanical properties and is hence likely to reduce ductility, strength and toughness.

Previous investigations of the texture gradient across the thickness of rolled sheets have demonstrated that the plastic strain is heterogeneous even when the sheets are made of a single metal [4]. The amplitude of shear parallel to the rolling plane varies as a function of the depth across the sheet and also depending on the rolling draught, i.e. on the ratio between the thickness of the sheet and the contact length with the rolls [5,6]. To the knowledge of the authors, such studies have never been extended to the case of roll bonded composite multilayers.

The present study demonstrates that “macroscopic” Finite Element (FE) modeling of the through-thickness strain heterogeneity during rolling can efficiently be combined with “microscopic” crystal plasticity based finite element modeling (CPFEM) [7] of the texture development. The confrontation of numerical predictions and experimental observations is aimed to draw the link between deformation heterogeneity and through-thickness texture gradient.

## Experimental samples

A 9-layer multilaminate was successfully roll-bonded at ambient temperature by stacking layers of 1050 aluminum alloy ($\sigma_{yield} = 75$MPa and $\sigma_{UTS} = 120$MPa) and Armco pure iron ($\sigma_{yield} = 250$MPa and $\sigma_{UTS} = 300$MPa). Before each rolling pass, all surfaces were degreased and brushed using a 0.3 mm diameter wire-brush, which removed oxide layers. Folding of the sheets impeded any relative motion between the layers [9]. The diameter of the rolls was 112 mm and no lubricant was used.

Two rolling passes were required to obtain the 9-layer stack. To start, a 3-layer was obtained by rolling a 0.8 mm-thick Iron sheet wrapped inside a 0.4 aluminum sheet folded in two. The first rolling pass involved 61% thickness reduction to ensure bonding. Two such 3-layers, stacked

<figure><img src="image_1655.png" /></figure>
together with a central 0.2mm Fe layer, were then wrapped inside a 0.2 mm thick Fe sheet. The 9-layer multilaminate was heat treated (500°C for 30 minutes) before the second rolling pass of 62% thickness reduction.

Electron Back Scatter Diffraction (EBSD) was used to probe the crystallographic texture. A high level of surface finish was achieved by polishing up to grid 1200 with SiC paper, followed by diamond suspension up to 1 micron and OPS.

Figure 1 shows a section of the sample studied. Grey contrast in the optical microscope clearly identifies the 9 layers made of the two different metals. Four different zones were measured by EBSD, each one covering the plate thickness. One such orientation maps is shown in Figure 1 (blue square). Flat and elongated grains were found in all Fe layers whereas Al layers hosted a very refined equiaxed microstructure. Such microstructure is characteristic of continuous recrystallization [8,10]. The <111> pole figures shown in Fig. 1 are representative of individual layers. They were produced by combining suitable zones in the four orientation maps. Note that only the 7 central layers are represented as the indexing was poor (and the texture was weak) in the two peripheral layers.

## Numerical simulations

Figure 2 gives an illustration of the two numerical models, working at different length scales, which assisted the interpretation of experimental pole figures. The commercial finite element code Abaqus/Standard (d'Assault Systèmes, France) was used for these two simulations.

The macroscopic strain field was predicted based on finite element modeling of the second rolling pass at the scale of the multi-layer. Only the half of the thickness was modelled as the strain field was symmetrical relative to the rolling plane (disregarding the emergence of instabilities). A total of 54576 2D Plane strain elements (CPE4) were used. Armco Iron and Pure Al alloys proprieties were assigned to the corresponding elements. Isotropic $J_2$ plasticity with a power-law hardening was prescribed as the material responses (reproducing the experimental yield stress and UTS). A friction coefficient of 0.3 was used [12]. The bonding was not simulated: layers were considered tied. Important shear deformation parallel to the rolling plane was predicted in the two aluminum layers as shown in Figure 2. The most external Al layer was subjected to the most intense shear deformation. This resulted from the large contact length (about 10 times the thickness of the sample) between the rolls and the plate, and also because the yield stress of Al layers was much lower than in Fe layers. Friction played a role too by inducing important shear deformation close to the external surfaces. Limited shear deformation was observed inside the Iron layers.
together with a central 0.2mm Fe layer, were then wrapped inside a 0.2 mm thick Fe sheet. The 9-layer multilaminate was heat treated (500°C for 30 minutes) before the second rolling pass of 62% thickness reduction.

Electron Back Scatter Diffraction (EBSD) was used to probe the crystallographic texture. A high level of surface finish was achieved by polishing up to grid 1200 with SiC paper, followed by diamond suspension up to 1 micron and OPS.

Figure 1 shows a section of the sample studied. Grey contrast in the optical microscope clearly identifies the 9 layers made of the two different metals. Four different zones were measured by EBSD, each one covering the plate thickness. One such orientation maps is shown in Figure 1 (blue square). Flat and elongated grains were found in all Fe layers whereas Al layers hosted a very refined equiaxed microstructure. Such microstructure is characteristic of continuous recrystallization [8,10]. The <111> pole figures shown in Fig. 1 are representative of individual layers. They were produced by combining suitable zones in the four orientation maps. Note that only the 7 central layers are represented as the indexing was poor (and the texture was weak) in the two peripheral layers.

## Numerical simulations

Figure 2 gives an illustration of the two numerical models, working at different length scales, which assisted the interpretation of experimental pole figures. The commercial finite element code Abaqus/Standard (d'Assault Systèmes, France) was used for these two simulations.

The macroscopic strain field was predicted based on finite element modeling of the second rolling pass at the scale of the multi-layer. Only the half of the thickness was modelled as the strain field was symmetrical relative to the rolling plane (disregarding the emergence of instabilities). A total of 54576 2D Plane strain elements (CPE4) were used. Armco Iron and Pure Al alloys proprieties were assigned to the corresponding elements. Isotropic $J_2$ plasticity with a power-law hardening was prescribed as the material responses (reproducing the experimental yield stress and UTS). A friction coefficient of 0.3 was used [12]. The bonding was not simulated: layers were considered tied. Important shear deformation parallel to the rolling plane was predicted in the two aluminum layers as shown in Figure 2. The most external Al layer was subjected to the most intense shear deformation. This resulted from the large contact length (about 10 times the thickness of the sample) between the rolls and the plate, and also because the yield stress of Al layers was much lower than in Fe layers. Friction played a role too by inducing important shear deformation close to the external surfaces. Limited shear deformation was observed inside the Iron layers.
<figure><img src="image_1657.png" /><figcaption>Figure 1: Characterization of texture and microstructure based on EBSD and optical microscopy. <111> Pole figures (numbered according to the position in the stacking) present the texture developed inside each layer in the rolling direction (RD) pointing to the right and the normal direction (ND) pointing to the top.</figcaption></figure>

<figure><img src="image_1658.png" /><figcaption>Figure 2: Illustration of the two scales of the FE analysis. On the left, the macroscopic through-thickness strain distribution was predicted during the 2^nd^ rolling pass. The corresponding deformed mesh is displayed alongside the RD-ND shear strain map. On the right, the model polycrystal was subjected to (microscopic) periodic boundary conditions.</figcaption></figure>
<figure><img src="image_1656.png" /><figcaption>Figure 1: Characterization of texture and microstructure based on EBSD and optical microscopy. <111> Pole figures (numbered according to the position in the stacking) present the texture developed inside each layer in the rolling direction (RD) pointing to the right and the normal direction (ND) pointing to the top.</figcaption></figure>

<figure><img src="image_1657.png" /><figcaption>Figure 2: Illustration of the two scales of the FE analysis. On the left, the macroscopic through-thickness strain distribution was predicted during the 2^nd^ rolling pass. The corresponding deformed mesh is displayed alongside the RD-ND shear strain map. On the right, the model polycrystal was subjected to (microscopic) periodic boundary conditions.</figcaption></figure>
Crystal Plasticity Finite Element Modeling (CPFEM) [7] allowed us to predict the texture development, respectively, inside Al and Fe layers. Anisotropic plasticity at the level of individual grains was implemented as a user-defined material law (UMAT). As usually assumed, the fcc phase was subjected to {111}<110> dislocation glide whereas the bcc phase allowed slip on both {110}<111> and {112}<111>. A simplified microstructure with brick-shaped grains was used. The domain was meshed with 9x9x9 elements (C3D20), each element corresponding to one grain. A random crystallographic orientation was initially assigned to each grain. Figure 2 shows this simplified microstructure where each colored element corresponds to a different grain. Periodic boundary conditions were applied so that peripheral grains behaved as if they were in the bulk of the polycrystal. Such CPFEM strategy was already validated in numerous previous studies [7].

The deformation inside the different layers was considered to be a combination of three different deformation modes: ND-RD plane strain compression, shear parallel to the rolling plane and shear banding tilted at 45° from the rolling direction. These different modes were first investigated individually to assess their impact on the evolution of texture. A deformation allowing to achieve 50% thickness reduction of the rolled sheet was prescribed in all cases.

As shown in Figure 3, shear parallel to the rolling plane induces anti-clockwise rotation (~15°) of the pole figures that were predicted under orthotropic plane strain compression. Indeed, as highlighted in the figure, such shear is equivalent to compression along a direction that is inclined relative to RD. It is noteworthy that the (anticlockwise) rotation of the compression axis is opposite to the (clockwise) spin expected from shearing.

<figure><img src="image_1659.png" /><figcaption>Figure 3: Compression and shear deformation modes. Textures predicted to develop inside Fe or Al are presented as <111> pole figures.</figcaption></figure>
Crystal Plasticity Finite Element Modeling (CPFEM) [7] allowed us to predict the texture development, respectively, inside Al and Fe layers. Anisotropic plasticity at the level of individual grains was implemented as a user-defined material law (UMAT). As usually assumed, the fcc phase was subjected to {111}<110> dislocation glide whereas the bcc phase allowed slip on both {110}<111> and {112}<111>. A simplified microstructure with brick-shaped grains was used. The domain was meshed with 9x9x9 elements (C3D20), each element corresponding to one grain. A random crystallographic orientation was initially assigned to each grain. Figure 2 shows this simplified microstructure where each colored element corresponds to a different grain. Periodic boundary conditions were applied so that peripheral grains behaved as if they were in the bulk of the polycrystal. Such CPFEM strategy was already validated in numerous previous studies [7].

The deformation inside the different layers was considered to be a combination of three different deformation modes: ND-RD plane strain compression, shear parallel to the rolling plane and shear banding tilted at 45° from the rolling direction. These different modes were first investigated individually to assess their impact on the evolution of texture. A deformation allowing to achieve 50% thickness reduction of the rolled sheet was prescribed in all cases.

As shown in Figure 3, shear parallel to the rolling plane induces anti-clockwise rotation (~15°) of the pole figures that were predicted under orthotropic plane strain compression. Indeed, as highlighted in the figure, such shear is equivalent to compression along a direction that is inclined relative to RD. It is noteworthy that the (anticlockwise) rotation of the compression axis is opposite to the (clockwise) spin expected from shearing.

<figure><img src="image_1658.png" /><figcaption>Figure 3: Compression and shear deformation modes. Textures predicted to develop inside Fe or Al are presented as <111> pole figures.</figcaption></figure>
As shown in Fig. 4, the effect of shear bands inclined at +/-45° to the rolling direction is different. Such shear banding is equivalent to the superposition of orthotropic plane strain compression with rigid body rotation. The comparison of Fig. 3a and Fig. 4 shows that the pole figure rotation now conforms to the spin inside the shear band.

<figure><img src="image_1660.png" /><figcaption>Figure 4: Positive shear band and negative shear band modes. Textures developed inside Fe or Al microstructures are presented as <111> pole figures.</figcaption></figure>

The experimental pole figures shown in Fig. 1 are representative of the mean texture in each layer so that the local effects of shear bands are suppressed. Figure 5 indicates, in a qualitative manner, the type of shear expected, respectively, in the upper and lower layers of the multilaminate.

## Discussion

Textures in all the Fe layers, in Fig. 5, are close to conventional rolling textures predicted under plane strain compression. However, they are slightly rotated (~5°) in opposite directions on either side of the mid-thickness plane. This is consistent with the predictions of the macroscopic FE model: there is moderate shear parallel to the rolling plane in the hard phase. Note that the amplitude of rotation is much smaller than predicted in Figure 3 since there compression and shear occurred simultaneously in the real plate. The texture observed in the central Iron layer is orthorhombic indicating that shear was close to zero.

The EBSD measurements in aluminum reveal a weak “rotated cube” component. Such texture was already reported inside pure aluminum when continuous recrystallization occurred due to important shear deformation [8]. This too conforms to the prediction of the macroscopic FE model.
As shown in Fig. 4, the effect of shear bands inclined at +/-45° to the rolling direction is different. Such shear banding is equivalent to the superposition of orthotropic plane strain compression with rigid body rotation. The comparison of Fig. 3a and Fig. 4 shows that the pole figure rotation now conforms to the spin inside the shear band.

<figure><img src="image_1659.png" /><figcaption>Figure 4: Positive shear band and negative shear band modes. Textures developed inside Fe or Al microstructures are presented as <111> pole figures.</figcaption></figure>

The experimental pole figures shown in Fig. 1 are representative of the mean texture in each layer so that the local effects of shear bands are suppressed. Figure 5 indicates, in a qualitative manner, the type of shear expected, respectively, in the upper and lower layers of the multilaminate.

## Discussion

Textures in all the Fe layers, in Fig. 5, are close to conventional rolling textures predicted under plane strain compression. However, they are slightly rotated (~5°) in opposite directions on either side of the mid-thickness plane. This is consistent with the predictions of the macroscopic FE model: there is moderate shear parallel to the rolling plane in the hard phase. Note that the amplitude of rotation is much smaller than predicted in Figure 3 since there compression and shear occurred simultaneously in the real plate. The texture observed in the central Iron layer is orthorhombic indicating that shear was close to zero.

The EBSD measurements in aluminum reveal a weak “rotated cube” component. Such texture was already reported inside pure aluminum when continuous recrystallization occurred due to important shear deformation [8]. This too conforms to the prediction of the macroscopic FE model.
Crystal plasticity predictions of the texture development were less accurate in Al including a {111}<112>, not observed experimentally. The prediction of rolling textures in fcc metals remains a challenge.

<figure><img src="image_1661.png" /><figcaption>Figure 5: When considering shear parallel to the rolling plane, crystal plasticity predictions provide fair reproduction of the average texture inside each layer. The black arrows highlight the slight rotations relative to the orthorhombic texture expected under plane strain compression.</figcaption></figure>
Crystal plasticity predictions of the texture development were less accurate in Al including a {111}<112>, not observed experimentally. The prediction of rolling textures in fcc metals remains a challenge.

<figure><img src="image_1660.png" /><figcaption>Figure 5: When considering shear parallel to the rolling plane, crystal plasticity predictions provide fair reproduction of the average texture inside each layer. The black arrows highlight the slight rotations relative to the orthorhombic texture expected under plane strain compression.</figcaption></figure>
## Conclusions

* After two rolling passes of a 9-layer multilaminate, the hard Fe layers was fragmented with the emergence of inclined shear bands inside the softer Al layers.

* According to macroscopic FE modeling of the through-thickness deformation gradient, significant shear occurred parallel to the rolling plane

* Crystal plasticity predictions confirmed that individual layers underwent significant shear parallel to the rolling plane. The influence of inclined shear bands was not seen when averaging the texture over a whole layer.

* The texture predictions were more accurate in steel than in aluminum layers. The latter layers hosted a rotated-cube component which is characteristic of continuous recrystallization under intense shear deformation.

## Acknowledgements

Financial support for this work was provided by FNRS as well as by Eurofusion. LD is mandated by FNRS, Belgium. Computational resources were provided by the supercomputing facilities of the UCLouvain (CISM/UCLouvain) and the Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI) funded by the Fond de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under convention 2.5020.11 and by the Walloon Region.

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[4] Mishin, O. V., Bay, B., & Jensen, D. J. (2000). Through-thickness texture gradients in cold-rolled aluminum. Metallurgical and Materials Transactions A, 31, 1653-1662. https://doi.org/10.1007/s11661-000-0175-2

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[6] Schoenfeld, S.E., Asaro, R.J.: Through thickness texture gradients in rolled polycrystalline alloys. Int. J. Mech. Sci. 38(6), 661-683 (1996). https://doi.org/10.1016/S0020-7403(96)80008-7

[7] Lin, F., Marteleur, M., Jacques, P. J., & Delannay, L. (2018). Transmission of {332}<113> twins across grain boundaries in a metastable β-titanium alloy. International Journal of Plasticity, 105, 195-210. https://doi.org/10.1016/j.ijplas.2018.02.012

[8] Li, Z., Li, L. Y., Zhu, Y. B., Lin, K., Ren, Z. T., Yang, Y., ... & Langdon, T. G. (2022). Evidence for a stable single component sharp texture in high purity aluminum during tube high-pressure shearing at room temperature. Scientific Reports, 12(1), 17901. https://doi.org/10.1038/s41598-022-21717-z
## Conclusions

* After two rolling passes of a 9-layer multilaminate, the hard Fe layers was fragmented with the emergence of inclined shear bands inside the softer Al layers.

* According to macroscopic FE modeling of the through-thickness deformation gradient, significant shear occurred parallel to the rolling plane

* Crystal plasticity predictions confirmed that individual layers underwent significant shear parallel to the rolling plane. The influence of inclined shear bands was not seen when averaging the texture over a whole layer.

* The texture predictions were more accurate in steel than in aluminum layers. The latter layers hosted a rotated-cube component which is characteristic of continuous recrystallization under intense shear deformation.

## Acknowledgements

Financial support for this work was provided by FNRS as well as by Eurofusion. LD is mandated by FNRS, Belgium. Computational resources were provided by the supercomputing facilities of the UCLouvain (CISM/UCLouvain) and the Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI) funded by the Fond de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under convention 2.5020.11 and by the Walloon Region.

## References

[1] Hosseini, M., Pardis, N., Manesh, H. D., Abbasi, M., & Kim, D. I. (2017). Structural characteristics of Cu/Ti bimetal composite produced by accumulative roll-bonding (ARB). Materials & Design, 113, 128-136. https://doi.org/10.1016/j.matdes.2016.09.094

[2] McCabe, R. J., Nizolek, T. J., Li, N., Zhang, Y., Coughlin, D. R., Miller, C., & Carpenter, J. S. (2021). Evolution of microstructures and properties leading to layer instabilities during accumulative roll bonding of FeCu, FeAg, and FeAl. Materials & Design, 212, 110204. https://doi.org/10.1016/j.matdes.2021.110204

[3] Govindaraj, N. V., Frydendahl, J. G., & Holmedal, B. (2013). Layer continuity in accumulative roll bonding of dissimilar material combinations. Materials & Design (1980-2015), 52, 905-915. https://doi.org/10.1016/j.matdes.2013.06.031

[4] Mishin, O. V., Bay, B., & Jensen, D. J. (2000). Through-thickness texture gradients in cold-rolled aluminum. Metallurgical and Materials Transactions A, 31, 1653-1662. https://doi.org/10.1007/s11661-000-0175-2

[5] Engler, O., Tomé, C. N., & Huh, M. Y. (2000). A study of through-thickness texture gradients in rolled sheets. Metallurgical and materials transactions A, 31(9), 2299-2315. https://doi.org/10.1007/s11661-000-0146-7

[6] Schoenfeld, S.E., Asaro, R.J.: Through thickness texture gradients in rolled polycrystalline alloys. Int. J. Mech. Sci. 38(6), 661-683 (1996). https://doi.org/10.1016/S0020-7403(96)80008-7

[7] Lin, F., Marteleur, M., Jacques, P. J., & Delannay, L. (2018). Transmission of {332}<113> twins across grain boundaries in a metastable β-titanium alloy. International Journal of Plasticity, 105, 195-210. https://doi.org/10.1016/j.ijplas.2018.02.012

[8] Li, Z., Li, L. Y., Zhu, Y. B., Lin, K., Ren, Z. T., Yang, Y., ... & Langdon, T. G. (2022). Evidence for a stable single component sharp texture in high purity aluminum during tube high-pressure shearing at room temperature. Scientific Reports, 12(1), 17901. https://doi.org/10.1038/s41598-022-21717-z
[9] Hanon, G., Malet, L., Delannay, L. (2023). Texture and Microstructure After Roll-Bonding of an Fe-Al Multilaminate. In: Mocellin, K., Bouchard, P.O., Bigot, R., Balan, T. (eds) Proceedings of the 14th International Conference on the Technology of Plasticity - Current Trends in the Technology of Plasticity. ICTP 2023. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-40920-2_67

[10] Sidor, J., Petrov, R., & Kestens, L. (2014). Texture control in aluminum sheets by conventional and asymmetric rolling. In Comprehensive materials processing, vol. 3: advanced forming technologies (pp. 447-498). Elsevier. https://doi.org/10.1016/B978-0-08-096532-1.00324-1

[12] Tieu, A. K., & Liu, Y. J. (2004). Friction variation in the cold-rolling process. Tribology International, 37(2), 177-183. https://doi.org/10.1016/S0301-679X(03)00048-3
[9] Hanon, G., Malet, L., Delannay, L. (2023). Texture and Microstructure After Roll-Bonding of an Fe-Al Multilaminate. In: Mocellin, K., Bouchard, P.O., Bigot, R., Balan, T. (eds) Proceedings of the 14th International Conference on the Technology of Plasticity - Current Trends in the Technology of Plasticity. ICTP 2023. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-40920-2_67

[10] Sidor, J., Petrov, R., & Kestens, L. (2014). Texture control in aluminum sheets by conventional and asymmetric rolling. In Comprehensive materials processing, vol. 3: advanced forming technologies (pp. 447-498). Elsevier. https://doi.org/10.1016/B978-0-08-096532-1.00324-1

[12] Tieu, A. K., & Liu, Y. J. (2004). Friction variation in the cold-rolling process. Tribology International, 37(2), 177-183. https://doi.org/10.1016/S0301-679X(03)00048-3
# On the impact welding of dissimilar alloys for use in multimaterial skeletal fixation devices

SANGUEDOLCE Michela<fnref n="1" />, a*, ABDELMAOLA Mohammed<fnref n="2" />, b,
BORDA Francesco<fnref n="1" />, c, CHO Dae Hyun<fnref n="2" />, d, AVEY Thomas<fnref n="2" />, e,
OLIVAS-ALANIS Luis H.<fnref n="2" />, f, CHMIELEWSKA Agnieszka<fnref n="2" />, g, VIVEK Anupam<fnref n="2" />, h,
DAEHN Glenn<fnref n="2" />, i, LUO Alan A.<fnref n="2" />, l, PANTON Boyd<fnref n="2" />, m, FILICE Luigino<fnref n="1" />, n and
DEAN David<fnref n="2" />, 3, o

**Keywords:** Finite Element Analysis, Impact Welding, Dissimilar Welding, NiTi Alloys, Mg Alloys, Skeletal Fixation

**Abstract.** Multi-material skeletal fixators appear to be a promising approach to reduce failure due to the high stiffness of standard-of-care fixators. Nevertheless, joining different materials is challenging due to their different properties. High-velocity impact welding, a solid-state welding process, involves the collision of a “flyer” (moving) part with a stationary “target” at very high speed (i.e., hundreds of meters per second). In this paper we present a preliminary experimental campaign to use laser impact welding to join *NiTi* and *Mg* alloy *Mg-1.2Zn-0.5Ca-0.5Mn* (wt%) sheets and the parallel development of a finite element model to allow gathering further insights into the complex phenomena involved in the process. Preliminary results show the deposition of the *Mg* alloy on *NiTi* sheets by tuning the joining process conditions and promising results of the numerical model in terms of accordance with experiments: these findings provide the basis for further process optimization, numerical model calibration and the application of a valid protocol for multi-material skeletal fixation devices.

**Introduction**

Excessively stiff skeletal fixators are well known to interrupt physiological loading patterns of the contacting bone, bringing to the stress shielding phenomenon and early device failure, which hinders patients' quality of life [1]. The choice of materials with lower stiffness able to restore the physiological loading patterns into bone, as compared to the current and overly stiff gold standard *Ti6Al4V* alloy, represents a viable strategy to prevent these phenomena from occurring. Attempts to develop multi-material skeletal fixators are of current interest in this research field [2], including patent applications [3]. Within the investigated materials, *NiTi* alloys and *Mg* alloys play a major role due to the superelasticity and lower stiffness of the former - compared to the current standard of care alloys - and the resorbability of the latter.

A critical step in the design of these multi-material devices is the joining. Friction stir processes have been successfully employed to obtain *NiTi* – *Mg* alloy composites [4] and welds [5]. However, the current configuration of the process constraints the size of the parts that can be obtained, although micro-level friction stir processing is currently being studied [6]. Furthermore,

<figure><img src="image_1662.png" /></figure>

<footnote n="1">¹Dept. of Mechanical, Energy and Management Engineering, University of Calabria, Rende, CS, Italy</footnote>

<footnote n="2">²Dept. of Materials Science and Engineering, The Ohio State University, Columbus, OH, USA</footnote>

<footnote n="3">³Dept. of Plastic and Reconstructive Surgery, The Ohio State University, Columbus, OH, USA</footnote>

<footnote>aMichela.sanguedolce@unical.it, bAbdelmaola.1@osu.edu, cfrancesco.borda@unical.it,</footnote>

<footnote>dcho.1151@osu.edu, eavey.13@osu.edu, folivas.11@osu.edu, gachmielewska00@gmail.com,</footnote>

<footnote>hVivek.5@osu.edu, iDaehn.1@osu.edu, l Luo.445@osu.edu, mpanton.7@osu.edu,</footnote>

<footnote>nLuigino.filice@unical.it, odavid.dean@osumc.edu</footnote>
# On the impact welding of dissimilar alloys for use in multimaterial skeletal fixation devices

SANGUEDOLCE Michela<fnref n="1" />, a*, ABDELMAOLA Mohammed<fnref n="2" />, b,
BORDA Francesco<fnref n="1" />, c, CHO Dae Hyun<fnref n="2" />, d, AVEY Thomas<fnref n="2" />, e,
OLIVAS-ALANIS Luis H.<fnref n="2" />, f, CHMIELEWSKA Agnieszka<fnref n="2" />, g, VIVEK Anupam<fnref n="2" />, h,
DAEHN Glenn<fnref n="2" />, i, LUO Alan A.<fnref n="2" />, l, PANTON Boyd<fnref n="2" />, m, FILICE Luigino<fnref n="1" />, n and
DEAN David<fnref n="2" />, 3, o

**Keywords:** Finite Element Analysis, Impact Welding, Dissimilar Welding, NiTi Alloys, Mg Alloys, Skeletal Fixation

**Abstract.** Multi-material skeletal fixators appear to be a promising approach to reduce failure due to the high stiffness of standard-of-care fixators. Nevertheless, joining different materials is challenging due to their different properties. High-velocity impact welding, a solid-state welding process, involves the collision of a “flyer” (moving) part with a stationary “target” at very high speed (i.e., hundreds of meters per second). In this paper we present a preliminary experimental campaign to use laser impact welding to join *NiTi* and *Mg* alloy *Mg-1.2Zn-0.5Ca-0.5Mn* (wt%) sheets and the parallel development of a finite element model to allow gathering further insights into the complex phenomena involved in the process. Preliminary results show the deposition of the *Mg* alloy on *NiTi* sheets by tuning the joining process conditions and promising results of the numerical model in terms of accordance with experiments: these findings provide the basis for further process optimization, numerical model calibration and the application of a valid protocol for multi-material skeletal fixation devices.

## Introduction

Excessively stiff skeletal fixators are well known to interrupt physiological loading patterns of the contacting bone, bringing to the stress shielding phenomenon and early device failure, which hinders patients' quality of life [1]. The choice of materials with lower stiffness able to restore the physiological loading patterns into bone, as compared to the current and overly stiff gold standard *Ti6Al4V* alloy, represents a viable strategy to prevent these phenomena from occurring. Attempts to develop multi-material skeletal fixators are of current interest in this research field [2], including patent applications [3]. Within the investigated materials, *NiTi* alloys and *Mg* alloys play a major role due to the superelasticity and lower stiffness of the former - compared to the current standard of care alloys - and the resorbability of the latter.

A critical step in the design of these multi-material devices is the joining. Friction stir processes have been successfully employed to obtain *NiTi* – *Mg* alloy composites [4] and welds [5]. However, the current configuration of the process constraints the size of the parts that can be obtained, although micro-level friction stir processing is currently being studied [6]. Furthermore,

<figure><img src="image_1661.png" /></figure>

<footnote n="1">¹Dept. of Mechanical, Energy and Management Engineering, University of Calabria, Rende, CS, Italy</footnote>

<footnote n="2">²Dept. of Materials Science and Engineering, The Ohio State University, Columbus, OH, USA</footnote>

<footnote n="3">³Dept. of Plastic and Reconstructive Surgery, The Ohio State University, Columbus, OH, USA</footnote>

<footnote>aMichela.sanguedolce@unical.it, bAbdelmaola.1@osu.edu, cfrancesco.borda@unical.it,</footnote>

<footnote>dcho.1151@osu.edu, eavey.13@osu.edu, folivas.11@osu.edu, gachmielewska00@gmail.com,</footnote>

<footnote>hVivek.5@osu.edu, iDaehn.1@osu.edu, juo.445@osu.edu, mpanton.7@osu.edu,</footnote>

<footnote>nLuigino.filice@unical.it, odavid.dean@osumc.edu</footnote>
these alloys have extremely different physical and metallurgical properties, thus a good quality joint is difficult to obtain without compromising the materials' properties [7].

High-velocity impact welding, within the solid-state welding processes, involves the high-speed collision (100 – 1000 m/s [8]) of a flyer part onto a target part generating a jet able to clean and expose non-contaminated surfaces to obtain metallurgical bonding. It represents a viable solution to obtain dissimilar welding on the millimeter/micron-scale and, within a defined weldability window, preventing the formation of poor quality intermetallic compounds at the weld interface, due to the low welding temperatures involved and the fast welding speed [9]. Impact welding can be implemented on different length scales using different means to accelerate the flyer, depending on the size of the parts to be welded [8].

The possibility to obtain dissimilar welding of NiTi without affecting its superelasticity was demonstrated by using solid-state welding processes like Vaporizing Foil Actuator Welding [10] and Laser Impact Welding, by properly selecting processing conditions [11].

Laser impact welding (LIW), as a collision welding process, involves a moving part (flyer) launched towards a stationary part (target) following the formation of an expanding plasma. Plasma is generated by the vaporization of a sacrificial ablative layer and confined into a properly selected confinement layer (Fig.1). The flyer travels across a specific distance to gain the impact velocity that, under optimized conditions, allows the creation of a jet able to break down surface oxides and other contaminants and to verify the conditions necessary for welding. Typically, the welded joint exhibits a ring-like profile [9]. The combination of the laser system parameters (i.e. laser energy and spot size), the choice of the confinement and ablative layers and the travel distance of the flyer as well as its thickness all determine the velocity and the dynamic angle assumed by the flyer during the impact [12]. Generally, a low thickness of the flyer brings to a higher impact speed but implies a lower joint strength [12]. The effect of the flyer's thickness is more marked below specific threshold values determined by welding conditions, including the specifics of the laser system [13]. High surface roughness of the target (in the order of tenths of µm) was found to affect the collision angle and the weldability window [14], as also summarized in [15].

<figure><img src="image_1663.png" /><figcaption>Fig. 1 – Schematic of the laser impact welding process.</figcaption></figure>

Nevertheless, the mentioned processes require in-depth investigation as the final application of interest, i.e. bone fixators, further adds constraints to the materials to be used into the body and their conditions after processing [16]. Also, processes like laser impact welding still feature issues, in particular in terms of the shape of the joints that can be obtained and damage of the flyer and, often, the target, thus requiring extensive experimental campaigns and specific design of the parts to be welded to properly accommodate process constraints [12].

To reduce the effort devoted to extensive experimental testing, both in terms of time and costs, predictive models of the impact welding processes are currently being studied [17].
these alloys have extremely different physical and metallurgical properties, thus a good quality joint is difficult to obtain without compromising the materials' properties [7].

High-velocity impact welding, within the solid-state welding processes, involves the high-speed collision (100 – 1000 m/s [8]) of a flyer part onto a target part generating a jet able to clean and expose non-contaminated surfaces to obtain metallurgical bonding. It represents a viable solution to obtain dissimilar welding on the millimeter/micron-scale and, within a defined weldability window, preventing the formation of poor quality intermetallic compounds at the weld interface, due to the low welding temperatures involved and the fast welding speed [9]. Impact welding can be implemented on different length scales using different means to accelerate the flyer, depending on the size of the parts to be welded [8].

The possibility to obtain dissimilar welding of NiTi without affecting its superelasticity was demonstrated by using solid-state welding processes like Vaporizing Foil Actuator Welding [10] and Laser Impact Welding, by properly selecting processing conditions [11].

Laser impact welding (LIW), as a collision welding process, involves a moving part (flyer) launched towards a stationary part (target) following the formation of an expanding plasma. Plasma is generated by the vaporization of a sacrificial ablative layer and confined into a properly selected confinement layer (Fig.1). The flyer travels across a specific distance to gain the impact velocity that, under optimized conditions, allows the creation of a jet able to break down surface oxides and other contaminants and to verify the conditions necessary for welding. Typically, the welded joint exhibits a ring-like profile [9]. The combination of the laser system parameters (i.e. laser energy and spot size), the choice of the confinement and ablative layers and the travel distance of the flyer as well as its thickness all determine the velocity and the dynamic angle assumed by the flyer during the impact [12]. Generally, a low thickness of the flyer brings to a higher impact speed but implies a lower joint strength [12]. The effect of the flyer's thickness is more marked below specific threshold values determined by welding conditions, including the specifics of the laser system [13]. High surface roughness of the target (in the order of tenths of µm) was found to affect the collision angle and the weldability window [14], as also summarized in [15].

<figure><img src="image_1662.png" /><figcaption>Fig. 1 – Schematic of the laser impact welding process.</figcaption></figure>

Nevertheless, the mentioned processes require in-depth investigation as the final application of interest, i.e. bone fixators, further adds constraints to the materials to be used into the body and their conditions after processing [16]. Also, processes like laser impact welding still feature issues, in particular in terms of the shape of the joints that can be obtained and damage of the flyer and, often, the target, thus requiring extensive experimental campaigns and specific design of the parts to be welded to properly accommodate process constraints [12].

To reduce the effort devoted to extensive experimental testing, both in terms of time and costs, predictive models of the impact welding processes are currently being studied [17].
This work presents an experimental setup dedicated to employing the laser impact welding process to obtain spot joints between Nickel-Titanium and Magnesium alloy sheets. In parallel, a preliminary finite element model of the process to be used for weldability window prediction has been developed and described.

## Methodology

**Experimental tests.** A commercially available Nd:YAG laser source was used, in a Continuum PowerliteTM Precision II Scientific System. The maximum output energy of the laser is 3.1 J with a pulse width of 8.0 ns, and a wavelength of 1064 nm. The laser system can operate up to a 10 Hz pulse repetition rate or in single shot mode. In this work, a laser energy of 3.1 J was used.

*NiTi#1* alloy foils provided by Fort Wayne Metals (IN, USA) with a size or 25 mm x 7 mm and thickness of 250 µm and a customized Mg alloy *Mg-1.2Zn-0.5Ca-0.5Mn* (wt%) foils [18] with a size of 15 mm x 15 mm and a thickness of 200 µm were tested for welding. Welding parameters such as laser spot size and relative distance (stand-off) between flyer and target were varied. In particular two values of the laser spot size were tested: 3.00 mm and 5.22 mm. As stand-off distance, values of 300 and 500 µm have been used.

As suggested in literature [8] the alloy with higher ductility generally outperforms the less ductile material as flyer, thus *Mg* alloy foils were preliminary used as flyer, while *NiTi* alloy foils were used as target. The *NiTi* alloy foils were used in as-rolled conditions while the *Mg* alloy foils were ground up to #1200. Both flyer and target were sonicated in ethanol before welding.

The testing configuration is shown in Fig. 2. Target and flyer are placed on a metallic support (necessary as a support for the thin target [13]) locked on an holder to be located into the welding chamber. Black tape was used as backing to reduce rebound [18].

The stand-off distance between target and flyer was given by metallic ring spacers. The ablative layer in this case is given by a layer of GPN spread upon the flyer and covered by the confinement layer. Borosilicate glass and acrylic, with optimized thickness according to the laser system conditions, were both tested as confinement layers to tune the reaction force on the flyer [12].

<figure><img src="image_1664.png" /><figcaption>Fig. 2 – Images of the experimental setup for the laser impact welding tests.</figcaption></figure>

**Finite Element Model.** The numerical procedure was developed through a finite element model of the laser impact welding process using the SFTC Deform 2D® software, simplifying the experimental setup into 2D axisymmetric simulations, given the axisymmetric nature of the welded spot [19] and involving an automatic remeshing routine to handle the large plastic deformation involved. The target and flyer materials were modeled as plastic bodies and divided into 1500 elements each. The average element edge length was about 20 µm. Flow stress data for flyer and target were obtained from [20], considering the high strain rate induced by the process.

The bottom metallic support, spacer and top holder were defined as rigid bodies. The global heat transfer coefficient with the environment was established between the flyer and the target and
This work presents an experimental setup dedicated to employing the laser impact welding process to obtain spot joints between Nickel-Titanium and Magnesium alloy sheets. In parallel, a preliminary finite element model of the process to be used for weldability window prediction has been developed and described.

## Methodology

**Experimental tests.** A commercially available Nd:YAG laser source was used, in a Continuum PowerliteTM Precision II Scientific System. The maximum output energy of the laser is 3.1 J with a pulse width of 8.0 ns, and a wavelength of 1064 nm. The laser system can operate up to a 10 Hz pulse repetition rate or in single shot mode. In this work, a laser energy of 3.1 J was used.

*NiTi#1* alloy foils provided by Fort Wayne Metals (IN, USA) with a size or 25 mm x 7 mm and thickness of 250 µm and a customized Mg alloy *Mg-1.2Zn-0.5Ca-0.5Mn* (wt%) foils [18] with a size of 15 mm x 15 mm and a thickness of 200 µm were tested for welding. Welding parameters such as laser spot size and relative distance (stand-off) between flyer and target were varied. In particular two values of the laser spot size were tested: 3.00 mm and 5.22 mm. As stand-off distance, values of 300 and 500 µm have been used.

As suggested in literature [8] the alloy with higher ductility generally outperforms the less ductile material as flyer, thus *Mg* alloy foils were preliminary used as flyer, while *NiTi* alloy foils were used as target. The *NiTi* alloy foils were used in as-rolled conditions while the *Mg* alloy foils were ground up to #1200. Both flyer and target were sonicated in ethanol before welding.

The testing configuration is shown in Fig. 2. Target and flyer are placed on a metallic support (necessary as a support for the thin target [13]) locked on an holder to be located into the welding chamber. Black tape was used as backing to reduce rebound [18].

The stand-off distance between target and flyer was given by metallic ring spacers. The ablative layer in this case is given by a layer of GPN spread upon the flyer and covered by the confinement layer. Borosilicate glass and acrylic, with optimized thickness according to the laser system conditions, were both tested as confinement layers to tune the reaction force on the flyer [12].

<figure><img src="image_1663.png" /><figcaption>Fig. 2 – Images of the experimental setup for the laser impact welding tests.</figcaption></figure>

**Finite Element Model.** The numerical procedure was developed through a finite element model of the laser impact welding process using the SFTC Deform 2D® software, simplifying the experimental setup into 2D axisymmetric simulations, given the axisymmetric nature of the welded spot [19] and involving an automatic remeshing routine to handle the large plastic deformation involved. The target and flyer materials were modeled as plastic bodies and divided into 1500 elements each. The average element edge length was about 20 µm. Flow stress data for flyer and target were obtained from [20], considering the high strain rate induced by the process.

The bottom metallic support, spacer and top holder were defined as rigid bodies. The global heat transfer coefficient with the environment was established between the flyer and the target and
it was set equal to $10^5$ kW/(m²K) according to literature [21] while no thermal effects due to the plasma have been included in the simulation tests, due to the adiabatic nature of the process demonstrated in [15].

The plasma pressure distribution was simulated according to the equation (1) obtained by Sunny and Gleason et al [17, 23]:

$$p(x) = \frac{1}{\bar{\sigma}\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x}{\bar{\sigma}}\right)^2} \quad (1)$$

Where x is the radial distance from the spot center (Fig. 3) and $\bar{\sigma}$ represents the standard deviation of the experimental laser intensity distribution [22].

The impact speed on the top surface of the flyer reached right before the impact [10] and the friction at the interface between flyer and target were used as input to the finite element model while the interface temperature on the flyer was analyzed as output. As regards the impact speed, a range of values between 100 m/s and 900 m/s was tested which, based on literature experimental measurements, incorporates in a complex fashion the effects of the travel distance (stand-off), laser parameters set to accelerate the flyer, and the flyer thickness [8]. A 90% fraction of the plastic work was set to be converted to heat [15].

Due to the process characteristics, the friction at the interface between flyer and target was modeled using a constant shear friction based on [23], where the friction factor is defined by equation (2):

$$m = \frac{f_s}{k} \quad (2)$$

Being $f_s$ the frictional stress and k the shear yield stress. Values of the friction factor m between 0.1 and 0.3 have been tested [24], with steps of 0.05 since friction plays a pivotal role into local temperatures and process evolution [17]. Table 1 reports a summary of the studied values for impact speed v and friction m while Figure 3 shows the configuration of the finite element model.

<table><caption>Table 1 – Values investigated for friction factor m values and impact speed v.</caption><thead><tr><th>m [-]</th><th>v [m/s]</th></tr></thead><tbody><tr><td>0.10</td><td>100</td></tr><tr><td>0.15</td><td>200</td></tr><tr><td>0.20</td><td>300</td></tr><tr><td>0.25</td><td>600</td></tr><tr><td>0.30</td><td>900</td></tr></tbody></table>
it was set equal to $10^5$ kW/(m²K) according to literature [21] while no thermal effects due to the plasma have been included in the simulation tests, due to the adiabatic nature of the process demonstrated in [15].

The plasma pressure distribution was simulated according to the equation (1) obtained by Sunny and Gleason et al [17, 23]:

$$p(x) = \frac{1}{\bar{\sigma}\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x}{\bar{\sigma}}\right)^2} \quad (1)$$

Where x is the radial distance from the spot center (Fig. 3) and $\bar{\sigma}$ represents the standard deviation of the experimental laser intensity distribution [22].

The impact speed on the top surface of the flyer reached right before the impact [10] and the friction at the interface between flyer and target were used as input to the finite element model while the interface temperature on the flyer was analyzed as output. As regards the impact speed, a range of values between 100 m/s and 900 m/s was tested which, based on literature experimental measurements, incorporates in a complex fashion the effects of the travel distance (stand-off), laser parameters set to accelerate the flyer, and the flyer thickness [8]. A 90% fraction of the plastic work was set to be converted to heat [15].

Due to the process characteristics, the friction at the interface between flyer and target was modeled using a constant shear friction based on [23], where the friction factor is defined by equation (2):

$$m = \frac{f_s}{k} \quad (2)$$

Being $f_s$ the frictional stress and k the shear yield stress. Values of the friction factor m between 0.1 and 0.3 have been tested [24], with steps of 0.05 since friction plays a pivotal role into local temperatures and process evolution [17]. Table 1 reports a summary of the studied values for impact speed v and friction m while Figure 3 shows the configuration of the finite element model.

<table><caption>Table 1 – Values investigated for friction factor m values and impact speed v.</caption><thead><tr><th>m [-]</th><th>v [m/s]</th></tr></thead><tbody><tr><td>0.10</td><td>100</td></tr><tr><td>0.15</td><td>200</td></tr><tr><td>0.20</td><td>300</td></tr><tr><td>0.25</td><td>600</td></tr><tr><td>0.30</td><td>900</td></tr></tbody></table>
<figure><img src="image_1665.png" /><figcaption>Fig. 3 – Finite Element Model setup on SFTC Deform 2D®.</figcaption></figure>

## Results and Discussion

**Experimental tests.** The experimental campaign carried out confirmed the influence of the confinement layer material on the weld obtainment [8]. In fact, under the tested conditions, using the acrylic material brought to a higher quantity of deposited flyer material onto the target: Fig. 4 shows the successful deposition of the *Mg* alloy flyer onto the surface of the *NiTi* alloy target for a laser spot size of 5.22 mm using acrylic (Fig. 4b) as compared to borosilicate glass (Fig. 4a). Further assessments are necessary by finely tuning laser energy and spot size as less damage to the flyer was reported for increased laser spot size, and by lowering laser beam energy density [13].

<figure><img src="image_1666.png" /><figcaption>Fig. 4 – Macrographs of the NiTi targets, with varying confinement layer material a) absence of adhered material from the flyer by using borosilicate glass and b) adhered material from the flyer by using acrylic.</figcaption></figure>

**Finite Element Model.** A range of processing conditions bringing to local melting (around 650 °C [25]) of the *Mg* alloy material was detected at the interface between target and flyer (Fig. 6). This is in accordance with the possible interface melting during the process [8] due to the high plastic deformation and the high strain rates involved in impact welding, together with consequent increase in temperature due to plastic dissipative heating and friction [26]. Local temperature rise might prevent a successful joint if the temperature reached far exceeds the melting temperature of the alloy, thus highlighting the importance to predict the temperature trends.

Extremely high strain rates around 10⁶ - 10⁷ s⁻¹ were obtained for the flyer, as also reported in [18]. The presence of a range where local melting does not occur reflects the possibility to reach
<figure><img src="image_1664.png" /><figcaption>Fig. 3 – Finite Element Model setup on SFTC Deform 2D®.</figcaption></figure>

## Results and Discussion

**Experimental tests.** The experimental campaign carried out confirmed the influence of the confinement layer material on the weld obtainment [8]. In fact, under the tested conditions, using the acrylic material brought to a higher quantity of deposited flyer material onto the target: Fig. 4 shows the successful deposition of the *Mg* alloy flyer onto the surface of the *NiTi* alloy target for a laser spot size of 5.22 mm using acrylic (Fig. 4b) as compared to borosilicate glass (Fig. 4a). Further assessments are necessary by finely tuning laser energy and spot size as less damage to the flyer was reported for increased laser spot size, and by lowering laser beam energy density [13].

<figure><img src="image_1665.png" /><figcaption>Fig. 4 – Macrographs of the NiTi targets, with varying confinement layer material a) absence of adhered material from the flyer by using borosilicate glass and b) adhered material from the flyer by using acrylic.</figcaption></figure>

**Finite Element Model.** A range of processing conditions bringing to local melting (around 650 °C [25]) of the *Mg* alloy material was detected at the interface between target and flyer (Fig. 6). This is in accordance with the possible interface melting during the process [8] due to the high plastic deformation and the high strain rates involved in impact welding, together with consequent increase in temperature due to plastic dissipative heating and friction [26]. Local temperature rise might prevent a successful joint if the temperature reached far exceeds the melting temperature of the alloy, thus highlighting the importance to predict the temperature trends.

Extremely high strain rates around 10⁶ - 10⁷ s⁻¹ were obtained for the flyer, as also reported in [18]. The presence of a range where local melting does not occur reflects the possibility to reach
optimum conditions for the process and possibly define a weldability window [9]. Weldability can be defined based on the local stress reached at the interface and related to impact velocity, since bonding has been found to happen within a certain threshold range that varies as a function of the materials to be joined [27].

By both looking at the results in the 3D plot of maximum temperatures in Fig. 5, it is evident that the predictions are sensitive to the coefficient of friction [28], much more than to the impact velocity. Thus, highlighting the relevance of a correct estimation of the friction factor for the proper calibration of the model, being friction related to multiple factors such as cleanliness of the surfaces, oxides, local temperature and pressure conditions [29].

Fig. 6 shows the temperature distribution on the flyer for the conditions v 100 m/s and m 0.10, at the interface with the target, highlighting a maximum along a ring having the size of the laser spot. The significant rise in the temperature of the flyer was located within tenths of µm distance from the interface with the target.

<figure><img src="image_1667.png" /><figcaption>Fig. 5 – 3D plot of maximum temperature reached on the Mg flyer at the flyer-target interface.</figcaption></figure>
optimum conditions for the process and possibly define a weldability window [9]. Weldability can be defined based on the local stress reached at the interface and related to impact velocity, since bonding has been found to happen within a certain threshold range that varies as a function of the materials to be joined [27].

By both looking at the results in the 3D plot of maximum temperatures in Fig. 5, it is evident that the predictions are sensitive to the coefficient of friction [28], much more than to the impact velocity. Thus, highlighting the relevance of a correct estimation of the friction factor for the proper calibration of the model, being friction related to multiple factors such as cleanliness of the surfaces, oxides, local temperature and pressure conditions [29].

Fig. 6 shows the temperature distribution on the flyer for the conditions v 100 m/s and m 0.10, at the interface with the target, highlighting a maximum along a ring having the size of the laser spot. The significant rise in the temperature of the flyer was located within tenths of µm distance from the interface with the target.

<figure><img src="image_1666.png" /><figcaption>Fig. 5 – 3D plot of maximum temperature reached on the Mg flyer at the flyer-target interface.</figcaption></figure>
<figure><img src="image_1668.png" /><figcaption>Fig. 6 – Maximum temperature reached on the Mg flyer at the flyer-target interface for the configuration at v 100 m/s and m 0.10 (a) 2D view, (b) 3D view.</figcaption></figure>

## Summary and Outlook

The current work showed a parallel development of both experimental and numerical testing configurations for the joining of NiTi and Mg alloys towards the development of multi-material skeletal fixation devices. From an experimental point of view, preliminary results confirm the possibility to deposit Mg alloys on NiTi alloys as also anticipated from the literature. Nevertheless, the whole testing process involves a great amount of trials in order to locate a possible weldability window, able to preserve both flyer integrity and prevent local melting and materials mixing at the weld interface. The finite element modeling shows promising results in terms of accordance with literature trends.

Future development will involve the calibration of the model to be used as a predictive tool for future experimental campaigns. Modeling aspects concerning the effects of materials flow stress - also including the effects of alloying elements -, materials damage, welding criteria based on local stress increase as studied in [30] will be then included. Measurement of the experimental values of the impact velocity will also be addressed to connect experimental and simulation results.

## References

[1] A. Chmielewska and D. Dean, The role of stiffness-matching in avoiding stress shielding-induced bone loss and stress concentration-induced skeletal reconstruction device failure. Acta Biomater. 173. 51–65. (Jan. 2024). https://doi.org/10.1016/j.actbio.2023.11.011

[2] M. S. Gogheri, M. Kasiri-Asgarani, H. R. Bakhsheshi-Rad, H. Ghayour, and M. Rafiei, Mechanical properties, corrosion behavior and biocompatibility of orthopedic pure titanium–magnesium alloy screw prepared by friction welding. Trans. Nonferrous Met. Soc. China
<figure><img src="image_1667.png" /><figcaption>Fig. 6 – Maximum temperature reached on the Mg flyer at the flyer-target interface for the configuration at v 100 m/s and m 0.10 (a) 2D view, (b) 3D view.</figcaption></figure>

**Summary and Outlook**

The current work showed a parallel development of both experimental and numerical testing configurations for the joining of *NiTi* and *Mg* alloys towards the development of multi-material skeletal fixation devices. From an experimental point of view, preliminary results confirm the possibility to deposit *Mg* alloys on *NiTi* alloys as also anticipated from the literature. Nevertheless, the whole testing process involves a great amount of trials in order to locate a possible weldability window, able to preserve both flyer integrity and prevent local melting and materials mixing at the weld interface. The finite element modeling shows promising results in terms of accordance with literature trends.

Future development will involve the calibration of the model to be used as a predictive tool for future experimental campaigns. Modeling aspects concerning the effects of materials flow stress - also including the effects of alloying elements -, materials damage, welding criteria based on local stress increase as studied in [30] will be then included. Measurement of the experimental values of the impact velocity will also be addressed to connect experimental and simulation results.

**References**

[1] A. Chmielewska and D. Dean, The role of stiffness-matching in avoiding stress shielding-induced bone loss and stress concentration-induced skeletal reconstruction device failure. Acta Biomater. 173. 51–65. (Jan. 2024). https://doi.org/10.1016/j.actbio.2023.11.011

[2] M. S. Gogheri, M. Kasiri-Asgarani, H. R. Bakhsheshi-Rad, H. Ghayour, and M. Rafiei, Mechanical properties, corrosion behavior and biocompatibility of orthopedic pure titanium–magnesium alloy screw prepared by friction welding. Trans. Nonferrous Met. Soc. China
(English Ed. 30 (11). 2952-2966. (2020). https://doi.org/10.1016/S1003-6326(20)65434-6

[3] A. Chmielewska, A. Luo, B. Panton, L. Olivas, and D. Dean, Multimaterial Stiffness-Matched Implants (US Provisional Patent Application)

[4] N. Gangil, A. N. Siddiquee, S. Mufazzal, S. M. Muzakkir, and S. Maheshwari, Shape memory alloy based NiTi reinforced functionally graded material for vibration damping. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 235 (12). 2771-2782. (2021). https://doi.org/10.1177/14644207211035521

[5] S. Ji, W. Hu, Z. Ma, Q. Li, and X. Gong, Friction Stir Lap Welding of Mg/Ti Dissimilar Alloys Using a Slight Penetration Depth. JOM. 72 (4). 1589-1596. (Apr. 2020). https://doi.org/10.1007/s11837-019-03959-x

[6] M. Sen, S. Shankar, and S. Chattopadhyaya, Micro-friction stir welding (µFSW) – A review. Mater. Today Proc. 27. 2469-2473. (2020). https://doi.org/10.1016/j.matpr.2019.09.220

[7] A. Shamsolhodaei, J. P. Oliveira, B. Panton, B. Ballesteros, N. Schell, and Y. N. Zhou, Superelasticity preservation in dissimilar joint of NiTi shape memory alloy to biomedical PtIr. Materialia. 16 (November 2020). 101090. (2021). https://doi.org/10.1016/j.mtla.2021.101090

[8] W. Cai et al., A state-of-the-art review on solid-state metal joining. J. Manuf. Sci. Eng. Trans. ASME. 141 (3). (2019). https://doi.org/10.1115/1.4041182

[9] H. Wang and Y. Wang, High-velocity impact welding process: A review. Metals (Basel). 9 (2). (2019). https://doi.org/10.3390/met9020144

[10] F. Du, L. Deng, X. Wang, M. Zhang, J. Jin, and J. Zhang, Study on interfacial characteristics and properties of NiTi/Al-Mg joint by vaporizing foil actuator welding. J. Mater. Res. Technol. 20. 3429-3440. (Sep. 2022). https://doi.org/10.1016/j.jmrt.2022.08.090

[11] J. Li, B. Panton, S. Liang, A. Vivek, and G. Daehn, High strength welding of NiTi and stainless steel by impact: Process, structure and properties. Mater. Today Commun. 25 (March). 101306. (2020). https://doi.org/10.1016/j.mtcomm.2020.101306

[12] K. Wang, H. Wang, H. Zhou, W. Zheng, and A. Xu, Research status and prospect of laser impact welding. Metals (Basel). 10 (11). 1-18. (2020). https://doi.org/10.3390/met10111444

[13] H. Wang, A. Vivek, Y. Wang, G. Taber, and G. S. Daehn, Laser impact welding application in joining aluminum to titanium. J. Laser Appl. 28 (3). (2016). https://doi.org/10.2351/1.4946887

[14] J. Bellmann et al., Particle Ejection by Jetting and Related Effects in Impact Welding Processes. Metals (Basel). 10 (8). (2020). https://doi.org/10.3390/met10081108

[15] G. Gleason, K. Bailey, S. Sunny, A. Malik, and R. A. Bernal, Influence of surface roughness on the transient interfacial phenomena in laser impact welding. J. Manuf. Process. 80 (February). 480-490. (2022). https://doi.org/10.1016/j.jmapro.2022.06.022

[16] W. R. Wagner, S. E. Sakiyama-Elbert, G. Zhang, and M. J. Yaszemski, Biomaterials Science: An Introduction to Materials in Medicine. 2020. https://doi.org/10.1016/C2017-0-02323-6

[17] A. Nassiri, G. Chini, A. Vivek, G. Daehn, and B. Kinsey, Arbitrary Lagrangian-Eulerian finite element simulation and experimental investigation of wavy interfacial morphology during high velocity impact welding. Mater. Des. 88. 345-358. (2015). https://doi.org/10.1016/j.matdes.2015.09.005

[18] S. Sadeh et al., Simulation and experimental comparison of laser impact welding with a
(English Ed. 30 (11). 2952-2966. (2020). https://doi.org/10.1016/S1003-6326(20)65434-6

[3] A. Chmielewska, A. Luo, B. Panton, L. Olivas, and D. Dean, Multimaterial Stiffness-Matched Implants (US Provisional Patent Application)

[4] N. Gangil, A. N. Siddiquee, S. Mufazzal, S. M. Muzakkir, and S. Maheshwari, Shape memory alloy based NiTi reinforced functionally graded material for vibration damping. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 235 (12). 2771-2782. (2021). https://doi.org/10.1177/14644207211035521

[5] S. Ji, W. Hu, Z. Ma, Q. Li, and X. Gong, Friction Stir Lap Welding of Mg/Ti Dissimilar Alloys Using a Slight Penetration Depth. JOM. 72 (4). 1589-1596. (Apr. 2020). https://doi.org/10.1007/s11837-019-03959-x

[6] M. Sen, S. Shankar, and S. Chattopadhyaya, Micro-friction stir welding (µFSW) – A review. Mater. Today Proc. 27. 2469-2473. (2020). https://doi.org/10.1016/j.matpr.2019.09.220

[7] A. Shamsolhodaei, J. P. Oliveira, B. Panton, B. Ballesteros, N. Schell, and Y. N. Zhou, Superelasticity preservation in dissimilar joint of NiTi shape memory alloy to biomedical PtIr. Materialia. 16 (November 2020). 101090. (2021). https://doi.org/10.1016/j.mtla.2021.101090

[8] W. Cai et al., A state-of-the-art review on solid-state metal joining. J. Manuf. Sci. Eng. Trans. ASME. 141 (3). (2019). https://doi.org/10.1115/1.4041182

[9] H. Wang and Y. Wang, High-velocity impact welding process: A review. Metals (Basel). 9 (2). (2019). https://doi.org/10.3390/met9020144

[10] F. Du, L. Deng, X. Wang, M. Zhang, J. Jin, and J. Zhang, Study on interfacial characteristics and properties of NiTi/Al-Mg joint by vaporizing foil actuator welding. J. Mater. Res. Technol. 20. 3429-3440. (Sep. 2022). https://doi.org/10.1016/j.jmrt.2022.08.090

[11] J. Li, B. Panton, S. Liang, A. Vivek, and G. Daehn, High strength welding of NiTi and stainless steel by impact: Process, structure and properties. Mater. Today Commun. 25 (March). 101306. (2020). https://doi.org/10.1016/j.mtcomm.2020.101306

[12] K. Wang, H. Wang, H. Zhou, W. Zheng, and A. Xu, Research status and prospect of laser impact welding. Metals (Basel). 10 (11). 1-18. (2020). https://doi.org/10.3390/met10111444

[13] H. Wang, A. Vivek, Y. Wang, G. Taber, and G. S. Daehn, Laser impact welding application in joining aluminum to titanium. J. Laser Appl. 28 (3). (2016). https://doi.org/10.2351/1.4946887

[14] J. Bellmann et al., Particle Ejection by Jetting and Related Effects in Impact Welding Processes. Metals (Basel). 10 (8). (2020). https://doi.org/10.3390/met10081108

[15] G. Gleason, K. Bailey, S. Sunny, A. Malik, and R. A. Bernal, Influence of surface roughness on the transient interfacial phenomena in laser impact welding. J. Manuf. Process. 80 (February). 480-490. (2022). https://doi.org/10.1016/j.jmapro.2022.06.022

[16] W. R. Wagner, S. E. Sakiyama-Elbert, G. Zhang, and M. J. Yaszemski, Biomaterials Science: An Introduction to Materials in Medicine. 2020. https://doi.org/10.1016/C2017-0-02323-6

[17] A. Nassiri, G. Chini, A. Vivek, G. Daehn, and B. Kinsey, Arbitrary Lagrangian-Eulerian finite element simulation and experimental investigation of wavy interfacial morphology during high velocity impact welding. Mater. Des. 88. 345-358. (2015). https://doi.org/10.1016/j.matdes.2015.09.005

[18] S. Sadeh et al., Simulation and experimental comparison of laser impact welding with a
plasma pressure model. Metals (Basel). 9 (11). (2019). https://doi.org/10.3390/met9111196

[19] X. Wang, H. Zhang, Z. Shen, J. Li, Q. Qian, and H. Liu, Experimental and numerical investigation of laser shock synchronous welding and forming of Copper/Aluminum. Opt. Lasers Eng. 86. 291–302. (Nov. 2016). https://doi.org/10.1016/j.optlaseng.2016.06.018

[20] Y. V. R. K. Prasad, K. P. Rao, and S. Sasidhara, Hot Working Guide: A Compendium of Processing Maps. Second. ASM International, 2015. [Online]. Available: http://linkinghub.elsevier.com/retrieve/pii/B978008033454750019X

[21] P. J. Arrazola, T. Özel, D. Umbrello, M. Davies, and I. S. Jawahir, Recent advances in modelling of metal machining processes. CIRP Ann. - Manuf. Technol. 62 (2). 695–718. (2013). https://doi.org/10.1016/j.cirp.2013.05.006

[22] G. Gleason, S. Sunny, S. Sadeh, H. Yu, and A. Malik, Eulerian modeling of plasma-pressure driven laser impact weld processes. Procedia Manuf. 48 (2019). 204–214. (2020). https://doi.org/10.1016/j.promfg.2020.05.039

[23] S. Kalpakjian and S. R. Schmid, Tecnologia Meccanica. Sixth. Pearson, 2021.

[24] O. Majidi, M. Jahazi, and N. Bombardier, Finite element simulation of high-speed blow forming of an automotive component. Metals (Basel). 8 (11). (2018). https://doi.org/10.3390/met8110901

[25] M. M. Avedesian and H. Baker, Eds., ASM Specialty Handbook: Magnesium and Magnesium Alloys. ASM International, 1999.

[26] J. L. Robinson, The mechanics of wave formation in impact welding. Philos. Mag. A J. Theor. Exp. Appl. Phys. 31 (3). 587–597. (Mar. 1975). https://doi.org/10.1080/14786437508226540

[27] H. Wang, D. Liu, J. C. Lippold, and G. S. Daehn, Laser impact welding for joining similar and dissimilar metal combinations with various target configurations. J. Mater. Process. Tech. 278 (November 2019). 116498. (2020). https://doi.org/10.1016/j.jmatprotec.2019.116498

[28] T. Lee, A. Nassiri, T. Dittrich, A. Vivek, and G. Daehn, Microstructure development in impact welding of a model system. Scr. Mater. 178. 203–206. (2020). https://doi.org/10.1016/j.scriptamat.2019.11.031

[29] B. Bhushan, Introduction to tribology. Second Edi. John Wiley & Sons, Ltd., 2013.

[30] F. Gagliardi, T. Citrea, G. Ambrogio, and L. Filice, Influence of the process setup on the microstructure and mechanical properties evolution in porthole die extrusion. Mater. Des. 60. 274–281. (2014). https://doi.org/10.1016/j.matdes.2014.04.004
plasma pressure model. Metals (Basel). 9 (11). (2019). https://doi.org/10.3390/met9111196

[19] X. Wang, H. Zhang, Z. Shen, J. Li, Q. Qian, and H. Liu, Experimental and numerical investigation of laser shock synchronous welding and forming of Copper/Aluminum. Opt. Lasers Eng. 86. 291–302. (Nov. 2016). https://doi.org/10.1016/j.optlaseng.2016.06.018

[20] Y. V. R. K. Prasad, K. P. Rao, and S. Sasidhara, Hot Working Guide: A Compendium of Processing Maps. Second. ASM International, 2015. [Online]. Available: http://linkinghub.elsevier.com/retrieve/pii/B978008033454750019X

[21] P. J. Arrazola, T. Özel, D. Umbrello, M. Davies, and I. S. Jawahir, Recent advances in modelling of metal machining processes. CIRP Ann. - Manuf. Technol. 62 (2). 695–718. (2013). https://doi.org/10.1016/j.cirp.2013.05.006

[22] G. Gleason, S. Sunny, S. Sadeh, H. Yu, and A. Malik, Eulerian modeling of plasma-pressure driven laser impact weld processes. Procedia Manuf. 48 (2019). 204–214. (2020). https://doi.org/10.1016/j.promfg.2020.05.039

[23] S. Kalpakjian and S. R. Schmid, Tecnologia Meccanica. Sixth. Pearson, 2021.

[24] O. Majidi, M. Jahazi, and N. Bombardier, Finite element simulation of high-speed blow forming of an automotive component. Metals (Basel). 8 (11). (2018). https://doi.org/10.3390/met8110901

[25] M. M. Avedesian and H. Baker, Eds., ASM Specialty Handbook: Magnesium and Magnesium Alloys. ASM International, 1999.

[26] J. L. Robinson, The mechanics of wave formation in impact welding. Philos. Mag. A J. Theor. Exp. Appl. Phys. 31 (3). 587–597. (Mar. 1975). https://doi.org/10.1080/14786437508226540

[27] H. Wang, D. Liu, J. C. Lippold, and G. S. Daehn, Laser impact welding for joining similar and dissimilar metal combinations with various target configurations. J. Mater. Process. Tech. 278 (November 2019). 116498. (2020). https://doi.org/10.1016/j.jmatprotec.2019.116498

[28] T. Lee, A. Nassiri, T. Dittrich, A. Vivek, and G. Daehn, Microstructure development in impact welding of a model system. Scr. Mater. 178. 203–206. (2020). https://doi.org/10.1016/j.scriptamat.2019.11.031

[29] B. Bhushan, Introduction to tribology. Second Edi. John Wiley & Sons, Ltd., 2013.

[30] F. Gagliardi, T. Citrea, G. Ambrogio, and L. Filice, Influence of the process setup on the microstructure and mechanical properties evolution in porthole die extrusion. Mater. Des. 60. 274–281. (2014). https://doi.org/10.1016/j.matdes.2014.04.004
Lionel Fourment MS on
optimization and inverse analysis
in forming

<figure><img src="image_1669.png" /></figure>

ESAFORM
TOULOUSE
2024
Lionel Fourment MS on
optimization and inverse analysis
in forming

<figure><img src="image_1668.png" /></figure>

ESAFORM
TOULOUSE
2024
# Experimental investigation of heterogeneous mechanical tests

GONÇALVES Mafalda¹,ª*, GUEGAN Briag²,ᵇ, THUILLIER Sandrine²,ᶜ and
ANDRADE-CAMPOS António¹,ᵈ

¹Department of Mechanical Engineering, TEMA - Centre for Mechanical Technology and Automation, LASI - Intelligent Systems Associate Laboratory, University of Aveiro, Campus Universitario de Santiago, 3810-193 Aveiro, Portugal

²Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

ªmafalda.goncalves@ua.pt, ᵇbriag.guegan@grenoble-inp.org, ᶜsandrine.thuillier@univ-ubs.fr,

ᵈgilac@ua.pt

**Keywords:** Heterogeneous Test, Digital Image Correlation, Mechanical Behavior, Sheet Metal

**Abstract.** The virtualization of sheet metal forming processes requires a precise numerical model with an accurate description of the material behavior that is classically obtained by carrying out quasi-homogeneous mechanical tests. However, several alternatives to this time-consuming task are under study. Heterogeneous tests can provide a large quantity of mechanical information in a single experiment and, therefore, their potential needs to be investigated. This work aims to present an advanced mechanical test designed by topology optimization under experimental investigation. A numerical design methodology is described, leading to a specimen geometry that is subjected experimentally to uniaxial tensile loading up to rupture. A dual-phase DP600 steel is used. During the test, the strain field is extracted from the specimen surface using a stereo digital image correlation system, and the richness of the mechanical information is further analyzed.

**Introduction**

Numerical simulation has played a major role in the virtualization of sheet metal forming processes. This process leads to a reduced number of issues or failed processes, resulting in a reduction in cost, material waste, and development time of sheet metal forming parts. However, it requires an accurate numerical model, which in turn relies on an exact representation of the material mechanical behavior. This is only possible with an adequate constitutive model that is accurately calibrated, providing a good agreement between the real material behavior and the one predicted by the model equations. Therefore, model calibration procedures play a key role in the process.

Quasi-homogeneous mechanical tests are classically performed to obtain the required information to characterize material behavior; however, a large number of different tests are required to cover a wide range of strain and stress states. Heterogeneous mechanical tests are being used to reduce the experimental effort and speed up the model calibration and, therefore, the material behavior characterization task. The more complex boundary conditions and/or geometries can give them the ability to provide a larger variety of strain and stress states in a single experiment. To replace classical methods, full-field measurement techniques such as Digital Image Correlation (DIC) and inverse identification techniques are being used side by side with advanced mechanical tests to exploit their full potential [1]. These new techniques have been referred to as Material Testing 2.0 [2].

Several works have already addressed the calibration of material models using inverse methods with full-field data from heterogeneous mechanical tests [3-11]. However, it is important to note that some of these works remain on the use of numerical data from virtual tests. As for actual experiments, for example, Zhang et al. [9] used the heterogeneous strain field of a biaxial test to

<figure><img src="image_1670.png" /></figure>
# Experimental investigation of heterogeneous mechanical tests

GONÇALVES Mafalda¹,ª*, GUEGAN Briag²,ᵇ, THUILLIER Sandrine²,ᶜ and
ANDRADE-CAMPOS António¹,ᵈ

¹Department of Mechanical Engineering, TEMA - Centre for Mechanical Technology and Automation, LASI - Intelligent Systems Associate Laboratory, University of Aveiro, Campus Universitario de Santiago, 3810-193 Aveiro, Portugal

²Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

ªmafalda.goncalves@ua.pt, ᵇbriag.guegan@grenoble-inp.org, ᶜsandrine.thuillier@univ-ubs.fr,

ᵈgilac@ua.pt

**Keywords:** Heterogeneous Test, Digital Image Correlation, Mechanical Behavior, Sheet Metal

**Abstract.** The virtualization of sheet metal forming processes requires a precise numerical model with an accurate description of the material behavior that is classically obtained by carrying out quasi-homogeneous mechanical tests. However, several alternatives to this time-consuming task are under study. Heterogeneous tests can provide a large quantity of mechanical information in a single experiment and, therefore, their potential needs to be investigated. This work aims to present an advanced mechanical test designed by topology optimization under experimental investigation. A numerical design methodology is described, leading to a specimen geometry that is subjected experimentally to uniaxial tensile loading up to rupture. A dual-phase DP600 steel is used. During the test, the strain field is extracted from the specimen surface using a stereo digital image correlation system, and the richness of the mechanical information is further analyzed.

**Introduction**

Numerical simulation has played a major role in the virtualization of sheet metal forming processes. This process leads to a reduced number of issues or failed processes, resulting in a reduction in cost, material waste, and development time of sheet metal forming parts. However, it requires an accurate numerical model, which in turn relies on an exact representation of the material mechanical behavior. This is only possible with an adequate constitutive model that is accurately calibrated, providing a good agreement between the real material behavior and the one predicted by the model equations. Therefore, model calibration procedures play a key role in the process.

Quasi-homogeneous mechanical tests are classically performed to obtain the required information to characterize material behavior; however, a large number of different tests are required to cover a wide range of strain and stress states. Heterogeneous mechanical tests are being used to reduce the experimental effort and speed up the model calibration and, therefore, the material behavior characterization task. The more complex boundary conditions and/or geometries can give them the ability to provide a larger variety of strain and stress states in a single experiment. To replace classical methods, full-field measurement techniques such as Digital Image Correlation (DIC) and inverse identification techniques are being used side by side with advanced mechanical tests to exploit their full potential [1]. These new techniques have been referred to as Material Testing 2.0 [2].

Several works have already addressed the calibration of material models using inverse methods with full-field data from heterogeneous mechanical tests [3-11]. However, it is important to note that some of these works remain on the use of numerical data from virtual tests. As for actual experiments, for example, Zhang et al. [9] used the heterogeneous strain field of a biaxial test to

<figure><img src="image_1669.png" /></figure>
calibrate an advanced anisotropic yield function for an aluminum alloy and a dual-phase steel. Lou et al. [10] calibrated a yield function and a hardening law using designed specimens via an inverse experimental-numerical procedure. Aquino et al. [11] used a mechanical test designed by shape optimization to calibrate a yield criterion via a Finite Element Model Updating (FEMU) strategy.

To fill this gap and to experimentally investigate the potential of these new procedures, this work aims at designing an original mechanical test using topology optimization and validating its design experimentally. The design procedure attempts to increase the strain heterogeneities in the specimen by maximizing the heterogeneity of the displacement field. The optimal specimen geometry is experimentally subjected to a uniaxial tensile loading up to rupture. The mechanical information is extracted from the specimen surface using full-field measurements and evaluated to analyze the relevance of the test for improving model calibration procedures. The paper is split into three main parts, dedicated to the test design procedure, then to the experiments using the optimized geometry and a dual phase steel and finally to an analysis of the results.

## Test design procedure

**Framework.** The design of mechanical tests has been addressed in several works, trying to obtain a mechanical test from which a large variety of strain and stress states can be retrieved to improve model calibration. The use of optimization techniques in the design procedure has shown improved efficiency and quality of the solutions [12]. In this work, the test design procedure applies topology optimization to the design of compliant mechanisms [13]. While the former is a design method that aims to find the best material distribution for a design domain, the latter is known for deforming its flexible members as a consequence of the applied displacement. The main idea is to gather the potential of topology optimization to generate non-standard specimen geometries with heterogeneous displacement fields and the possibility of enhancing such heterogeneity by controlling the displacements in specific locations of the specimen according to the compliant mechanisms' theory. Depending on the applied displacements, specific strain/stress states can be induced on the specimen. Such heterogeneity of strain/stress states can lead to richer mechanical tests that will improve the accuracy and efficiency of model calibration procedures due to the high diversity of mechanical information.

In this work, the test design starts from a design domain shown in Figure 1. While a uniaxial tensile loading test is reproduced, the aim is to find the optimal material distribution that maximizes the displacement field heterogeneity. The load applied by the grips of the testing machine is represented by **F**<sub>in</sub>. To control the displacement field, two displacements are applied in different directions and locations of the specimen to impose different deformation states. To provide stiffness to the solution, **u**<sub>in</sub>, corresponds to the displacement of the grips during the test. The location and direction of the displacement **u**<sub>out</sub> determine the way the specimen behaves. To avoid numerical issues during the design optimization, two springs of stiffness, K<sub>in</sub> and K<sub>out</sub>, are implemented at the input and output locations, respectively. The design optimization is only performed in one quarter of the test geometry since double symmetry conditions are assumed.

**Problem formulation.** The design domain can be described by topology optimization as a finite element mesh. Each element is represented by a design variable, X<sub>e</sub>, that corresponds to the element's relative density. Depending on the quantity of material, it can assume values between 0 and 1 if the element is respectively void or full of material. At the beginning, 35% of the total material is associated to each element (X<sub>e</sub> = 0.35, e = 1, ..., M and M the total number of elements). During the optimization, according to the objective-function evolution, material is added or removed from each element in order to find the optimum material layout. The material distribution is here represented by **X**.
calibrate an advanced anisotropic yield function for an aluminum alloy and a dual-phase steel. Lou et al. [10] calibrated a yield function and a hardening law using designed specimens via an inverse experimental-numerical procedure. Aquino et al. [11] used a mechanical test designed by shape optimization to calibrate a yield criterion via a Finite Element Model Updating (FEMU) strategy.

To fill this gap and to experimentally investigate the potential of these new procedures, this work aims at designing an original mechanical test using topology optimization and validating its design experimentally. The design procedure attempts to increase the strain heterogeneities in the specimen by maximizing the heterogeneity of the displacement field. The optimal specimen geometry is experimentally subjected to a uniaxial tensile loading up to rupture. The mechanical information is extracted from the specimen surface using full-field measurements and evaluated to analyze the relevance of the test for improving model calibration procedures. The paper is split into three main parts, dedicated to the test design procedure, then to the experiments using the optimized geometry and a dual phase steel and finally to an analysis of the results.

## Test design procedure

**Framework.** The design of mechanical tests has been addressed in several works, trying to obtain a mechanical test from which a large variety of strain and stress states can be retrieved to improve model calibration. The use of optimization techniques in the design procedure has shown improved efficiency and quality of the solutions [12]. In this work, the test design procedure applies topology optimization to the design of compliant mechanisms [13]. While the former is a design method that aims to find the best material distribution for a design domain, the latter is known for deforming its flexible members as a consequence of the applied displacement. The main idea is to gather the potential of topology optimization to generate non-standard specimen geometries with heterogeneous displacement fields and the possibility of enhancing such heterogeneity by controlling the displacements in specific locations of the specimen according to the compliant mechanisms' theory. Depending on the applied displacements, specific strain/stress states can be induced on the specimen. Such heterogeneity of strain/stress states can lead to richer mechanical tests that will improve the accuracy and efficiency of model calibration procedures due to the high diversity of mechanical information.

In this work, the test design starts from a design domain shown in Figure 1. While a uniaxial tensile loading test is reproduced, the aim is to find the optimal material distribution that maximizes the displacement field heterogeneity. The load applied by the grips of the testing machine is represented by **F**<sub>in</sub>. To control the displacement field, two displacements are applied in different directions and locations of the specimen to impose different deformation states. To provide stiffness to the solution, **u**<sub>in</sub>, corresponds to the displacement of the grips during the test. The location and direction of the displacement **u**<sub>out</sub> determine the way the specimen behaves. To avoid numerical issues during the design optimization, two springs of stiffness, K<sub>in</sub> and K<sub>out</sub>, are implemented at the input and output locations, respectively. The design optimization is only performed in one quarter of the test geometry since double symmetry conditions are assumed.

**Problem formulation.** The design domain can be described by topology optimization as a finite element mesh. Each element is represented by a design variable, X<sub>e</sub>, that corresponds to the element's relative density. Depending on the quantity of material, it can assume values between 0 and 1 if the element is respectively void or full of material. At the beginning, 35% of the total material is associated to each element (X<sub>e</sub> = 0.35, e = 1, ..., M and M the total number of elements). During the optimization, according to the objective-function evolution, material is added or removed from each element in order to find the optimum material layout. The material distribution is here represented by **X**.
<figure><img src="image_1671.png" /><figcaption>Fig. 1. Schematic representation of the test design domain subjected to a uniaxial tensile loading. Double symmetry conditions are assumed and only a quarter of the specimen is optimized.</figcaption></figure>

It is proposed to find the specimen geometry (material distribution) that presents the most heterogeneous displacement field. While the input displacement gives stiffness to the solution, the output displacement is responsible for creating heterogeneity. Therefore, to increase the potential of the specimen, it is proposed to maximize the ratio between the output and input displacements. The problem formulation is here represented:

$$ \begin{array}{ll} \underset{\mathbf{X}}{\operatorname{maximize}} & T(\mathbf{X}) = \frac{u_{\text{out}}(\mathbf{X})}{u_{\text{in}}(\mathbf{X})}, \\ \text{subject to} & \mathbf{R} = \mathbf{0}, \end{array} \tag{1} $$

$$ \frac{\sum_{e=1}^{M} X_e V_e}{\sum_{e=1}^{M} V_e} - V^* \leq 0, $$

$$ 0 \leq \rho_{\min} \leq X_e \leq 1, e = 1, 2, \dots, M. $$

The optimization problem is solved by taking into account three constraints: system equilibrium, design domain volume fraction and the limits of the design variables. The first one requires solving a nonlinear finite element analysis at each iteration until the equilibrium between the internal and external loads is achieved ($\mathbf{R} = \mathbf{0}$) [14]. Nonlinear geometric and material nonlinearities are considered in the design procedure. The volume constraint establishes a limit for the maximum volume fraction of the design domain, $V^*$, meaning the sum of each element's volume fraction, $V_e$. For each element, a lower limit for the relative density, $\rho_{\min}$, is imposed to avoid numerical issues.
<figure><img src="image_1670.png" /><figcaption>Fig. 1. Schematic representation of the test design domain subjected to a uniaxial tensile loading. Double symmetry conditions are assumed and only a quarter of the specimen is optimized.</figcaption></figure>

It is proposed to find the specimen geometry (material distribution) that presents the most heterogeneous displacement field. While the input displacement gives stiffness to the solution, the output displacement is responsible for creating heterogeneity. Therefore, to increase the potential of the specimen, it is proposed to maximize the ratio between the output and input displacements. The problem formulation is here represented:

$$ \begin{array}{ll} \underset{\mathbf{X}}{\operatorname{maximize}} & T(\mathbf{X}) = \frac{u_{\text{out}}(\mathbf{X})}{u_{\text{in}}(\mathbf{X})}, \\ \text{subject to} & \mathbf{R} = \mathbf{0}, \end{array} \qquad (1) $$

$$ \frac{\sum_{e=1}^{M} X_e V_e}{\sum_{e=1}^{M} V_e} - V^* \le 0, $$

$$ 0 \le \rho_{\min} \le X_e \le 1, e = 1, 2, \dots, M. $$

The optimization problem is solved by taking into account three constraints: system equilibrium, design domain volume fraction and the limits of the design variables. The first one requires solving a nonlinear finite element analysis at each iteration until the equilibrium between the internal and external loads is achieved ($\mathbf{R} = \mathbf{0}$) [14]. Nonlinear geometric and material nonlinearities are considered in the design procedure. The volume constraint establishes a limit for the maximum volume fraction of the design domain, $V^*$, meaning the sum of each element's volume fraction, $V_e$. For each element, a lower limit for the relative density, $\rho_{\min}$, is imposed to avoid numerical issues.
**Obtained geometry.** Based on a previous work [13], the initial configuration of the design domain as well as the volume fraction were established. A mesh of 50x50 elements is used to discretize the design domain and a volume fraction of 35% is imposed. While the input displacement corresponds to the displacement of the grips, the output displacement is applied in the left symmetry boundary pointing downwards. The elastic and Swift's hardening law parameters presented in Table 1 were used to reproduce the elastoplastic material behavior.

<table><caption>Table 1. Elastic and Swift hardening law parameters for a DP600 steel.</caption><thead><tr><th colspan="2">Elastic</th><th colspan="3">Swift hardening law</th></tr><tr><th><i>E</i> [GPa]</th><th><i>ν</i> [-]</th><th><i>K</i> [MPa]</th><th><i>ε</i><sub>0</sub> [-]</th><th><i>n</i> [-]</th></tr></thead><tbody><tr><td>210</td><td>0.3</td><td>979.46</td><td>0.00535</td><td>0.194</td></tr></tbody></table>

As mentioned, the optimization process starts from a uniform material distribution, in which each element presents the same material quantity. From then on, depending on the objective-function evolution, the material is distributed according to the necessity of each element, leading to a final material layout. The evolution of the material layout, from the initial design domain until the final configuration, is represented in Figure 2. Only one quarter of the specimen is represented in five of the 37 iterations of the optimization process.

<figure><img src="image_1672.png" /><figcaption>Fig. 2. Evolution of the material layout of one quarter of the specimen since the initial design domain (iteration 1) until the final material layout (iteration 37).</figcaption></figure>

Although only one quarter is obtained, double symmetry is assumed and, therefore, the numerical design obtained by a topology optimization design procedure, referred to as TopOpt, is represented in Figure 3. From the numerical design, a smoothing of the specimen boundaries was made due to machining requirements and, therefore, an approximation of the original shapes was made. The height of the grips must be 40 cm and, therefore, the specimen was scaled, maintaining the ratio between the width and height of the specimen. The dimensions of the final configuration of the specimen are presented in Figure 4 and more details can be found in [15]. This geometry was investigated numerically in [15] alongside other advanced mechanical tests, in which their potential to improve model calibration procedures was analyzed.
**Obtained geometry.** Based on a previous work [13], the initial configuration of the design domain as well as the volume fraction were established. A mesh of 50x50 elements is used to discretize the design domain and a volume fraction of 35% is imposed. While the input displacement corresponds to the displacement of the grips, the output displacement is applied in the left symmetry boundary pointing downwards. The elastic and Swift's hardening law parameters presented in Table 1 were used to reproduce the elastoplastic material behavior.

<table><caption>Table 1. Elastic and Swift hardening law parameters for a DP600 steel.</caption><thead><tr><th colspan="2">Elastic</th><th colspan="3">Swift hardening law</th></tr><tr><th><i>E</i> [GPa]</th><th><i>ν</i> [-]</th><th><i>K</i> [MPa]</th><th><i>ε</i><sub>0</sub> [-]</th><th><i>n</i> [-]</th></tr></thead><tbody><tr><td>210</td><td>0.3</td><td>979.46</td><td>0.00535</td><td>0.194</td></tr></tbody></table>

As mentioned, the optimization process starts from a uniform material distribution, in which each element presents the same material quantity. From then on, depending on the objective-function evolution, the material is distributed according to the necessity of each element, leading to a final material layout. The evolution of the material layout, from the initial design domain until the final configuration, is represented in Figure 2. Only one quarter of the specimen is represented in five of the 37 iterations of the optimization process.

<figure><img src="image_1671.png" /><figcaption>Fig. 2. Evolution of the material layout of one quarter of the specimen since the initial design domain (iteration 1) until the final material layout (iteration 37).</figcaption></figure>

Although only one quarter is obtained, double symmetry is assumed and, therefore, the numerical design obtained by a topology optimization design procedure, referred to as TopOpt, is represented in Figure 3. From the numerical design, a smoothing of the specimen boundaries was made due to machining requirements and, therefore, an approximation of the original shapes was made. The height of the grips must be 40 cm and, therefore, the specimen was scaled, maintaining the ratio between the width and height of the specimen. The dimensions of the final configuration of the specimen are presented in Figure 4 and more details can be found in [15]. This geometry was investigated numerically in [15] alongside other advanced mechanical tests, in which their potential to improve model calibration procedures was analyzed.
<figure><img src="image_1673.png" /><figcaption>Fig. 3. Numerical design obtained by a topology optimization design procedure.</figcaption></figure>

<figure><img src="image_1674.png" /><figcaption>Fig. 4. TopOpt specimen geometry and general dimensions in mm.</figcaption></figure>

**Experimental procedure**

**Specimen design.** For the experimental procedure, the specimen depicted in Figure 4 was used. A rectangular section of 55 cm length was added to each side of the geometry for the grips. Specimens are cut by water jet and the Y axis of the specimen frame is inclined at 45° with respect to the rolling direction. Due to the inaccuracy of the cutting technique, some measured dimensions are not strictly equal to the theoretical ones. The theoretical and measured dimensions as well as the deviation between them are presented in Table 2.
<figure><img src="image_1672.png" /><figcaption>Fig. 3. Numerical design obtained by a topology optimization design procedure.</figcaption></figure>

<figure><img src="image_1673.png" /><figcaption>Fig. 4. TopOpt specimen geometry and general dimensions in mm.</figcaption></figure>

## Experimental procedure

**Specimen design.** For the experimental procedure, the specimen depicted in Figure 4 was used. A rectangular section of 55 cm length was added to each side of the geometry for the grips. Specimens are cut by water jet and the Y axis of the specimen frame is inclined at 45° with respect to the rolling direction. Due to the inaccuracy of the cutting technique, some measured dimensions are not strictly equal to the theoretical ones. The theoretical and measured dimensions as well as the deviation between them are presented in Table 2.
<table><caption>Table 2. Theoretical and measured dimensions as well as the deviation between them.</caption><thead><tr><th></th><th>L1</th><th>L2</th><th>R1</th><th>R2</th><th>R3</th><th>R4</th><th>R5</th><th>t</th></tr></thead><tbody><tr><th>Theoretical [mm]</th><td>40</td><td>93</td><td>81.69</td><td>5</td><td>4.30</td><td>107.96</td><td>200</td><td>0.80</td></tr><tr><th>Measured [mm]</th><td>39.95</td><td>88.07</td><td>78.02</td><td>4.90</td><td>4.22</td><td>102.15</td><td>189.62</td><td>0.81</td></tr><tr><th>Deviation [%]</th><td>0.13</td><td>5.30</td><td>4.49</td><td>2</td><td>1.86</td><td>5.38</td><td>5.19</td><td>1.25</td></tr></tbody></table>

**Material.** The material investigated in this work is a dual-phase DP600 steel of thickness 0.8 mm. The mechanical properties of the material, measured with tensile tests at several orientations to the rolling direction, are given in Table 3 such as the ultimate yield stress, $R_m$, and the average and standard deviation of the plastic anisotropy coefficients, $\bar{r}$ and $\Delta r$, respectively.

<table><caption>Table 3. Mechanical properties of DP600 steel.</caption><thead><tr><th>R<sub>m</sub> [MPa]</th><th>r&#773;</th><th>&Delta;r</th></tr></thead><tbody><tr><td>661.9</td><td>0.999</td><td>0.085</td></tr></tbody></table>

**Test and DIC setup.** The specimen geometries are submitted to a uniaxial tensile loading, using an Instron 5969 machine equipped with a load cell of 50 kN maximum capacity. The specimen is clamped on both sides and a test speed of 3 mm/min is set.

To measure the deformation from the specimen surface during the test, Digital Image Correlation is used as a full-field measurement technique. The optical measurement technique requires the application of a random speckle pattern on the specimen surface that follows the deformation of the test. In this work, a spray-painting technique was used consisting in a mate white coat uniformly distributed and a layer of random black dots. Throughout the test, by taking and correlating two consecutive pictures of the specimen, it is possible to obtain the displacement and strain fields. The experimental acquisition was made with the commercial software VIC 3D developed by Correlated Solutions. In order to improve the accuracy of the results by taking advantage of the potential of the cameras sensor, the two cameras were placed vertically. The following analysis to compute the mechanical fields was performed with MatchID. The hardware configuration and the DIC analysis settings are presented in Table 4. More details on the full-field measurement technique can be found in [15].

## Results

Due to the complex geometry of the specimen, a buckling phenomenon is noticed during the test leading to an out-of-plane behavior. The buckling effect starts to be non-negligible after a grips' displacement of 1 mm where the out-of-plane displacement is almost 4 mm. At the moment just before rupture, after 18.43 mm of displacement of the grips, the specimen configuration is represented in Figure 5. This behavior, associated with plastic buckling, was observed in all tests performed, making this test repeatable and suitable for material testing.
<table><caption>Table 2. Theoretical and measured dimensions as well as the deviation between them.</caption><thead><tr><th></th><th>L1</th><th>L2</th><th>R1</th><th>R2</th><th>R3</th><th>R4</th><th>R5</th><th>t</th></tr></thead><tbody><tr><td>Theoretical [mm]</td><td>40</td><td>93</td><td>81.69</td><td>5</td><td>4.30</td><td>107.96</td><td>200</td><td>0.80</td></tr><tr><td>Measured [mm]</td><td>39.95</td><td>88.07</td><td>78.02</td><td>4.90</td><td>4.22</td><td>102.15</td><td>189.62</td><td>0.81</td></tr><tr><td>Deviation [%]</td><td>0.13</td><td>5.30</td><td>4.49</td><td>2</td><td>1.86</td><td>5.38</td><td>5.19</td><td>1.25</td></tr></tbody></table>

**Material.** The material investigated in this work is a dual-phase DP600 steel of thickness 0.8 mm. The mechanical properties of the material, measured with tensile tests at several orientations to the rolling direction, are given in Table 3 such as the ultimate yield stress, $R_m$, and the average and standard deviation of the plastic anisotropy coefficients, $\bar{r}$ and $\Delta r$, respectively.

<table><caption>Table 3. Mechanical properties of DP600 steel.</caption><thead><tr><th>R<sub>m</sub> [MPa]</th><th>r&#773;</th><th>&Delta;r</th></tr></thead><tbody><tr><td>661.9</td><td>0.999</td><td>0.085</td></tr></tbody></table>

**Test and DIC setup.** The specimen geometries are submitted to a uniaxial tensile loading, using an Instron 5969 machine equipped with a load cell of 50 kN maximum capacity. The specimen is clamped on both sides and a test speed of 3 mm/min is set.

To measure the deformation from the specimen surface during the test, Digital Image Correlation is used as a full-field measurement technique. The optical measurement technique requires the application of a random speckle pattern on the specimen surface that follows the deformation of the test. In this work, a spray-painting technique was used consisting in a mate white coat uniformly distributed and a layer of random black dots. Throughout the test, by taking and correlating two consecutive pictures of the specimen, it is possible to obtain the displacement and strain fields. The experimental acquisition was made with the commercial software VIC 3D developed by Correlated Solutions. In order to improve the accuracy of the results by taking advantage of the potential of the cameras sensor, the two cameras were placed vertically. The following analysis to compute the mechanical fields was performed with MatchID. The hardware configuration and the DIC analysis settings are presented in Table 4. More details on the full-field measurement technique can be found in [15].

## Results

Due to the complex geometry of the specimen, a buckling phenomenon is noticed during the test leading to an out-of-plane behavior. The buckling effect starts to be non-negligible after a grips' displacement of 1 mm where the out-of-plane displacement is almost 4 mm. At the moment just before rupture, after 18.43 mm of displacement of the grips, the specimen configuration is represented in Figure 5. This behavior, associated with plastic buckling, was observed in all tests performed, making this test repeatable and suitable for material testing.
<table><caption>Table 4. DIC system configuration and analysis' settings.</caption><thead><tr><th></th><th>TopOpt</th></tr></thead><tbody><tr><td>Camera 0 (noise)</td><td>Basler ac A2440-35um 5MP (0.71%)</td></tr><tr><td>Image resolution</td><td>2048 x 2448</td></tr><tr><td>Camera 1 (noise)</td><td>Basler ac A2440-35um 5MP (0.75%)</td></tr><tr><td>Image resolution</td><td>2048 x 2448</td></tr><tr><td>Focal length</td><td>35 µm</td></tr><tr><td>Stereo angle</td><td>25</td></tr><tr><td>FOV [mm]</td><td>173.6 x 207.3</td></tr><tr><td>Distance [mm]</td><td>860</td></tr><tr><td>Correlation algorithm</td><td>ZNSSD</td></tr><tr><td>Interpolation</td><td>Local bicubic splines</td></tr><tr><td>Subset shape function</td><td>Quadratic</td></tr><tr><td>Subset and step size</td><td>19/8</td></tr><tr><td>Image prefiltering</td><td>Gaussian</td></tr><tr><td>Strain window and convention</td><td>7/Log. Euler-Almansi</td></tr><tr><td>Progress history</td><td>Spatial + Update reference</td></tr><tr><td>Noise floor (displacement and strain)</td><td>1.44 x 10<sup>-3</sup>, 1.36 x 10<sup>-4</sup></td></tr></tbody></table>

<figure><img src="image_1675.png" /><figcaption>Fig. 5. Final configuration of the specimen at the moment just before rupture. The colormap represents the Z-coordinate over the specimen, highlighting the out-of-plane behavior of the test.</figcaption></figure>

Due to this behavior, the mechanical information presented by one surface is different from the other. Therefore, a larger variety of information than the one presented by one surface can be extracted from a single experiment. The mechanical test was performed with two similar DIC setups, one of each side of the specimen, to be possible to extract all the information that can be provided by the specimen. Therefore, Figure 6 represents the principal maximum and minimum strain fields on the specimen surfaces (from DIC systems 1 and 2) just before rupture.
<table><caption>Table 4. DIC system configuration and analysis' settings.</caption><thead><tr><th></th><th>TopOpt</th></tr></thead><tbody><tr><td>Camera 0 (noise)</td><td>Basler ac A2440-35um 5MP (0.71%)</td></tr><tr><td>Image resolution</td><td>2048 x 2448</td></tr><tr><td>Camera 1 (noise)</td><td>Basler ac A2440-35um 5MP (0.75%)</td></tr><tr><td>Image resolution</td><td>2048 x 2448</td></tr><tr><td>Focal length</td><td>35 µm</td></tr><tr><td>Stereo angle</td><td>25</td></tr><tr><td>FOV [mm]</td><td>173.6 x 207.3</td></tr><tr><td>Distance [mm]</td><td>860</td></tr><tr><td>Correlation algorithm</td><td>ZNSSD</td></tr><tr><td>Interpolation</td><td>Local bicubic splines</td></tr><tr><td>Subset shape function</td><td>Quadratic</td></tr><tr><td>Subset and step size</td><td>19/8</td></tr><tr><td>Image prefiltering</td><td>Gaussian</td></tr><tr><td>Strain window and convention</td><td>7/Log. Euler-Almansi</td></tr><tr><td>Progress history</td><td>Spatial + Update reference</td></tr><tr><td>Noise floor (displacement and strain)</td><td>1.44 x 10<sup>-3</sup>, 1.36 x 10<sup>-4</sup></td></tr></tbody></table>

<figure><img src="image_1674.png" /><figcaption>Fig. 5. Final configuration of the specimen at the moment just before rupture. The colormap represents the Z-coordinate over the specimen, highlighting the out-of-plane behavior of the test.</figcaption></figure>

Due to this behavior, the mechanical information presented by one surface is different from the other. Therefore, a larger variety of information than the one presented by one surface can be extracted from a single experiment. The mechanical test was performed with two similar DIC setups, one of each side of the specimen, to be possible to extract all the information that can be provided by the specimen. Therefore, Figure 6 represents the principal maximum and minimum strain fields on the specimen surfaces (from DIC systems 1 and 2) just before rupture.
<figure><img src="image_1676.png" /><figcaption>Fig. 6. Experimental principal maximum and minimum strain fields on the specimen surfaces from DIC systems 2: (a) and (b) and 1: (c) and (d).</figcaption></figure>

It can be noted that high strain magnitude values are achieved, however the peak values are very localized and near the specimen boundaries. The peak values may be missed by the measurement technique due to their location and weakness of DIC of extracting information close to the edges. Moreover, some data was not possible to be read by system 2 causing the white area on the bottom corner of Figures 6 (a) and (c).

The main purpose of the test is to provide a large variety of strain states due to the heterogeneities present in the strain field. The strain state heterogeneity on the specimen can be better investigated by analyzing the principal strain diagram that is represented in Figure 7. The information provided by both DIC systems is represented. It is worth noting that, in addition to the points in uniaxial tension, an interesting quantity of material points are placed between uniaxial and plane strain compression. Moreover, even with lower strain magnitudes, some material points are under shear and uniaxial compression. To level the information provided by this heterogeneous mechanical test, several quasi-homogeneous mechanical tests would have to be performed, supporting the works dedicated to Material Testing 2.0.
<figure><img src="image_1675.png" /><figcaption>Fig. 6. Experimental principal maximum and minimum strain fields on the specimen surfaces from DIC systems 2: (a) and (b) and 1: (c) and (d).</figcaption></figure>

It can be noted that high strain magnitude values are achieved, however the peak values are very localized and near the specimen boundaries. The peak values may be missed by the measurement technique due to their location and weakness of DIC of extracting information close to the edges. Moreover, some data was not possible to be read by system 2 causing the white area on the bottom corner of Figures 6 (a) and (c).

The main purpose of the test is to provide a large variety of strain states due to the heterogeneities present in the strain field. The strain state heterogeneity on the specimen can be better investigated by analyzing the principal strain diagram that is represented in Figure 7. The information provided by both DIC systems is represented. It is worth noting that, in addition to the points in uniaxial tension, an interesting quantity of material points are placed between uniaxial and plane strain compression. Moreover, even with lower strain magnitudes, some material points are under shear and uniaxial compression. To level the information provided by this heterogeneous mechanical test, several quasi-homogeneous mechanical tests would have to be performed, supporting the works dedicated to Material Testing 2.0.
<figure><img src="image_1677.png" /><figcaption>Fig. 7. Principal strain diagram with information from DIC systems 1 and 2 just before fracture.</figcaption></figure>

## Concluding remarks

This work addressed the numerical design of a mechanical test using topology optimization as a design method and its validation through mechanical experiments. With the aim of covering a wide range of strain and stress states, a specimen geometry was obtained presenting a high heterogeneous displacement field. This specimen was subjected experimentally to uniaxial tensile loading up to rupture. Two Digital Image Correlation systems were used to extract the full-field information from the specimen surfaces during the test. A large range of strain states was recorded on the specimen surfaces, overcoming the ones presented by standard and quasi-homogeneous mechanical tests.

**Acknowledgments.** This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. This paper was also supported by the projects UIDB/00481/2020 and UIDP/00481/2020 - Fundação para a Ciência e a Tecnologia, DOI 10.54499/UIDB/00481/2020 and DOI 10.54499/UIDP/00481/2020. M. Gonçalves is grateful to the FCT for the Ph.D. grant Ref. UI/BD/151257/2021 and to the ESAFORM association for the mobility grant.

**Disclaimer.** The results reflect only the authors' view, and the European Commission is not responsible for any use that may be made of the information it contains.

## References

[1] J.M.P. Martins, S. Thuillier, A. Andrade-Campos, Calibration of a modified Johnson-Cook model using the Virtual Fields Method and a heterogeneous thermo-mechanical tensile test, Int J Mech Sci 106511 (2021) 202-203. https://doi.org/10.1016/j.ijmecsci.2021.106511

[2] F. Pierron, M. Grédiac, Towards Material Testing 2.0. A review of test design for identification of constitutive parameters from full-field measurements, Strain 57 (2021) 1-22. https://doi.org/10.1111/str.12370

[3] Y. Zhang, S. Gothivarekar, M. Conde, A. Van de Velde, B. Paermentier, A. Andrade-Campos, et al. Enhancing the information-richness of sheet metal specimens for inverse identification of
<figure><img src="image_1676.png" /><figcaption>Fig. 7. Principal strain diagram with information from DIC systems 1 and 2 just before fracture.</figcaption></figure>

**Concluding remarks**

This work addressed the numerical design of a mechanical test using topology optimization as a
design method and its validation through mechanical experiments. With the aim of covering a wide
range of strain and stress states, a specimen geometry was obtained presenting a high
heterogeneous displacement field. This specimen was subjected experimentally to uniaxial tensile
loading up to rupture. Two Digital Image Correlation systems were used to extract the full-field
information from the specimen surfaces during the test. A large range of strain states was recorded
on the specimen surfaces, overcoming the ones presented by standard and quasi-homogeneous
mechanical tests.

**Acknowledgments.** This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. This paper was also supported by the projects UIDB/00481/2020 and UIDP/00481/2020 - Fundação para a Ciência e a Tecnologia, DOI 10.54499/UIDB/00481/2020 and DOI 10.54499/UIDP/00481/2020. M. Gonçalves is grateful to the FCT for the Ph.D. grant Ref. UI/BD/151257/2021 and to the ESAFORM association for the mobility grant.

**Disclaimer.** The results reflect only the authors' view, and the European Commission is not
responsible for any use that may be made of the information it contains.

**References**

[1] J.M.P. Martins, S. Thuillier, A. Andrade-Campos, Calibration of a modified Johnson-Cook model using the Virtual Fields Method and a heterogeneous thermo-mechanical tensile test, Int J Mech Sci 106511 (2021) 202-203. https://doi.org/10.1016/j.ijmecsci.2021.106511

[2] F. Pierron, M. Grédiac, Towards Material Testing 2.0. A review of test design for identification of constitutive parameters from full-field measurements, Strain 57 (2021) 1–22. https://doi.org/10.1111/str.12370

[3] Y. Zhang, S. Gothivarekar, M. Conde, A. Van de Velde, B. Paermentier, A. Andrade-Campos, et al. Enhancing the information-richness of sheet metal specimens for inverse identification of
plastic anisotropy through strain fields, Int J Mech Sci 214 (2022) 106891.
https://doi.org/10.1016/j.ijmecsci.2021.106891

[4] M. Conde, Y. Zhang, J. Henriques, S. Coppieters, A. Andrade-Campos. Design and validation of a heterogeneous interior notched specimen for inverse material parameter identification, Finite Elem Anal Des 214 (2023) 103866. https://doi.org/10.1016/j.finel.2022.103866

[5] J.M.P. Martins, A. Andrade-Campos, S. Thuillier. Calibration of anisotropic plasticity models using a biaxial test and the virtual fields method, Int J Solids Struct 172-173 (2019) 21-37.
https://doi.org/10.1016/j.ijsolstr.2019.05.019

[6] A. Maček, B. Starman, N. Mole, M. Halilovič. Calibration of advanced yield criteria using uniaxial and heterogeneous tensile test data, Metals 10 (2020).
https://doi.org/10.3390/met10040542

[7] J.H. Kim, F. Barlat, F. Pierron, M.G. Lee. Determination of Anisotropic Plastic Constitutive Parameters Using the Virtual Fields Method, Exp Mech 54 (2014) 1189-204.
https://doi.org/10.1007/s11340-014-9879-x

[8] M. Rossi, F. Pierron, M. Štamborská. Application of the virtual fields method to large strain anisotropic plasticity. Int J Solids Struct 97-98 (2016) 322-35.
https://doi.org/10.1016/j.ijsolstr.2016.07.015

[9] S. Zhang, L. Léotoing, D. Guines, S. Thuillier, Potential of the cross biaxial test for anisotropy characterization based on heterogeneous strain field, Exp Mech 55 (2015) 817-835,
https://doi.org/10.1016/j.ijsolstr.2016.07.015

[10] Y. Lou, S. Zhang, J.W. Yoon, Strength modeling of sheet metals from shear to plane strain tension, Int J Plast 134 (2020) 102813. https://doi.org/10.1016/j.ijplas.2020.102813

[11] J. Aquino, A. Andrade-Campos, J.M.P. Martins, S. Thuillier. Design of heterogeneous mechanical tests: Numerical methodology and experimental validation, Strain 55 (2019) 1-18.
https://doi.org/10.1111/str.12313

[12] N. Souto, A. Andrade-Campos, S. Thuillier. A numerical methodology to design heterogeneous mechanical tests. Int J Mech Sci 107 (2016) 264-76.
https://doi.org/10.1016/j.ijmecsci.2016.01.021

[13] M. Gonçalves, A. Andrade-Campos, B. Barroqueiro, On the design of mechanical heterogeneous specimens using multilevel topology optimization, Adv Eng Softw 175 (2023) 103314. https://doi.org/10.1016/j.advengsoft.2022.103314

[14] M. Gonçalves, A. Andrade-Campos, S. Thuillier, On the topology design of a mechanical heterogeneous specimen using geometric and material nonlinearities, IOP Conf. Ser.: Mater. Sci. Eng. 1238 (2022) 012055. https://doi.org/10.1088/1757-899x/1238/1/012055

[15] M. Grédiac, B. Blaysat, F. Sur. A Critical Comparison of Some Metrological Parameters Characterizing Local Digital Image Correlation and Grid Method, Exp Mech 57 (2017) 871-903.
https://doi.org/10.1007/s11340-017-0279-x
plastic anisotropy through strain fields, Int J Mech Sci 214 (2022) 106891.
https://doi.org/10.1016/j.ijmecsci.2021.106891

[4] M. Conde, Y. Zhang, J. Henriques, S. Coppieters, A. Andrade-Campos. Design and validation of a heterogeneous interior notched specimen for inverse material parameter identification, Finite Elem Anal Des 214 (2023) 103866. https://doi.org/10.1016/j.finel.2022.103866

[5] J.M.P. Martins, A. Andrade-Campos, S. Thuillier. Calibration of anisotropic plasticity models using a biaxial test and the virtual fields method, Int J Solids Struct 172-173 (2019) 21-37.
https://doi.org/10.1016/j.ijsolstr.2019.05.019

[6] A. Maček, B. Starman, N. Mole, M. Halilovič. Calibration of advanced yield criteria using uniaxial and heterogeneous tensile test data, Metals 10 (2020).
https://doi.org/10.3390/met10040542

[7] J.H. Kim, F. Barlat, F. Pierron, M.G. Lee. Determination of Anisotropic Plastic Constitutive Parameters Using the Virtual Fields Method, Exp Mech 54 (2014) 1189-204.
https://doi.org/10.1007/s11340-014-9879-x

[8] M. Rossi, F. Pierron, M. Štamborská. Application of the virtual fields method to large strain anisotropic plasticity. Int J Solids Struct 97-98 (2016) 322-35.
https://doi.org/10.1016/j.ijsolstr.2016.07.015

[9] S. Zhang, L. Léotoing, D. Guines, S. Thuillier, Potential of the cross biaxial test for anisotropy characterization based on heterogeneous strain field, Exp Mech 55 (2015) 817-835,
https://doi.org/10.1016/j.ijsolstr.2016.07.015

[10] Y. Lou, S. Zhang, J.W. Yoon, Strength modeling of sheet metals from shear to plane strain tension, Int J Plast 134 (2020) 102813. https://doi.org/10.1016/j.ijplas.2020.102813

[11] J. Aquino, A. Andrade-Campos, J.M.P. Martins, S. Thuillier. Design of heterogeneous mechanical tests: Numerical methodology and experimental validation, Strain 55 (2019) 1-18.
https://doi.org/10.1111/str.12313

[12] N. Souto, A. Andrade-Campos, S. Thuillier. A numerical methodology to design heterogeneous mechanical tests. Int J Mech Sci 107 (2016) 264-76.
https://doi.org/10.1016/j.ijmecsci.2016.01.021

[13] M. Gonçalves, A. Andrade-Campos, B. Barroqueiro, On the design of mechanical heterogeneous specimens using multilevel topology optimization, Adv Eng Softw 175 (2023) 103314. https://doi.org/10.1016/j.advengsoft.2022.103314

[14] M. Gonçalves, A. Andrade-Campos, S. Thuillier, On the topology design of a mechanical heterogeneous specimen using geometric and material nonlinearities, IOP Conf. Ser.: Mater. Sci. Eng. 1238 (2022) 012055. https://doi.org/10.1088/1757-899x/1238/1/012055

[15] M. Grédiac, B. Blaysat, F. Sur. A Critical Comparison of Some Metrological Parameters Characterizing Local Digital Image Correlation and Grid Method, Exp Mech 57 (2017) 871-903.
https://doi.org/10.1007/s11340-017-0279-x
# Inverse identification of the heterogeneous strain hardening of the friction stir welded aluminum sheets using virtual fields method

KIM Chanyang<fnref n="1" />, a *, LEE Jinwoo<fnref n="2" />, b *, KIM Daeyong<fnref n="3" />, c and LEE Myoung-Gyu ⁴, d *

**Keywords:** FSW, VFM, Inverse Method, Local Plastic Strain Hardening, Aluminum Alloys

**Abstract.** In this paper, inverse identification of the local heterogeneous strain hardening in friction stir welded aluminum alloy sheets using a single tensile test is presented. During friction stir welding, the mechanical properties of the jointed aluminum sheets are altered from their original states. The finite element-based virtual fields method (FE-VFM) was applied to measure this change in the mechanical properties during the joining. Special numerical schemes were adopted in the FE-VFM for the accurate and numerically efficient identification of the local properties, including a two-step VFM calculation procedure, a sub-zone division approach in the weld-affected zone, quadratic interpolation of the constitutive parameters, and a special type of virtual fields based on the Gauss distribution function. The newly developed method was applied to measure strain-hardening distributions of the friction stir welded AA6061-T6 sheets. The measured distributions of the local strain hardening well captured decreased strength in the narrow thermo-mechanically affected zone (TMAZ). Finally, inversely measured mechanical properties were validated based on finite element analysis.

**Introduction**

Friction stir welding (FSW) is a solid-state joining process of metals, invented by The Welding Institute (TSW) [1]. In this process, the FSW tool rotates in between two workpieces, and frictional heat is generated. Materials soften owing to the uprising temperature in solid states, and mechanically stired and joined. The joining of the materials is proceeded by the translation forward of the FSW tools with the rotations. Microstructures of the materials in the weld-affected zone (WAZ) change majorly due to the heat, and mechanical properties can also be altered from the base material properties, subsequently. Particularly for the aluminum alloys, mechanical properties in the WAZ may decrease since the distribution distance and size of the precipitation deviate from the optimal state that was achieved in the previous heat treatment process in the base sheet material manufacturing [2]. This decrease in strength in the FSW of the aluminum alloys is critical for the evaluation or prediction of the performance in the post-FSW forming process.

As per the mechanical properties in the WAZ, assessing plastic strain hardening locally varied in the WAS is important. Distributions of the (micro) hardness are widely adopted conventionally, to evaluate strength changes in the WAZ [3,4]. However, Hardness distributions only can give relative changes in strength compared to the original base material properties, not the absolute information like the full stress-strain curves. A reliable method that can directly measure local full stress-strain curves in the WAZ is conducting miniature or micro tensile tests [5,6], but it demands multiple tests and relatively complex specimen fabrications. To overcome this issue, inverse

<figure><img src="image_1678.png" /></figure>

<footnote n="1">¹Korea Institute of Materials Science, 797, Changwon-daero, Seongsan-gu, Changwon-si, Gyeongsangnam-do 51508, Republic of Korea</footnote>

<footnote n="2">²School of Mechanical Engineering, University of Ulsan, 93, Daehak-ro, Nam-gu, Ulsan 44610, Republic of Korea</footnote>

<footnote n="3">³Department of Intelligent Mobility, Chonnam National University, 77 Yongbong-ro, Buk-gu, Gwangju 61186, Republic of Korea</footnote>

<footnote>ªc.kim@snu.ac.kr, ᵇjinwoolee@ulsan.ac.kr, ᶜdaeyong.kim@jnu.ac.kr, ᵈmyounglee@snu.ac.kr</footnote>
# Inverse identification of the heterogeneous strain hardening of the friction stir welded aluminum sheets using virtual fields method

KIM Chanyang<sup>1,a*</sup>, LEE Jinwoo<sup>2,b*</sup>, KIM Daeyong<sup>3,c</sup> and LEE Myoung-Gyu<sup>4,d*</sup>

<sup>1</sup>Korea Institute of Materials Science, 797, Changwon-daero, Seongsan-gu, Changwon-si, Gyeongsangnam-do 51508, Republic of Korea

<sup>2</sup>School of Mechanical Engineering, University of Ulsan, 93, Daehak-ro, Nam-gu, Ulsan 44610, Republic of Korea

<sup>3</sup>Department of Intelligent Mobility, Chonnam National University, 77 Yongbong-ro, Buk-gu, Gwangju 61186, Republic of Korea

<sup>a</sup>c.kim@snu.ac.kr, <sup>b</sup>jinwoolee@ulsan.ac.kr, <sup>c</sup>daeyong.kim@jnu.ac.kr, <sup>d</sup>myounglee@snu.ac.kr

**Keywords:** FSW, VFM, Inverse Method, Local Plastic Strain Hardening, Aluminum Alloys

**Abstract.** In this paper, inverse identification of the local heterogeneous strain hardening in friction stir welded aluminum alloy sheets using a single tensile test is presented. During friction stir welding, the mechanical properties of the jointed aluminum sheets are altered from their original states. The finite element-based virtual fields method (FE-VFM) was applied to measure this change in the mechanical properties during the joining. Special numerical schemes were adopted in the FE-VFM for the accurate and numerically efficient identification of the local properties, including a two-step VFM calculation procedure, a sub-zone division approach in the weld-affected zone, quadratic interpolation of the constitutive parameters, and a special type of virtual fields based on the Gauss distribution function. The newly developed method was applied to measure strain-hardening distributions of the friction stir welded AA6061-T6 sheets. The measured distributions of the local strain hardening well captured decreased strength in the narrow thermo-mechanically affected zone (TMAZ). Finally, inversely measured mechanical properties were validated based on finite element analysis.

**Introduction**

Friction stir welding (FSW) is a solid-state joining process of metals, invented by The Welding Institute (TSW) [1]. In this process, the FSW tool rotates in between two workpieces, and frictional heat is generated. Materials soften owing to the uprising temperature in solid states, and mechanically stired and joined. The joining of the materials is proceeded by the translation forward of the FSW tools with the rotations. Microstructures of the materials in the weld-affected zone (WAZ) change majorly due to the heat, and mechanical properties can also be altered from the base material properties, subsequently. Particularly for the aluminum alloys, mechanical properties in the WAZ may decrease since the distribution distance and size of the precipitation deviate from the optimal state that was achieved in the previous heat treatment process in the base sheet material manufacturing [2]. This decrease in strength in the FSW of the aluminum alloys is critical for the evaluation or prediction of the performance in the post-FSW forming process.

As per the mechanical properties in the WAZ, assessing plastic strain hardening locally varied in the WAS is important. Distributions of the (micro) hardness are widely adopted conventionally, to evaluate strength changes in the WAZ [3,4]. However, Hardness distributions only can give relative changes in strength compared to the original base material properties, not the absolute information like the full stress-strain curves. A reliable method that can directly measure local full stress-strain curves in the WAZ is conducting miniature or micro tensile tests [5,6], but it demands multiple tests and relatively complex specimen fabrications. To overcome this issue, inverse

<figure><img src="image_1677.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.
identification approaches were applied for the evaluation of the local plastic hardening in the WAZ such as the uniform or non-uniform stress method (USM or nUSM) [7,8], finite element model updating (FEMU) [9], and the virtual fields method (VFM) [8,10,11,12,13]. These methods all utilize strain distributions in the test specimen measured by the full-field measurement technique, such as the digital image correlation (DIC).

To measure the spatially changing mechanical properties within the WAZ using VFM, it is necessary to define the spatial dependency at the level of the constitutive model or its parameters. In the previous research, two major approaches were used. The straightforward approach is dividing an analyzed area of interest (AOI) in the VFM into sub-zones with equal step lengths [10,11,12,13]. Then, constitutive parameters in each sub-zone are assumed to be constant. This approach may be easy, but there are two disadvantages. In terms of resolution in the inverse measurement result, a small step length is favorable. However, the error included in the results is also increased for the smaller step lengths that are coming from inevitable noise in the non-contact type full-field measurements. In other words, accuracy and resolution can be in a trade-off relation depending on the step length size. Also, all the sub-zone material properties in the entire AOI were identified in the previous works. However, most of the welded materials remain as base material, and only the materials in the limited WAZ change from their original properties. Hence, it is numerically inefficient to measure already known, or easily obtainable base material properties. The other approach was predefining sub-zones considering the materials’ microstructure [8]. The WAZ in the FSWed metals can be divided into fusion zone (FZ), thermo-mechanically affected zone (TMAZ), and heat affected zone (HAZ) [1]. These zones have distinctive characteristics on the microstructure. Hence, predefining a sub-zone is possible by the characterization of welded metals’ microstructure, or through the hardness distribution measurements, indirectly. This approach enables to defining of good sub-zone divisions that coincide with the material characteristics, but it requires the preliminary procedure before applying the VFM. Nevertheless, both approaches assume the constant constitutive parameters within each sub-zone, and discontinuous material properties were assumed as a result.

In this study, the VFM was applied in measuring local strain hardening distributions in the WAZ of the friction stir welded (FSWed) AA6061-T6 sheets with a single tensile test [14]. In particular, finite element-based VFM (FE-VFM), which is an advanced formulation of the conventional VFM by adapting finite element schemes, was used [15]. Advanced numerical schemes in the VFM introduced in this research are as follows. A two-step identification procedure was applied, in which, the WAZ boundaries that material properties were varied from the base materials were inversely identified. Then, variations of the local strain hardening behavior (or the full stress-strain curves) were measured within the pre-identified WAZ, for the sake of efficient inverse measurements process. The VFM analyzed AOI was divided into several sub-zones, and the parameter distributions of the strain hardening laws were defined as a piecewise quadratic interpolation function. By doing so, continuous and quadratically varied distributions of the mechanical properties can be obtained within each sub-zone, rather than discontinuous and single constant values. Lastly, a special type of virtual field based on the Gaussian normal distribution function was introduced, which is for spatially employing strain (or stress) data at a certain range and uniquely obtaining mechanical property variations within each sub-zone. Finally, an in-depth discussion of the obtained distributions of the local strain hardening behavior and validation of the inverse measurements are presented in the result and discussion section.

Finite element-based virtual fields method (FE-VFM)

The VFM is an inverse technique to measure the mechanical properties of materials from full-field
d deformation measurements, theoretically relying on the principle of virtual works in Eq. (1).
identification approaches were applied for the evaluation of the local plastic hardening in the WAZ such as the uniform or non-uniform stress method (USM or nUSM) [7,8], finite element model updating (FEMU) [9], and the virtual fields method (VFM) [8,10,11,12,13]. These methods all utilize strain distributions in the test specimen measured by the full-field measurement technique, such as the digital image correlation (DIC).

To measure the spatially changing mechanical properties within the WAZ using VFM, it is necessary to define the spatial dependency at the level of the constitutive model or its parameters. In the previous research, two major approaches were used. The straightforward approach is dividing an analyzed area of interest (AOI) in the VFM into sub-zones with equal step lengths [10,11,12,13]. Then, constitutive parameters in each sub-zone are assumed to be constant. This approach may be easy, but there are two disadvantages. In terms of resolution in the inverse measurement result, a small step length is favorable. However, the error included in the results is also increased for the smaller step lengths that are coming from inevitable noise in the non-contact type full-field measurements. In other words, accuracy and resolution can be in a trade-off relation depending on the step length size. Also, all the sub-zone material properties in the entire AOI were identified in the previous works. However, most of the welded materials remain as base material, and only the materials in the limited WAZ change from their original properties. Hence, it is numerically inefficient to measure already known, or easily obtainable base material properties. The other approach was predefining sub-zones considering the materials’ microstructure [8]. The WAZ in the FSWed metals can be divided into fusion zone (FZ), thermo-mechanically affected zone (TMAZ), and heat affected zone (HAZ) [1]. These zones have distinctive characteristics on the microstructure. Hence, predefining a sub-zone is possible by the characterization of welded metals’ microstructure, or through the hardness distribution measurements, indirectly. This approach enables to defining of good sub-zone divisions that coincide with the material characteristics, but it requires the preliminary procedure before applying the VFM. Nevertheless, both approaches assume the constant constitutive parameters within each sub-zone, and discontinuous material properties were assumed as a result.

In this study, the VFM was applied in measuring local strain hardening distributions in the WAZ of the friction stir welded (FSWed) AA6061-T6 sheets with a single tensile test [14]. In particular, finite element-based VFM (FE-VFM), which is an advanced formulation of the conventional VFM by adapting finite element schemes, was used [15]. Advanced numerical schemes in the VFM introduced in this research are as follows. A two-step identification procedure was applied, in which, the WAZ boundaries that material properties were varied from the base materials were inversely identified. Then, variations of the local strain hardening behavior (or the full stress-strain curves) were measured within the pre-identified WAZ, for the sake of efficient inverse measurements process. The VFM analyzed AOI was divided into several sub-zones, and the parameter distributions of the strain hardening laws were defined as a piecewise quadratic interpolation function. By doing so, continuous and quadratically varied distributions of the mechanical properties can be obtained within each sub-zone, rather than discontinuous and single constant values. Lastly, a special type of virtual field based on the Gaussian normal distribution function was introduced, which is for spatially employing strain (or stress) data at a certain range and uniquely obtaining mechanical property variations within each sub-zone. Finally, an in-depth discussion of the obtained distributions of the local strain hardening behavior and validation of the inverse measurements are presented in the result and discussion section.

## Finite element-based virtual fields method (FE-VFM)

The VFM is an inverse technique to measure the mechanical properties of materials from full-field deformation measurements, theoretically relying on the principle of virtual works in Eq. (1).
$$ \int_{\Omega} \sigma(k, \epsilon) \cdot \frac{\partial \delta u}{\partial x} dV = \int_{\partial \Omega_F} f^S \cdot \delta u dS \quad (1) $$

In Eq. (1), the left and right terms are called internal and external virtual works. The virtual displacement fields $\delta u$ and its derivatives $\frac{\partial \delta u}{\partial x}$ can be set by the VFM user, and actual strain fields and external forces $\epsilon$ and $f^S$ are measurable in the mechanical tests. Then, the only unknown in Eq. (1) is $k$, which denotes the material parameters in constitutive law $\sigma(k, \epsilon)$. In other words, unknown constitutive parameters can be obtained inversely by solving the principle of virtual work. For the linear elastic materials, Eq. (1) can be directly solved by constructing a system of linear equations using several independent virtual fields with the same number of unknown material constants [16]. However, for the elastic-plastic materials, Eq. (1) cannot be directly solved due to the nature of non-linear problems. Hence, non-linear minimization of the residuals between internal and external virtual works is performed to obtain the constitutive parameters [15].

The FE-VFM is an advanced formulation of conventional VFM borrowing the two major numerical approaches from the finite element method (FEM). The first important concept is mapping the displacements on the high-order finite element meshes rather than directly utilizing full-field measurements. The mapping procedure is as follows. First, the real initial coordinates of a target specimen are transformed on the natural coordinates which have regularized space in the -1 to 1 range, using the interpolation function depending on global initial coordinate values,

$$ r = M(X_0)r^N \quad (2) $$

where $r$ and $X_0$ denote natural and initial global coordinates, $M(X_0)$ denotes high-order interpolation function, and uppercase N denotes nodal values. Then, displacements in each finite element can be described using the interpolation functions depending on the natural coordinate values,

$$ u = N(r)u^N. \quad (3) $$

In Eq. (3), nodal displacements in each time step are unknown initially, and these can be obtained using the Eqs. (2) and (3) and full displacement fields by the linear least square optimization.

Now then, all the upcoming numerical calculations can proceed just with the mapped displacements. Moreover, integration of the internal virtual work using the Gauss quadrature is possible, owing to the coordinate transformation from initial global to natural (or element) coordinates, and this is the second major scheme borrowing from the FEM. In summary, FE-VFM can achieve a numerically accurate and efficient process, owing to the high-order finite element interpolation and high-order Gauss quadrature.

## Advanced numerical schemes in FE-VFM

Fig. 1(a) illustrates a tensile specimen for applying the FE-VFM in this work. In the specimen, the weld line was positioned at the center, and the specimen length direction was perpendicular to the weld line. Fig. 1(b) is a 3rd other finite element meshes used in the FE-VFM. The blue-marked region in Fig. 1(a) was analyzed by the FE-VFM. The length (or height) of an element was 3 mm, and each low of elements was assumed as a single sub-zone. Note that numerical analysis in the FE-VFM was applied on the unit of a sub-zone independently, not the whole AOI simultaneously. The total number of constitutive parameters in the whole WAZ is a lot larger compared to the homogeneous material case, to describe spatial distributions. Hence, constitutive parameters might not be uniquely optimized if all the constitutive parameters for all the sub-zones are identified together.
$$ \int_{\Omega} \sigma(k, \varepsilon) \cdot \frac{\partial \delta u}{\partial x} dV = \int_{\partial \Omega_F} f^S \cdot \delta u dS \quad (1) $$

In Eq. (1), the left and right terms are called internal and external virtual works. The virtual displacement fields $\delta u$ and its derivatives $\frac{\partial \delta u}{\partial x}$ can be set by the VFM user, and actual strain fields and external forces $\varepsilon$ and $f^S$ are measurable in the mechanical tests. Then, the only unknown in Eq. (1) is $k$, which denotes the material parameters in constitutive law $\sigma(k, \varepsilon)$. In other words, unknown constitutive parameters can be obtained inversely by solving the principle of virtual work. For the linear elastic materials, Eq. (1) can be directly solved by constructing a system of linear equations using several independent virtual fields with the same number of unknown material constants [16]. However, for the elastic-plastic materials, Eq. (1) cannot be directly solved due to the nature of non-linear problems. Hence, non-linear minimization of the residuals between internal and external virtual works is performed to obtain the constitutive parameters [15].

The FE-VFM is an advanced formulation of conventional VFM borrowing the two major numerical approaches from the finite element method (FEM). The first important concept is mapping the displacements on the high-order finite element meshes rather than directly utilizing full-field measurements. The mapping procedure is as follows. First, the real initial coordinates of a target specimen are transformed on the natural coordinates which have regularized space in the -1 to 1 range, using the interpolation function depending on global initial coordinate values,

$$ \mathbf{r} = \mathbf{M}(\mathbf{X}_0)\mathbf{r}^N \quad (2) $$

where $\mathbf{r}$ and $\mathbf{X}_0$ denote natural and initial global coordinates, $\mathbf{M}(\mathbf{X}_0)$ denotes high-order interpolation function, and uppercase N denotes nodal values. Then, displacements in each finite element can be described using the interpolation functions depending on the natural coordinate values,

$$ \mathbf{u} = \mathbf{N}(\mathbf{r})\mathbf{u}^N. \quad (3) $$

In Eq. (3), nodal displacements in each time step are unknown initially, and these can be obtained using the Eqs. (2) and (3) and full displacement fields by the linear least square optimization.

Now then, all the upcoming numerical calculations can proceed just with the mapped displacements. Moreover, integration of the internal virtual work using the Gauss quadrature is possible, owing to the coordinate transformation from initial global to natural (or element) coordinates, and this is the second major scheme borrowing from the FEM. In summary, FE-VFM can achieve a numerically accurate and efficient process, owing to the high-order finite element interpolation and high-order Gauss quadrature.

## Advanced numerical schemes in FE-VFM

Fig. 1(a) illustrates a tensile specimen for applying the FE-VFM in this work. In the specimen, the weld line was positioned at the center, and the specimen length direction was perpendicular to the weld line. Fig. 1(b) is a 3rd other finite element meshes used in the FE-VFM. The blue-marked region in Fig. 1(a) was analyzed by the FE-VFM. The length (or height) of an element was 3 mm, and each low of elements was assumed as a single sub-zone. Note that numerical analysis in the FE-VFM was applied on the unit of a sub-zone independently, not the whole AOI simultaneously. The total number of constitutive parameters in the whole WAZ is a lot larger compared to the homogeneous material case, to describe spatial distributions. Hence, constitutive parameters might not be uniquely optimized if all the constitutive parameters for all the sub-zones are identified together.
Fig. 1(c) schematically illustrates the two-step identification process introduced in this work. Prior to the FE-VFM analysis, uniaxial tensile tests for the base sheet metals are conducted, and the mechanical properties of the base materials are obtained. Then, virtual work residuals are calculated using the base material's constitutive parameters in each sub-zone, in STEP1. If materials are affected during the FSW, then residuals will be larger compared to those in not affected area, and boundaries of the WAZ can be easily identified. In STEP2, distributions of the hardening law parameters are optimized by minimizing virtual work residuals, for each sub-domain independently. The distributions of constitutive parameters within each sub-zone are quadratically interpolated,

$$ \bar{k} = Q(Y)\bar{k}^N = \begin{bmatrix} 1 & Y & Y^2 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \\ \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{2}{9} & \frac{-4}{9} & \frac{2}{9} \end{bmatrix} \begin{bmatrix} \frac{-1}{k} \\ \frac{2}{k} \\ \frac{-3}{k} \end{bmatrix} \quad (4) $$

where $\bar{k}$ is constitutive parameters normalized by its based material values, and Y is coordinate value in the length direction of the specimen which has zero values at the center of each sub-zone, and the nodal position in parameter interpolation is located Y=1.5, 0, and -1.5 for node number (uppercase in Eq. (4)) 1,2, and 3.

<figure><img src="image_1679.png" /><figcaption>Fig. 1. (a) Specimen geometries for applying the FE-VFM. (b) Cubic quadrilateral high-order finite element (FE) meshes used in the FE-VFM. (c) Two-step identification procedure.</figcaption></figure>

The isotropic constitutive model was used in this research, the generalized Hooke's law and the von Mises yield criteria were used. The Swift hardening law was used,

$$ \bar{\sigma} = K(\bar{\epsilon} + e_0)^n. \quad (5) $$

Material constants of the base material AA6061-T6 sheets are listed in Table 1. In inverse measurements using the FE-VFM, only Swift hardening law parameters were optimized in STEP2, considering that the elastic constants are not significantly changed owing to the thermal histories. Also, note that normalized-nodal constitutive parameters of the Swift law were identified in STEP2 (denoted $\bar{k}$ in Eq. (4)), and real parameter distributions can be described by multiplying the values of the base materials in Table 1 and Eq. (5).