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

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

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

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

The Al1.2Mn alloy [22] is used as a reference to create the microstructure model. The alloy is in the cast and homogenized condition and consists of equiaxed grains with an average diameter of 60 µm with no texture. The constituent particles are predominantly distributed along the grain and dendrite boundaries and are mostly rounded, with a low number of elongated particles. The grain morphology and particle distribution may be seen in Fig. 1. The area fraction of the particles is 2.4 %, with a mean maximum length of 1.1 µm and a standard deviation of 1.2 µm. The alloy also contains a large number of dispersoids with diameter below 1 µm. The numerical model, created in the Abaqus FEA software, reproduces these essential features of the microstructure. The Neper software is used to generate a 450 × 450 µm tessellation, containing 50 grains, based on a grain growth algorithm [24], corresponding to an average grain diameter of 60 µm. The spherical particles with diameter of 2 µm are evenly distributed along the grain boundaries at 8 µm distance from each other, so that the area fraction is approximately 2.4 %. Due to uncertainty in size and area fraction of the dendrite particles, they are not represented in the model. The model is shown alongside the microstructure in Fig. 1. The model is a 2D representation of the material, with plane strain boundary conditions. The left edge of the model is fixed and the right one is moving with a speed smoothly ramped up to a constant value, simulating plane strain tension.

<figure><img src="image_2069.png" /><figcaption>Figure 1. The microstructure of Al1.2Mn alloy: a) constitutive particle size and distribution, b) grain morphology, c) numerical model with 50 grains in the centre and the isotropic plasticity boundary layers on the sides, marked by grey colour d) element mesh near the boundary layer.</figcaption></figure>

The coupled damage CP model is implemented as an Abaqus FEA user-defined material model, as described in [14]. The plastic flow is described by a rate-dependent constitutive relation:

$$ \dot{\gamma}^{\alpha} = \dot{\gamma}_0 \left| \frac{\tau^{(\alpha)}}{(1-\omega)\tau_c^{(\alpha)}} \right|^{\frac{1}{m}} \operatorname{sgn}(\tau^{(\alpha)}), \quad (1) $$

where $\dot{\gamma}^{\alpha}$ is the slip rate on slip system $\alpha$, $\dot{\gamma}_0$ is the reference slip rate, $m$ is the instantaneous rate sensitivity parameter, $\tau^{(\alpha)}$ is the resolved shear stress, $\tau_c^{(\alpha)}$ is the critical resolved shear stress and
Finite element model

The Al1.2Mn alloy [22] is used as a reference to create the microstructure model. The alloy is in the cast and homogenized condition and consists of equiaxed grains with an average diameter of 60 µm with no texture. The constituent particles are predominantly distributed along the grain and dendrite boundaries and are mostly rounded, with a low number of elongated particles. The grain morphology and particle distribution may be seen in Fig. 1. The area fraction of the particles is 2.4 %, with a mean maximum length of 1.1 µm and a standard deviation of 1.2 µm. The alloy also contains a large number of dispersoids with diameter below 1 µm. The numerical model, created in the Abaqus FEA software, reproduces these essential features of the microstructure. The Neper software is used to generate a 450 × 450 µm tessellation, containing 50 grains, based on a grain growth algorithm [24], corresponding to an average grain diameter of 60 µm. The spherical particles with diameter of 2 µm are evenly distributed along the grain boundaries at 8 µm distance from each other, so that the area fraction is approximately 2.4 %. Due to uncertainty in size and area fraction of the dendrite particles, they are not represented in the model. The model is shown alongside the microstructure in Fig. 1. The model is a 2D representation of the material, with plane strain boundary conditions. The left edge of the model is fixed and the right one is moving with a speed smoothly ramped up to a constant value, simulating plane strain tension.

<figure><img src="image_2061.png" /><figcaption>Figure 1. The microstructure of Al1.2Mn alloy: a) constitutive particle size and distribution, b) grain morphology, c) numerical model with 50 grains in the centre and the isotropic plasticity boundary layers on the sides, marked by grey colour d) element mesh near the boundary layer.</figcaption></figure>

The coupled damage CP model is implemented as an Abaqus FEA user-defined material model, as described in [14]. The plastic flow is described by a rate-dependent constitutive relation:

$$ \dot{\gamma}^{\alpha} = \dot{\gamma}_0 \left| \frac{\tau^{(\alpha)}}{(1-\omega)\tau_c^{(\alpha)}} \right|^{\frac{1}{m}} \operatorname{sgn}(\tau^{(\alpha)}), \quad (1) $$

where $\dot{\gamma}^{\alpha}$ is the slip rate on slip system $\alpha$, $\dot{\gamma}_0$ is the reference slip rate, $m$ is the instantaneous rate sensitivity parameter, $\tau^{(\alpha)}$ is the resolved shear stress, $\tau_c^{(\alpha)}$ is the critical resolved shear stress and
$\omega$ is the damage parameter. The critical resolved shear stress evolution is described by the Voce type equation:

$$ \dot{t}_c^{(\alpha)} = \sum_{\beta=1}^{N} \sum_{k=1}^{2} \theta_k \exp \left( -\frac{\theta_k}{\tau_k} \Gamma \right) q_{\alpha\beta}, \quad (2) $$

where $q_{\alpha\beta}$ is the latent hardening matrix, $\theta_k$ and $\tau_k$ are the hardening parameters and $\Gamma$ is the accumulated plastic shear strain. The initial value of critical resolved shear stress is denoted as $\tau_0$. The damage evolution is described by equation:

$$ \dot{\omega} = \frac{3}{4} q_1 q_2 \omega (1 - \omega) \sinh\left(\frac{3}{2} q_2 T\right) \dot{\Gamma}, \quad (3) $$

where $q_1$ and $q_2$ are the damage parameters and $T$ is the stress triaxiality ratio. The initial and critical values of the damage parameter are denoted $\omega_0$ and $\omega_c$. The elastic constants, rate sensitivity, reference slip rate and latent hardening parameter are taken from [14]. The critical slip and work hardening parameters are obtained by fitting the slip system resistance equation multiplied by the Taylor factor equal to 3 (approximately corresponding to the notexture value) to the equivalent stress-strain curve of the alloy taken from [22]. The result is approximate, but considering the nature of the study it is sufficient to represent the alloys work hardening behaviour and as demonstrated in [25] the local strain distribution and localization are not overly sensitive to the work hardening parameters. The parameters used in the study are given in Table 1.

<table><caption>Table 1: Parameters of the CP model.</caption><thead><tr><th>τ<sub>0</sub>[MPa]</th><th>θ<sub>1</sub>[MPa]</th><th>τ<sub>1</sub>[MPa]</th><th>θ<sub>2</sub>[MPa]</th><th>τ<sub>2</sub>[MPa]</th><th>ω<sub>0</sub></th><th>ω<sub>c</sub></th><th>q<sub>1</sub></th><th>q<sub>2</sub></th></tr></thead><tbody><tr><td>13.0</td><td>191.0</td><td>16.5</td><td>24.0</td><td>34.0</td><td>0.006</td><td>0.12</td><td>1.5</td><td>1.0</td></tr></tbody></table>

A set of 50 random crystallographic orientations was assigned to the 50 grains of the FE model. The left and right boundary layers of the model, indicated by grey colour in Fig. 1 are assigned an isotropic elastic-plastic material with the same equivalent stress-strain curve. This creates a more natural and relaxed boundary at the edges and forces localization in the central part of the model, avoiding necking and shear bands at the fixed edges. The particles are assigned an isotropic elastic material with Young's modulus equal to 210 GPa and Poisson's ratio equal to 0.3.

The fracture model parameters for the coupled damage CP model are taken from [14], where it was calibrated for the population of constituent particles in the cast and homogenized AA6063 alloy. Considering that both Al alloys are cast and homogenized, the effect of dispersoids and constituent particles on fracture in the Al1.2Mn alloy is practically impossible to separate, and the study is largely qualitative in nature, no further adjustments were made to the fracture parameters. The constituent particles are assigned a separate simple fracture criterion. When the von Mises stress in the particle reaches a certain threshold value, the corresponding element is eroded. The threshold stress is one of the parameters that were investigated in the study. It was assigned several increasing values: 100 MPa for what is later referred to as “weak particles”, 500 MPa for “medium strength particles”, 1000 MPa for “strong particles” and a very large value for “elastic particles”. The threshold stress may be interpreted as either the strength of the particles themselves or the particle-matrix interface.

The size of the modelled microstructure was chosen based on computational limitations. The mesh resolution should allow for representing the particles by at least several elements, while maintaining a similar element size for the whole oligocrystal model. Using a finer mesh around the particles or grain boundaries (GBs) and a coarser mesh inside the grains was found to affect the localization and fracture, thus a homogeneous element size was used. At the same time the total number of elements should not be too large, making the FE simulation time unfeasibly long. The additional limitation is that the simulation is run until global fracture or until some instability arises in the most deformed element, which for a ductile Al1.2Mn alloy leads to a large final global displacement and increased computation time. The resulting model and the corresponding mesh
$\omega$ is the damage parameter. The critical resolved shear stress evolution is described by the Voce type equation:

$$ \dot{t}_c^{(\alpha)} = \sum_{\beta=1}^{N} \sum_{k=1}^{2} \theta_k \exp \left( -\frac{\theta_k}{\tau_k} \Gamma \right) q_{\alpha\beta}, \quad (2) $$

where $q_{\alpha\beta}$ is the latent hardening matrix, $\theta_k$ and $\tau_k$ are the hardening parameters and $\Gamma$ is the accumulated plastic shear strain. The initial value of critical resolved shear stress is denoted as $\tau_0$. The damage evolution is described by equation:

$$ \dot{\omega} = \frac{3}{4} q_1 q_2 \omega (1 - \omega) \sinh\left(\frac{3}{2} q_2 T\right) \dot{\Gamma}, \quad (3) $$

where $q_1$ and $q_2$ are the damage parameters and $T$ is the stress triaxiality ratio. The initial and critical values of the damage parameter are denoted $\omega_0$ and $\omega_c$. The elastic constants, rate sensitivity, reference slip rate and latent hardening parameter are taken from [14]. The critical slip and work hardening parameters are obtained by fitting the slip system resistance equation multiplied by the Taylor factor equal to 3 (approximately corresponding to the notexture value) to the equivalent stress-strain curve of the alloy taken from [22]. The result is approximate, but considering the nature of the study it is sufficient to represent the alloys work hardening behaviour and as demonstrated in [25] the local strain distribution and localization are not overly sensitive to the work hardening parameters. The parameters used in the study are given in Table 1.

<table><caption>Table 1: Parameters of the CP model.</caption><thead><tr><th>τ<sub>0</sub>[MPa]</th><th>θ<sub>1</sub>[MPa]</th><th>τ<sub>1</sub>[MPa]</th><th>θ<sub>2</sub>[MPa]</th><th>τ<sub>2</sub>[MPa]</th><th>ω<sub>0</sub></th><th>ω<sub>c</sub></th><th>q<sub>1</sub></th><th>q<sub>2</sub></th></tr></thead><tbody><tr><td>13.0</td><td>191.0</td><td>16.5</td><td>24.0</td><td>34.0</td><td>0.006</td><td>0.12</td><td>1.5</td><td>1.0</td></tr></tbody></table>

A set of 50 random crystallographic orientations was assigned to the 50 grains of the FE model. The left and right boundary layers of the model, indicated by grey colour in Fig. 1 are assigned an isotropic elastic-plastic material with the same equivalent stress-strain curve. This creates a more natural and relaxed boundary at the edges and forces localization in the central part of the model, avoiding necking and shear bands at the fixed edges. The particles are assigned an isotropic elastic material with Young's modulus equal to 210 GPa and Poisson's ratio equal to 0.3.

The fracture model parameters for the coupled damage CP model are taken from [14], where it was calibrated for the population of constituent particles in the cast and homogenized AA6063 alloy. Considering that both Al alloys are cast and homogenized, the effect of dispersoids and constituent particles on fracture in the Al1.2Mn alloy is practically impossible to separate, and the study is largely qualitative in nature, no further adjustments were made to the fracture parameters. The constituent particles are assigned a separate simple fracture criterion. When the von Mises stress in the particle reaches a certain threshold value, the corresponding element is eroded. The threshold stress is one of the parameters that were investigated in the study. It was assigned several increasing values: 100 MPa for what is later referred to as “weak particles”, 500 MPa for “medium strength particles”, 1000 MPa for “strong particles” and a very large value for “elastic particles”. The threshold stress may be interpreted as either the strength of the particles themselves or the particle-matrix interface.

The size of the modelled microstructure was chosen based on computational limitations. The mesh resolution should allow for representing the particles by at least several elements, while maintaining a similar element size for the whole oligocrystal model. Using a finer mesh around the particles or grain boundaries (GBs) and a coarser mesh inside the grains was found to affect the localization and fracture, thus a homogeneous element size was used. At the same time the total number of elements should not be too large, making the FE simulation time unfeasibly long. The additional limitation is that the simulation is run until global fracture or until some instability arises in the most deformed element, which for a ductile Al1.2Mn alloy leads to a large final global displacement and increased computation time. The resulting model and the corresponding mesh
are shown in Fig. 1. The mesh of the oligocrystal part contains 120 000 linear hexahedral fully integrated elements, same type as used in [14]. The boundary layers of the model were meshed with 3000 wedge type elements. The simulations were run with the Abaqus/Explicit solver. Mass-scaling was applied to decrease the computation time and the kinetic energy was controlled at every step to ensure that it was small compared to the total energy and that the simulation remained quasi-static. The simulation time until fracture was around 5-7 days on 8 cores of a computer cluster node.

To investigate the effect of particles on the fracture simulations, the following variations were tested. Firstly, simulations with the CP model without damage were run. A baseline model contained no particles, then the same oligocrystal, but with elastic, strong, medium, and weak particles was simulated. Secondly, the coupled damage CP simulations were run, with a baseline oligocrystal and then the oligocrystal with particles. Several different oligocrystal morphologies and random orientation sets were tested, but they all yielded the same trends, so only one is shown further.

## Results and discussion

The contours of the von Mises equivalent plastic strain for the simulations with the CP model are presented in Fig. 2. The baseline simulation shows a typical shear band type localization, originating in softer grains with favourable orientations and propagating through the whole model. The introduction of elastic particles does not significantly affect the results and the same localization band arises, although with slightly reduced intensity. When strong particles (with a finite although high strength) are introduced, large strains in the shear band and the corresponding stress concentrations manage to erode some of the particles, thus creating a softer material locally and increasing the localization intensity. Decreasing the particle strength further leads to faster erosion of particles at the GBs with the highest stress concentrations, producing a large soft zone and concentrating the strain there, interrupting further concentration in the shear band. The weak particles quickly erode, leaving a crystal with voids. The result is strain concentration around the growing voids, which prevents the shear band from propagating out of the weaker grain where it originated. Instead, the voids grow quickly at several favourably oriented (normal to tensile direction) GBs on softer grains, leading to a quick softening effect of the global force-displacement curve. Fig. 3 shows contour plots of the equivalent plastic strain at the end of the simulations with the coupled damage CP model. Fracture occurs first in the shear band, but after initiation it does not propagate along the band, it follows its own path instead. The ductile crack propagation may be traced more clearly in Fig. 4, which shows plots of the damage distribution at the end of the simulations. The fracture may initiate and propagate either through softer grains with large plastic strains, or along the GBs due to higher stress concentrations. The introduction of elastic particles changes the fracture development. Additional stress concentrations around the particles lead to earlier fracture in the crystal, more concentrated damage, and a change of the crack propagation path.
are shown in Fig. 1. The mesh of the oligocrystal part contains 120 000 linear hexahedral fully integrated elements, same type as used in [14]. The boundary layers of the model were meshed with 3000 wedge type elements. The simulations were run with the Abaqus/Explicit solver. Mass-scaling was applied to decrease the computation time and the kinetic energy was controlled at every step to ensure that it was small compared to the total energy and that the simulation remained quasi-static. The simulation time until fracture was around 5-7 days on 8 cores of a computer cluster node.

To investigate the effect of particles on the fracture simulations, the following variations were tested. Firstly, simulations with the CP model without damage were run. A baseline model contained no particles, then the same oligocrystal, but with elastic, strong, medium, and weak particles was simulated. Secondly, the coupled damage CP simulations were run, with a baseline oligocrystal and then the oligocrystal with particles. Several different oligocrystal morphologies and random orientation sets were tested, but they all yielded the same trends, so only one is shown further.

## Results and discussion

The contours of the von Mises equivalent plastic strain for the simulations with the CP model are presented in Fig. 2. The baseline simulation shows a typical shear band type localization, originating in softer grains with favourable orientations and propagating through the whole model. The introduction of elastic particles does not significantly affect the results and the same localization band arises, although with slightly reduced intensity. When strong particles (with a finite although high strength) are introduced, large strains in the shear band and the corresponding stress concentrations manage to erode some of the particles, thus creating a softer material locally and increasing the localization intensity. Decreasing the particle strength further leads to faster erosion of particles at the GBs with the highest stress concentrations, producing a large soft zone and concentrating the strain there, interrupting further concentration in the shear band. The weak particles quickly erode, leaving a crystal with voids. The result is strain concentration around the growing voids, which prevents the shear band from propagating out of the weaker grain where it originated. Instead, the voids grow quickly at several favourably oriented (normal to tensile direction) GBs on softer grains, leading to a quick softening effect of the global force-displacement curve. Fig. 3 shows contour plots of the equivalent plastic strain at the end of the simulations with the coupled damage CP model. Fracture occurs first in the shear band, but after initiation it does not propagate along the band, it follows its own path instead. The ductile crack propagation may be traced more clearly in Fig. 4, which shows plots of the damage distribution at the end of the simulations. The fracture may initiate and propagate either through softer grains with large plastic strains, or along the GBs due to higher stress concentrations. The introduction of elastic particles changes the fracture development. Additional stress concentrations around the particles lead to earlier fracture in the crystal, more concentrated damage, and a change of the crack propagation path.
<figure><img src="image_2070.png" /><figcaption>Figure 2: Equivalent plastic strain contours for the CP model at the end of the simulation for: a) baseline oligocrystal, b) model with elastic particles, c) model with strong particles, d) model with medium particles, and e) model with weak particles.</figcaption></figure>

<figure><img src="image_2071.png" /><figcaption>Figure 3: Equivalent plastic strain contours for the coupled damage CP model at the end of simulation for: a) baseline oligocrystal, b) model with elastic particles, c) model with strong particles, d) model with medium particles, and e) model with weak particles.</figcaption></figure>

As the particle strength is decreased, some of the particles get eroded and produce voids. The voids lead to a combination of stress concentrations and large plastic strains, as they grow, which enhances the damage development in the crystal around them. As a result, strain and damage tend to concentrate more and more on the GBs around the particles and voids. The damage variable is more dispersed inside the grains in the simulations with no particles and elastic particles, while in the simulations with medium and weak particles the damage is extremely concentrated in the GB zone. Nevertheless, even in the simulation with weak particles, some transgranular fracture occurs, with certain combinations of softer grains and GB orientations. It is observed that two grains are fractured through the thickness by the ductile crack, with a very concentrated damage field around it.
<figure><img src="image_2062.png" /><figcaption>Figure 2: Equivalent plastic strain contours for the CP model at the end of the simulation for: a) baseline oligocrystal, b) model with elastic particles, c) model with strong particles, d) model with medium particles, and e) model with weak particles.</figcaption></figure>

<figure><img src="image_2063.png" /><figcaption>Figure 3: Equivalent plastic strain contours for the coupled damage CP model at the end of simulation for: a) baseline oligocrystal, b) model with elastic particles, c) model with strong particles, d) model with medium particles, and e) model with weak particles.</figcaption></figure>

As the particle strength is decreased, some of the particles get eroded and produce voids. The voids lead to a combination of stress concentrations and large plastic strains, as they grow, which enhances the damage development in the crystal around them. As a result, strain and damage tend to concentrate more and more on the GBs around the particles and voids. The damage variable is more dispersed inside the grains in the simulations with no particles and elastic particles, while in the simulations with medium and weak particles the damage is extremely concentrated in the GB zone. Nevertheless, even in the simulation with weak particles, some transgranular fracture occurs, with certain combinations of softer grains and GB orientations. It is observed that two grains are fractured through the thickness by the ductile crack, with a very concentrated damage field around it.
<figure><img src="image_2072.png" /><figcaption>Figure 4: Damage variable for the coupled damage CP model at the end of simulation on the undeformed configuration for: a) baseline oligocrystal, b) model with elastic particles, c) model with strong particles, d) model with medium particles, and e) model with weak particles.</figcaption></figure>

<figure><img src="image_2073.png" /><figcaption>Figure 5: Engineering stress-strain curves obtained from the simulations with the CP model on the left and the coupled damage CP model on the right.</figcaption></figure>

The engineering stress-strain curves for the simulations are presented in Fig. 5. In the baseline case of no damage in the CP model, the introduction of elastic particles just reinforces the material. The point at which the strong particles begin to erode may be identified on the stress-strain curve by the onset of softening. Comparing the curves for different particle strength it may be seen that the strain at which the material begins to soften is approximately proportional to the particle strength. A similar trend is observed in the simulations with the coupled damage CP model, although the stress reduction is much more abrupt. The results show that the effect of particle strength on the stress-strain curve and damage is quite significant and cannot be neglected, e.g., when fitting the work hardening and damage parameters of the CP model.
<figure><img src="image_2064.png" /><figcaption>Figure 4: Damage variable for the coupled damage CP model at the end of simulation on the undeformed configuration for: a) baseline oligocrystal, b) model with elastic particles, c) model with strong particles, d) model with medium particles, and e) model with weak particles.</figcaption></figure>

<figure><img src="image_2065.png" /><figcaption>Figure 5: Engineering stress-strain curves obtained from the simulations with the CP model on the left and the coupled damage CP model on the right.</figcaption></figure>

The engineering stress-strain curves for the simulations are presented in Fig. 5. In the baseline case of no damage in the CP model, the introduction of elastic particles just reinforces the material. The point at which the strong particles begin to erode may be identified on the stress-strain curve by the onset of softening. Comparing the curves for different particle strength it may be seen that the strain at which the material begins to soften is approximately proportional to the particle strength. A similar trend is observed in the simulations with the coupled damage CP model, although the stress reduction is much more abrupt. The results show that the effect of particle strength on the stress-strain curve and damage is quite significant and cannot be neglected, e.g., when fitting the work hardening and damage parameters of the CP model.
The fractography of the Al1.2Mn in [22] shows two populations of dimples, coarse and fine, with coarse dimples often containing constituent particles. This is interpreted in [22] as the evidence for coarse dimples originating on the constituent particles and correspondingly the fine ones on the dispersoids. The simulations show that the proportion of coarse dimples in the ductile crack can be correlated, among other things with the particle strength or the particle-matrix interface strength. The longitudinal cross-section of the fracture zone in [22] also may be used to evaluate how much void formation occurred in the vicinity of the ductile crack and compare it to the simulation results, which show a significant variation in the damage distribution, depending on the particle strength.

The most prominent aspect of the simulations is the critical role of localization in the response of the material. The strain localizes either due to crystallographic softening or void growth and coalescence, showing quite distinct distributions of strain and damage in the two cases, which transform one into the other as the particle strength decreases. The shear bands tend to be oriented at approximately a 45-degree angle to the tensile axis, while for the case of weak particles the localization is dominated by the zones around the GBs normal to the tensile axis. The orientation of the fracture surface from the experiments may be compared to the orientation of the fracture surface from the simulations.

## Conclusions

The ductile fracture in the Al1.2Mn alloy was modelled using a 2D FE model with a coupled damage CP model for the grains and an elastic model for the particles. The qualitative effect of particles, their strength and the presence of a second population of smaller particles were investigated. The threshold stress for particle fracture had a pronounced effect on the stress-strain curve, the type and intensity of the localization in the material, the distribution of damage in the coupled damage CP model, the proportion of fracture in the coarse and fine particles, and the path of the ductile crack through the material. These results may be compared to the experimentally observed features of ductile fracture in the Al1.2Mn alloy.

## References

[1] R. J. Asaro, "Crystal plasticity," *J Appl Mech*, 50 (1983) 921-934.
https://doi.org/10.1115/1.3167205

[2] F. Di Gioacchino and J. Q. da Fonseca, "An experimental study of the polycrystalline plasticity of austenitic stainless steel," *Int J Plasticity*, 74 (2015) 92-109.
https://doi.org/10.1016/j.ijplas.2015.05.012

[3] A. Guery, F. Hild, F. Latourte, and S. Roux, "Slip activities in polycrystals determined by coupling DIC measurements with crystal plasticity calculations," *Int J Plasticity*, 81 (2016) 249-266.
https://doi.org/10.1016/j.ijplas.2016.01.008

[4] H. Lim, J. D. Carroll, C. C. Battaile, B. L. Boyce, and C. R. Weinberger, "Quantitative comparison between experimental measurements and CP-FEM predictions of plastic deformation in a tantalum oligocrystal," *Int J Mech Sci*, vol. 92, pp. 98-108, 2015.
https://doi.org/10.1016/j.ijmecsci.2014.12.010

[5] D. Depriester, J. Goulmy, and L. Barrallier, "Crystal Plasticity simulations of in situ tensile tests: A two-step inverse method for identification of CP parameters, and assessment of CPFEM capabilities," *Int J Plasticity*, 168 (2023)103695.
https://doi.org/10.1016/j.ijplas.2023.103695

[6] T. F. Morgeneyer et al., "On crystallographic aspects of heterogeneous plastic flow during ductile tearing: 3D measurements and crystal plasticity simulations for AA7075-T651," *Int J Plasticity*, 144 (2021) 103028.
https://doi.org/10.1016/j.ijplas.2021.103028
The fractography of the Al1.2Mn in [22] shows two populations of dimples, coarse and fine, with coarse dimples often containing constituent particles. This is interpreted in [22] as the evidence for coarse dimples originating on the constituent particles and correspondingly the fine ones on the dispersoids. The simulations show that the proportion of coarse dimples in the ductile crack can be correlated, among other things with the particle strength or the particle-matrix interface strength. The longitudinal cross-section of the fracture zone in [22] also may be used to evaluate how much void formation occurred in the vicinity of the ductile crack and compare it to the simulation results, which show a significant variation in the damage distribution, depending on the particle strength.

The most prominent aspect of the simulations is the critical role of localization in the response of the material. The strain localizes either due to crystallographic softening or void growth and coalescence, showing quite distinct distributions of strain and damage in the two cases, which transform one into the other as the particle strength decreases. The shear bands tend to be oriented at approximately a 45-degree angle to the tensile axis, while for the case of weak particles the localization is dominated by the zones around the GBs normal to the tensile axis. The orientation of the fracture surface from the experiments may be compared to the orientation of the fracture surface from the simulations.

## Conclusions

The ductile fracture in the Al1.2Mn alloy was modelled using a 2D FE model with a coupled damage CP model for the grains and an elastic model for the particles. The qualitative effect of particles, their strength and the presence of a second population of smaller particles were investigated. The threshold stress for particle fracture had a pronounced effect on the stress-strain curve, the type and intensity of the localization in the material, the distribution of damage in the coupled damage CP model, the proportion of fracture in the coarse and fine particles, and the path of the ductile crack through the material. These results may be compared to the experimentally observed features of ductile fracture in the Al1.2Mn alloy.

## References

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[2] F. Di Gioacchino and J. Q. da Fonseca, "An experimental study of the polycrystalline plasticity of austenitic stainless steel," *Int J Plasticity*, 74 (2015) 92-109.
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[3] A. Guery, F. Hild, F. Latourte, and S. Roux, "Slip activities in polycrystals determined by coupling DIC measurements with crystal plasticity calculations," *Int J Plasticity*, 81 (2016) 249-266.
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[4] H. Lim, J. D. Carroll, C. C. Battaile, B. L. Boyce, and C. R. Weinberger, "Quantitative comparison between experimental measurements and CP-FEM predictions of plastic deformation in a tantalum oligocrystal," *Int J Mech Sci*, vol. 92, pp. 98-108, 2015.
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[7] K. M. Min, H. Lee, H.-D. Joo, H. N. Han, and M.-G. Lee, "Crystal Plasticity Simulation of Shear Band Formation and Recrystallization Texture in Grain-Oriented Electrical Steel," *Available at SSRN* 4467215.

[8] A. L. Gurson, "Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media," *J Eng Mater*, 99 (1977) 2-15.
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[13] M. Khadyko, B. H. Frodal, and O. S. Hopperstad, "Finite element simulation of ductile fracture in polycrystalline materials using a regularized porous crystal plasticity model," *International Journal of Fracture*, 228 (2021) 15-31.
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[14] B. H. Frodal, S. Thomesen, T. Børvik, and O. S. Hopperstad, "On the coupling of damage and single crystal plasticity for ductile polycrystalline materials," *Int J Plasticity*, 142 (2021) 102996.
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[15] A. Li, W. Hu, H. Li, Z. Zhan, and Q. Meng, "A crystal plasticity-based microdamage model and its application on the tensile failure process analysis of 7075 aluminum alloy," *Mat Sci Eng*, 884 (2023) 145541.
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[16] B. H. Frodal, L. Lodgaard, Y. Langsrud, T. Børvik, and O. S. Hopperstad, "Influence of Local Microstructural Variations on the Bendability of Aluminum Extrusions: Experiments and Crystal Plasticity Analyses," *J Appl Mech*, 90 (2023) 041006.
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[17] C. K. Cocke et al., "Implementation and experimental validation of nonlocal damage in a large-strain elasto-viscoplastic FFT-based framework for predicting ductile fracture in 3D polycrystalline materials," *Int J Plasticity*, 162 (2023) 103508.
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[18] T. Yalçinkaya, İ. T. Tandoğan, and İ. Özdemir, "Void growth based inter-granular ductile fracture in strain gradient polycrystalline plasticity," *Int J Plasticity*, 147 (2021) 103123.
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[19] J.-M. Scherer, J. Besson, S. Forest, J. Hure, and B. Tanguy, "A strain gradient plasticity model of porous single crystal ductile fracture," *J Mech Phys Solids*, 156 (2021) 104606.
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[20] P. Gao, M. Fei, M. Zhan, and M. Fu, "Microstructure-and damage-nucleation-based crystal plasticity finite element modeling for the nucleation of multi-type voids during plastic deformation of Al alloys," *Int J Plasticity*, 165 (2023) 103609.
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[7] K. M. Min, H. Lee, H.-D. Joo, H. N. Han, and M.-G. Lee, "Crystal Plasticity Simulation of Shear Band Formation and Recrystallization Texture in Grain-Oriented Electrical Steel," *Available at SSRN* 4467215.

[8] A. L. Gurson, "Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media," *J Eng Mater*, 99 (1977) 2-15.
https://doi.org/10.1115/1.3443401

[9] J. R. Rice and D. M. Tracey, "On the ductile enlargement of voids in triaxial stress fields," *J Mech Phys Solids*, 17 (1969) 201-217.
https://doi.org/10.1016/0022-5096(69)90033-7

[10] V. Tvergaard and A. Needleman, "Analysis of the cup-cone fracture in a round tensile bar," *Acta Metall Mater*, 32 (1984) 157-169.
https://doi.org/10.1016/0001-6160(84)90213-X

[11] L. E. B. Dæhli, J. Faleskog, T. Børvik, and O. S. Hopperstad, "Unit cell simulations and porous plasticity modelling for strongly anisotropic FCC metals," *Eur J Mech A-Solid*, 65 (2017) 360-383.
https://doi.org/10.1016/j.euromechsol.2017.05.004

[12] X. Han, J. Besson, S. Forest, B. Tanguy, and S. Bugat, "A yield function for single crystals containing voids," *Int. J Solids Struct*, 50 (2013) 2115-2131.
https://doi.org/10.1016/j.ijsolstr.2013.02.005

[13] M. Khadyko, B. H. Frodal, and O. S. Hopperstad, "Finite element simulation of ductile fracture in polycrystalline materials using a regularized porous crystal plasticity model," *International Journal of Fracture*, 228 (2021) 15-31.
https://doi.org/10.1007/s10704-020-00503-w

[14] B. H. Frodal, S. Thomesen, T. Børvik, and O. S. Hopperstad, "On the coupling of damage and single crystal plasticity for ductile polycrystalline materials," *Int J Plasticity*, 142 (2021) 102996.
https://doi.org/10.1016/j.ijplas.2021.102996

[15] A. Li, W. Hu, H. Li, Z. Zhan, and Q. Meng, "A crystal plasticity-based microdamage model and its application on the tensile failure process analysis of 7075 aluminum alloy," *Mat Sci Eng*, 884 (2023) 145541.
https://doi.org/10.1016/j.msea.2023.145541

[16] B. H. Frodal, L. Lodgaard, Y. Langsrud, T. Børvik, and O. S. Hopperstad, "Influence of Local Microstructural Variations on the Bendability of Aluminum Extrusions: Experiments and Crystal Plasticity Analyses," *J Appl Mech*, 90 (2023) 041006.
https://doi.org/10.1115/1.4056429

[17] C. K. Cocke et al., "Implementation and experimental validation of nonlocal damage in a large-strain elasto-viscoplastic FFT-based framework for predicting ductile fracture in 3D polycrystalline materials," *Int J Plasticity*, 162 (2023) 103508.
https://doi.org/10.1016/j.ijplas.2022.103508

[18] T. Yalçinkaya, İ. T. Tandoğan, and İ. Özdemir, "Void growth based inter-granular ductile fracture in strain gradient polycrystalline plasticity," *Int J Plasticity*, 147 (2021) 103123.
https://doi.org/10.1016/j.ijplas.2021.103123

[19] J.-M. Scherer, J. Besson, S. Forest, J. Hure, and B. Tanguy, "A strain gradient plasticity model of porous single crystal ductile fracture," *J Mech Phys Solids*, 156 (2021) 104606.
https://doi.org/10.1016/j.jmps.2021.104606

[20] P. Gao, M. Fei, M. Zhan, and M. Fu, "Microstructure-and damage-nucleation-based crystal plasticity finite element modeling for the nucleation of multi-type voids during plastic deformation of Al alloys," *Int J Plasticity*, 165 (2023) 103609.
https://doi.org/10.1016/j.ijplas.2023.103609
[21] F. Qayyum et al., "Influence of non-metallic inclusions on local deformation and damage behavior of modified 16MnCrS5 steel," *Crystals*, 12 (2022) 281. https://doi.org/10.3390/cryst12020281

[22] I. Westermann, K. O. Pedersen, T. Furu, T. Børvik, and O. S. Hopperstad, "Effects of particles and solutes on strength, work-hardening and ductile fracture of aluminium alloys," *Mechanics of Materials*, 79 (2014) 58-72. https://doi.org/10.1016/j.mechmat.2014.08.006

[23] J. Liu, Z. Li, M. Huang, J. Zhu, L. Zhao, and Y. Zhu, "Crystallographic texture effect on statistical microvoid growth in heterogeneous polycrystals," *Int. J Solids Struct*, 281 (2023) 112435. https://doi.org/10.1016/j.ijsolstr.2023.112435

[24] R. Quey and M. Kasemer, "The neper/fepx project: free/open-source polycrystal generation, deformation simulation, and post-processing," in *IOP Conference Series: Materials Science and Engineering*, 2022, vol. 1249, no. 1: IOP Publishing, p. 012021. https://doi.org/10.1088/1757-899X/1249/1/012021

[25] M. Khadyko, J. Sturdy, S. Dumoulin, L. R. Hellevik, and O. S. Hopperstad, "Uncertainty quantification and sensitivity analysis of material parameters in crystal plasticity finite element models," *Journal of Mechanics of Materials and Structures*, 13 (2018) 379-400. https://doi.org/10.2140/jomms.2018.13.379
[21] F. Qayyum et al., "Influence of non-metallic inclusions on local deformation and damage behavior of modified 16MnCrS5 steel," *Crystals*, 12 (2022) 281. https://doi.org/10.3390/cryst12020281

[22] I. Westermann, K. O. Pedersen, T. Furu, T. Børvik, and O. S. Hopperstad, "Effects of particles and solutes on strength, work-hardening and ductile fracture of aluminium alloys," *Mechanics of Materials*, 79 (2014) 58-72. https://doi.org/10.1016/j.mechmat.2014.08.006

[23] J. Liu, Z. Li, M. Huang, J. Zhu, L. Zhao, and Y. Zhu, "Crystallographic texture effect on statistical microvoid growth in heterogeneous polycrystals," *Int. J Solids Struct*, 281 (2023) 112435. https://doi.org/10.1016/j.ijsolstr.2023.112435

[24] R. Quey and M. Kasemer, "The neper/fepx project: free/open-source polycrystal generation, deformation simulation, and post-processing," in *IOP Conference Series: Materials Science and Engineering*, 2022, vol. 1249, no. 1: IOP Publishing, p. 012021. https://doi.org/10.1088/1757-899X/1249/1/012021

[25] M. Khadyko, J. Sturdy, S. Dumoulin, L. R. Hellevik, and O. S. Hopperstad, "Uncertainty quantification and sensitivity analysis of material parameters in crystal plasticity finite element models," *Journal of Mechanics of Materials and Structures*, 13 (2018) 379-400. https://doi.org/10.2140/jomms.2018.13.379
# Computational thermo-mechanical process design by integrating crystal plasticity and phase field model

MIN Kyung Mun<sup>1,2,a</sup>, BONG Hyuk Jong<sup>2,b*</sup> and LEE Myoung-Gyu<sup>1,c*</sup>

**Keywords:** Crystal Plasticity, Phase Field Model, Recrystallization, Anisotropy, Formability

**Abstract.** An integrated model, merging the crystal plasticity finite element model (CPFEM) and the phase field model (PFM), is introduced for simulating the thermo-mechanical processing of ultra-low carbon steels. CPFEM serves as the mechanical simulation tool, forecasting deformation inconsistencies such as local stress concentration, inhomogeneous dislocation distribution, and shear bands. Meanwhile, PFM is utilized for predicting microstructural evolution, particularly nucleation and growth during heat treatments. To seamlessly integrate CPFEM and PFM, which are based on the finite element and finite difference methods respectively, an optimized coupling algorithm is utilized to avoid excessive computational cost. Importantly, a generalized strain energy release maximization model is integrated into the PFM, which leverages the analytical outcomes of CPFEM to predict the recrystallization texture of steels, factoring in multiple slip activities under mechanical loading conditions. The proposed model is applied to evaluate the anisotropy and formability of the thermo-mechanically processed ultra-low carbon steel through virtual mechanical experiments.

**Introduction**

Low carbon steels are FeC alloys with a BCC structure where the carbon concentration is typically maintained below ~0.1 wt%. Ultra-low carbon steels, a subset of low carbon steels, have even lower carbon concentrations, typically less than ~0.02 wt% [1]. These steels find use in a variety of applications, both in their hot-rolled state and after further thermo-mechanical processing steps such as cold-rolling, annealing, and potentially coating. Modern processing techniques allow for a high degree of control over both the chemical composition and manufacturing processes of low carbon steel, enabling the production of a wide range of grades with varying combinations of strength and formability. Precise control of dimensions such as gauge, width, shape, and flatness is also achievable [2].

However, it is the formability of low carbon steel that ensures its continued prominence as a major engineering material. This material can be economically and satisfactorily formed into a wide array of complex shapes without issues such as splitting, necking, or wrinkling, which are common problems in the forming processes of other materials [2].

Sheet formability can be evaluated from two perspectives: drawability and stretchability. Drawability refers to the ability of the material to flow easily in the plane of the sheet and resist thinning in the thickness direction [3]. It is primarily characterized by sheet anisotropy, as measured by Lankford coefficients (r-values), and flow stresses. These parameters are significantly influenced by the developed crystallographic orientations that evolves during thermo-mechanical processes. In contrast, stretchability represents the capacity of the material to resist localized

<figure><img src="image_2074.png" /></figure>

<footnote n="1"><sup>1</sup>Department of Materials Science and Engineering and Research Institute of Advanced Materials (RIAM), Seoul National University, Seoul 08826, Republic of Korea</footnote>

<footnote n="2"><sup>2</sup>Department of Materials Processing, Materials Digital Platform Division, Korea Institute of Materials Science (KIMS), Changwon 51508, Republic of Korea</footnote>

<footnote><sup>a</sup>kmmmin@kims.re.kr, <sup>b</sup>hjbong@kims.re.kr, <sup>c</sup>myounglee@snu.ac.kr</footnote>
# Computational thermo-mechanical process design by integrating crystal plasticity and phase field model

MIN Kyung Mun<sup>1,2,a</sup>, BONG Hyuk Jong<sup>2,b*</sup> and LEE Myoung-Gyu<sup>1,c*</sup>

**Keywords:** Crystal Plasticity, Phase Field Model, Recrystallization, Anisotropy, Formability

**Abstract.** An integrated model, merging the crystal plasticity finite element model (CPFEM) and the phase field model (PFM), is introduced for simulating the thermo-mechanical processing of ultra-low carbon steels. CPFEM serves as the mechanical simulation tool, forecasting deformation inconsistencies such as local stress concentration, inhomogeneous dislocation distribution, and shear bands. Meanwhile, PFM is utilized for predicting microstructural evolution, particularly nucleation and growth during heat treatments. To seamlessly integrate CPFEM and PFM, which are based on the finite element and finite difference methods respectively, an optimized coupling algorithm is utilized to avoid excessive computational cost. Importantly, a generalized strain energy release maximization model is integrated into the PFM, which leverages the analytical outcomes of CPFEM to predict the recrystallization texture of steels, factoring in multiple slip activities under mechanical loading conditions. The proposed model is applied to evaluate the anisotropy and formability of the thermo-mechanically processed ultra-low carbon steel through virtual mechanical experiments.

**Introduction**

Low carbon steels are FeC alloys with a BCC structure where the carbon concentration is typically maintained below ~0.1 wt%. Ultra-low carbon steels, a subset of low carbon steels, have even lower carbon concentrations, typically less than ~0.02 wt% [1]. These steels find use in a variety of applications, both in their hot-rolled state and after further thermo-mechanical processing steps such as cold-rolling, annealing, and potentially coating. Modern processing techniques allow for a high degree of control over both the chemical composition and manufacturing processes of low carbon steel, enabling the production of a wide range of grades with varying combinations of strength and formability. Precise control of dimensions such as gauge, width, shape, and flatness is also achievable [2].

However, it is the formability of low carbon steel that ensures its continued prominence as a major engineering material. This material can be economically and satisfactorily formed into a wide array of complex shapes without issues such as splitting, necking, or wrinkling, which are common problems in the forming processes of other materials [2].

Sheet formability can be evaluated from two perspectives: drawability and stretchability. Drawability refers to the ability of the material to flow easily in the plane of the sheet and resist thinning in the thickness direction [3]. It is primarily characterized by sheet anisotropy, as measured by Lankford coefficients (r-values), and flow stresses. These parameters are significantly influenced by the developed crystallographic orientations that evolves during thermo-mechanical processes. In contrast, stretchability represents the capacity of the material to resist localized

<figure><img src="image_2066.png" /></figure>

<footnote n="1"><sup>1</sup>Department of Materials Science and Engineering and Research Institute of Advanced Materials (RIAM), Seoul National University, Seoul 08826, Republic of Korea</footnote>

<footnote n="2"><sup>2</sup>Department of Materials Processing, Materials Digital Platform Division, Korea Institute of Materials Science (KIMS), Changwon 51508, Republic of Korea</footnote>

<footnote><sup>a</sup>kmmmin@kims.re.kr, <sup>b</sup>hjbong@kims.re.kr, <sup>c</sup>myounglee@snu.ac.kr</footnote>
necking and consequently withstand complex non-uniform deformation. It is typically characterized by factors such as the strain hardening and strain rate hardening [4].

During the thermo-mechanical processes where static recrystallization (SRX) occurs, the crystallographic orientations of low carbon steel evolve. These orientations can be represented by two main characteristic fibers: α-fiber (<110>//RD, where RD stands for the rolling direction) and γ-fiber ({111}//ND, where ND stands for the normal direction). For instance, during the cold rolling process, α-fiber is predominantly developed, with only minor development of γ-fiber. However, during the subsequent annealing process, the γ-fiber becomes the primary component, providing unique anisotropic properties to low carbon steel. Simultaneously, cold-rolled low carbon steel, which has high stored energy due to dislocation accumulation during deformation, can recover its original strength and ductility by releasing this stored energy through SRX, affecting material hardening. As both sheet anisotropy and material hardening are influenced by these microstructure evolutions, it is necessary to take them into account in any comprehensive evaluation of the formability of the thermo-mechanically processed low carbon steel.

In this work, a virtual identification scheme has been proposed as a complementary strategy to the conventional experiment-based approach, with the aim of predicting the anisotropy and formability of thermo-mechanically processed ultra-low carbon steel sheets. This scheme employs the proposed integrated crystal plasticity and phase field model, taking into consideration the effect of microstructural characteristics on the mechanical performance of the final product. Local inhomogeneities during cold rolling are predicted using the CPFEM, while the PFM is used to analyze microstructure evolution during the annealing process. Upon simulating the sheet manufacturing process of ultra-low carbon steel, the representative volume element (RVE) is generated. The RVE, which encompass deformation history and recrystallized microstructural data, serves as the basis for evaluating sheet anisotropy and formability during the virtual experiments. Plastic anisotropy, characterized by r-values and yield stresses, is predicted using virtual experiments and then compared with experimental values. Various anisotropic yield functions are identified based on the predicted anisotropic mechanical properties. When calculating the forming limit diagram (FLD) using the M-K model [5] to evaluate sheet formability, this study discusses not only the conventional approach using phenomenological anisotropic yield functions, but also a crystal plasticity-based approach [6] designed to consider texture evolution during deformation. Both approaches are quantitatively compared to provide a comprehensive understanding for evaluating formability. The anisotropy and formability predictions from these approaches are then validated experimentally, confirming the accuracy of the proposed integrated crystal plasticity and phase field model.

## Integrated Crystal Plasticity and Phase Field Model

A flow chart depicting the proposed integrated crystal plasticity and phase field model is presented in Fig. 1. The crystal plasticity model was formulated using the user material subroutine UMAT provided in Abaqus/Standard, while the phase field model was implemented in the Intel Visual Fortran program®. The proposed multi-scale approach encompasses the following four steps, outlined step-by-step with the main inputs and outputs at each stage:

Step 1. CPFEM for mechanical loading: Boundary value problems are addressed using finite element simulation that incorporates crystal plasticity constitutive laws. Instead of realistic boundary conditions of cold rolling, which include contact between tools and workpieces, a simpler RVE approach is employed for computational efficiency while maintaining reproducible microstructure features. In this step, local deformation inhomogeneities such as crystallographic orientations (rolling texture), slip activities, and dislocation densities (or stored energy) are calculated from single crystal properties, orientations, and grain sizes.

Step 2. Numerical mapping to connect CPFEM and PFM: Due to differences in discretization methods between CPFEM and PFM, which are based on the finite element and finite difference
necking and consequently withstand complex non-uniform deformation. It is typically characterized by factors such as the strain hardening and strain rate hardening [4].

During the thermo-mechanical processes where static recrystallization (SRX) occurs, the crystallographic orientations of low carbon steel evolve. These orientations can be represented by two main characteristic fibers: α-fiber (<110>//RD, where RD stands for the rolling direction) and γ-fiber ({111}//ND, where ND stands for the normal direction). For instance, during the cold rolling process, α-fiber is predominantly developed, with only minor development of γ-fiber. However, during the subsequent annealing process, the γ-fiber becomes the primary component, providing unique anisotropic properties to low carbon steel. Simultaneously, cold-rolled low carbon steel, which has high stored energy due to dislocation accumulation during deformation, can recover its original strength and ductility by releasing this stored energy through SRX, affecting material hardening. As both sheet anisotropy and material hardening are influenced by these microstructure evolutions, it is necessary to take them into account in any comprehensive evaluation of the formability of the thermo-mechanically processed low carbon steel.

In this work, a virtual identification scheme has been proposed as a complementary strategy to the conventional experiment-based approach, with the aim of predicting the anisotropy and formability of thermo-mechanically processed ultra-low carbon steel sheets. This scheme employs the proposed integrated crystal plasticity and phase field model, taking into consideration the effect of microstructural characteristics on the mechanical performance of the final product. Local inhomogeneities during cold rolling are predicted using the CPFEM, while the PFM is used to analyze microstructure evolution during the annealing process. Upon simulating the sheet manufacturing process of ultra-low carbon steel, the representative volume element (RVE) is generated. The RVE, which encompass deformation history and recrystallized microstructural data, serves as the basis for evaluating sheet anisotropy and formability during the virtual experiments. Plastic anisotropy, characterized by r-values and yield stresses, is predicted using virtual experiments and then compared with experimental values. Various anisotropic yield functions are identified based on the predicted anisotropic mechanical properties. When calculating the forming limit diagram (FLD) using the M-K model [5] to evaluate sheet formability, this study discusses not only the conventional approach using phenomenological anisotropic yield functions, but also a crystal plasticity-based approach [6] designed to consider texture evolution during deformation. Both approaches are quantitatively compared to provide a comprehensive understanding for evaluating formability. The anisotropy and formability predictions from these approaches are then validated experimentally, confirming the accuracy of the proposed integrated crystal plasticity and phase field model.

## Integrated Crystal Plasticity and Phase Field Model

A flow chart depicting the proposed integrated crystal plasticity and phase field model is presented in Fig. 1. The crystal plasticity model was formulated using the user material subroutine UMAT provided in Abaqus/Standard, while the phase field model was implemented in the Intel Visual Fortran program®. The proposed multi-scale approach encompasses the following four steps, outlined step-by-step with the main inputs and outputs at each stage:

Step 1. CPFEM for mechanical loading: Boundary value problems are addressed using finite element simulation that incorporates crystal plasticity constitutive laws. Instead of realistic boundary conditions of cold rolling, which include contact between tools and workpieces, a simpler RVE approach is employed for computational efficiency while maintaining reproducible microstructure features. In this step, local deformation inhomogeneities such as crystallographic orientations (rolling texture), slip activities, and dislocation densities (or stored energy) are calculated from single crystal properties, orientations, and grain sizes.

Step 2. Numerical mapping to connect CPFEM and PFM: Due to differences in discretization methods between CPFEM and PFM, which are based on the finite element and finite difference
methods, respectively, suitable numerical mapping must be introduced to transfer state variables between the two models. The orientation and stored energy distributions calculated in the first step are transferred to the PFM using the mapping algorithm. A second mapping is performed from the predicted microstructure to the RVE model for virtual mechanical testing, transferring updated single crystal properties, orientations, and grain sizes to the integration points of finite element model.

Step 3. PFM for microstructure evolution: Using the mapped state variables of the finite element integration points, multi-phase field simulations [8,9] are carried out to compute grain boundary migration in SRX. Grain boundary characteristics, such as grain boundary energy and mobility, serve as primary inputs for this step.

Step 4. CPFEM for virtual mechanical testing: In this step, various mechanical tests can be virtually conducted using the calculated microstructure information, such as updated grain size, crystallographic orientations, and mechanical properties. The focus of this step is on the evaluation of anisotropy and formability of thermo-mechanically processed sheet metals. For validation purposes, anisotropy, including r-values and flow stresses along different material orientations, and formability, represented by FLD, are calculated and compared with experimental results.

In addition to these main steps, two complementary steps for PFM are emphasized during the primary Step 3:

Step 3-1. Calculation of recrystallization texture: Dislocation densities in activated slip systems and the orientation of each grain, determined from the mechanical simulation step, are utilized to analyze the orientations of the newly recrystallized nucleus, i.e., recrystallization texture. In this study, strain energy release maximization (SERM) theory proposed by Lee [10–12] is expanded to accommodate the general multi-axial stress state and cubic crystals, resulting in the generalized SERM (GSERM) [13].

Step 3-2. Nucleation Criterion: The nucleation criterion is formulated by taking into account the grain boundary mobility and stored energy, using empirical expressions from Humphreys and Hatherly [14] and Zurob et al. [15] to reproduce the nucleation step.

<figure><img src="image_2075.png" /><figcaption>Fig. 1. Overall flow chart of the integrated crystal plasticity and phase field model.</figcaption></figure>

## Results and Discussion

Simulation of Cold-rolled Ultra-low Carbon Steel. The RVE approach was adopted to simulate the mechanical response of the steel during the cold rolling process. The RVE model, consisting of 498 grains, was generated based on the microstructural information from the hot-rolled state. The side length of the RVE model was set to 90 µm, and it comprised 27,000 (30 × 30 × 30) elements.
methods, respectively, suitable numerical mapping must be introduced to transfer state variables between the two models. The orientation and stored energy distributions calculated in the first step are transferred to the PFM using the mapping algorithm. A second mapping is performed from the predicted microstructure to the RVE model for virtual mechanical testing, transferring updated single crystal properties, orientations, and grain sizes to the integration points of finite element model.

**Step 3.** PFM for microstructure evolution: Using the mapped state variables of the finite element integration points, multi-phase field simulations [8,9] are carried out to compute grain boundary migration in SRX. Grain boundary characteristics, such as grain boundary energy and mobility, serve as primary inputs for this step.

**Step 4.** CPFEM for virtual mechanical testing: In this step, various mechanical tests can be virtually conducted using the calculated microstructure information, such as updated grain size, crystallographic orientations, and mechanical properties. The focus of this step is on the evaluation of anisotropy and formability of thermo-mechanically processed sheet metals. For validation purposes, anisotropy, including r-values and flow stresses along different material orientations, and formability, represented by FLD, are calculated and compared with experimental results.

In addition to these main steps, two complementary steps for PFM are emphasized during the primary Step 3:

**Step 3-1.** Calculation of recrystallization texture: Dislocation densities in activated slip systems and the orientation of each grain, determined from the mechanical simulation step, are utilized to analyze the orientations of the newly recrystallized nucleus, i.e., recrystallization texture. In this study, strain energy release maximization (SERM) theory proposed by Lee [10–12] is expanded to accommodate the general multi-axial stress state and cubic crystals, resulting in the generalized SERM (GSERM) [13].

**Step 3-2.** Nucleation Criterion: The nucleation criterion is formulated by taking into account the grain boundary mobility and stored energy, using empirical expressions from Humphreys and Hatherly [14] and Zurob et al. [15] to reproduce the nucleation step.

<figure><img src="image_2067.png" /><figcaption>Fig. 1. Overall flow chart of the integrated crystal plasticity and phase field model.</figcaption></figure>

## Results and Discussion

Simulation of Cold-rolled Ultra-low Carbon Steel. The RVE approach was adopted to simulate the mechanical response of the steel during the cold rolling process. The RVE model, consisting of 498 grains, was generated based on the microstructural information from the hot-rolled state. The side length of the RVE model was set to 90 µm, and it comprised 27,000 (30 × 30 × 30) elements.
For the finite element simulations, the eight-node continuum elements with reduced integration were used. Fig. 2(a) presents the RVE model for the hot-rolled ultra-low carbon steel.

<figure><img src="image_2076.png" /><figcaption>Fig. 2. (a) Representative volume element (RVE) for hot-rolled ultra-low carbon steel and (b) deformed RVE and its equivalent plastic strain after a thickness reduction ratio of 85.5%.</figcaption></figure>

For the simulation, a plane strain compression condition was imposed as a simplified loading equivalent to the typical cold rolling process. The displacement in the transverse direction (TD) was fixed and compressive loading was applied along the ND. The thickness reduction ratio was set to 85.5%. Additionally, periodic boundary conditions (PBCs) were also applied to the RVE to ensure its physically continuous behavior. Fig. 2(b) displays the deformed RVE and its equivalent plastic strain with a thickness reduction ratio of 85.5%. Due to the large deformation from plane strain compression with a thickness reduction ratio of 85.5%, the equivalent plastic strain reached ~2.3.

<figure><img src="image_2077.png" /><figcaption>Fig. 3. Deformed representation volume element and its total dislocation densities (unit: m⁻²) after a thickness reduction ratio of 85.5% rolling before (with CPFEM) and after mapping to PFM grids.</figcaption></figure>

Fig. 3 illustrates the distributions of total dislocation densities after the crystal plasticity simulation and the corresponding dislocation densities mapped for the PFM simulations. For this mapping process, approximately 3000 grids per grain were utilized. In CPFEM, dislocation densities and other state variables were calculated at the integration points and then appropriately
For the finite element simulations, the eight-node continuum elements with reduced integration were used. Fig. 2(a) presents the RVE model for the hot-rolled ultra-low carbon steel.

<figure><img src="image_2068.png" /><figcaption>Fig. 2. (a) Representative volume element (RVE) for hot-rolled ultra-low carbon steel and (b) deformed RVE and its equivalent plastic strain after a thickness reduction ratio of 85.5%.</figcaption></figure>

For the simulation, a plane strain compression condition was imposed as a simplified loading equivalent to the typical cold rolling process. The displacement in the transverse direction (TD) was fixed and compressive loading was applied along the ND. The thickness reduction ratio was set to 85.5%. Additionally, periodic boundary conditions (PBCs) were also applied to the RVE to ensure its physically continuous behavior. Fig. 2(b) displays the deformed RVE and its equivalent plastic strain with a thickness reduction ratio of 85.5%. Due to the large deformation from plane strain compression with a thickness reduction ratio of 85.5%, the equivalent plastic strain reached ~2.3.

<figure><img src="image_2069.png" /><figcaption>Fig. 3. Deformed representation volume element and its total dislocation densities (unit: m⁻²) after a thickness reduction ratio of 85.5% rolling before (with CPFEM) and after mapping to PFM grids.</figcaption></figure>

Fig. 3 illustrates the distributions of total dislocation densities after the crystal plasticity simulation and the corresponding dislocation densities mapped for the PFM simulations. For this mapping process, approximately 3000 grids per grain were utilized. In CPFEM, dislocation densities and other state variables were calculated at the integration points and then appropriately
mapped onto the PFM grids. This study employed a regularized Wigner-Seitz cell approach, as described in the work of Min et al. [13].

<figure><img src="image_2078.png" /><figcaption>Fig. 4. Predicted rolling texture of ultra-low carbon steel, represented by φ₂=45° orientation distribution function section.</figcaption></figure>

The calculated rolling texture of ultra-low carbon steel are presented in Fig. 4, which depicts the φ₂=45° orientation distribution function (ODF) section after a thickness reduction ratio of 85.5% rolling (equivalently, the plane strain compression). The calculated cold-rolling texture of ultra-low carbon steel showed mixture of the major α-fiber (<110>//RD) and minor γ-fiber (<111>//ND), which was in agreement with experimentally measured texture components [16,17]. In particular, three main experimentally reported cold-rolling texture components, {665}<110> (approximated by {111}<110>), {558}<110> (approximated by {112}<110>), and {001}<110>, could be perfectly predicted from the proposed CPFEM.

Simulation of Recrystallized Ultra-low Carbon Steel. The process of static recrystallization, or annealing, was simulated using the PFM. This was performed by utilizing the state variables derived from the CPFEM results for the cold rolling simulation.

Fig. 5 illustrates the progression of static recrystallization over time during the annealing process, as predicted by the PFM. Grains were color-coded to represent deformed (gray) and recrystallized (white) states. The orientations of the newly nucleated grains were determined based on predictions from the GSERM model, which calculates orientation probability upon nucleation. To prevent the formation of idealistically oriented grains through theoretical calculations, random orientations were also assigned to nucleation orientation probability, maintaining a ratio of 7:3 between calculated and random orientations.

The recrystallized grains can be clearly distinguished from the remaining deformed grains due to their equiaxed morphology. The residual deformed grains were elongated along the RD in the form of fibers, which is a characteristic feature of metals subjected to cold rolling.

In Fig. 5, the original parent grain boundaries (elongated to the RD) were fixed. This was primarily due to the large difference in stored energy between the recrystallized grain boundaries and their corresponding parent grains This assumption was also beneficial in terms of reducing the computational cost.

As depicted in the figure, nucleation was initiated in regions with a higher level of stored energy (refer to Fig. 3 for the distribution of dislocation density after rolling simulation). The calculated results from the annealing simulation effectively reproduced the experimentally observed characteristics of static recrystallization. Consequently, nucleation took place in areas of the high dislocation density, and it was delayed in the areas where there was less stored energy. As recrystallization progresses, the matrix was gradually consumed by growing nuclei originating from highly recrystallized regions. Furthermore, no additional nucleation occurred within the previously recrystallized grains; rather, they undergo coarsening due to small misorientation differences with adjacent grains.
mapped onto the PFM grids. This study employed a regularized Wigner-Seitz cell approach, as described in the work of Min et al. [13].

<figure><img src="image_2070.png" /><figcaption>Fig. 4. Predicted rolling texture of ultra-low carbon steel, represented by φ₂=45° orientation distribution function section.</figcaption></figure>

The calculated rolling texture of ultra-low carbon steel are presented in Fig. 4, which depicts the φ₂=45° orientation distribution function (ODF) section after a thickness reduction ratio of 85.5% rolling (equivalently, the plane strain compression). The calculated cold-rolling texture of ultra-low carbon steel showed mixture of the major α-fiber (<110>//RD) and minor γ-fiber (<111>//ND), which was in agreement with experimentally measured texture components [16,17]. In particular, three main experimentally reported cold-rolling texture components, {665}<110> (approximated by {111}<110>), {558}<110> (approximated by {112}<110>), and {001}<110>, could be perfectly predicted from the proposed CPFEM.

Simulation of Recrystallized Ultra-low Carbon Steel. The process of static recrystallization, or annealing, was simulated using the PFM. This was performed by utilizing the state variables derived from the CPFEM results for the cold rolling simulation.

Fig. 5 illustrates the progression of static recrystallization over time during the annealing process, as predicted by the PFM. Grains were color-coded to represent deformed (gray) and recrystallized (white) states. The orientations of the newly nucleated grains were determined based on predictions from the GSERM model, which calculates orientation probability upon nucleation. To prevent the formation of idealistically oriented grains through theoretical calculations, random orientations were also assigned to nucleation orientation probability, maintaining a ratio of 7:3 between calculated and random orientations.

The recrystallized grains can be clearly distinguished from the remaining deformed grains due to their equiaxed morphology. The residual deformed grains were elongated along the RD in the form of fibers, which is a characteristic feature of metals subjected to cold rolling.

In Fig. 5, the original parent grain boundaries (elongated to the RD) were fixed. This was primarily due to the large difference in stored energy between the recrystallized grain boundaries and their corresponding parent grains This assumption was also beneficial in terms of reducing the computational cost.

As depicted in the figure, nucleation was initiated in regions with a higher level of stored energy (refer to Fig. 3 for the distribution of dislocation density after rolling simulation). The calculated results from the annealing simulation effectively reproduced the experimentally observed characteristics of static recrystallization. Consequently, nucleation took place in areas of the high dislocation density, and it was delayed in the areas where there was less stored energy. As recrystallization progresses, the matrix was gradually consumed by growing nuclei originating from highly recrystallized regions. Furthermore, no additional nucleation occurred within the previously recrystallized grains; rather, they undergo coarsening due to small misorientation differences with adjacent grains.
<figure><img src="image_2079.png" /><figcaption>Fig. 5. Predicted microstructure evolution during static recrystallization of ultra-low carbon steel.</figcaption></figure>

The recrystallization texture derived from the PFM simulation is depicted in Fig. 6, represented by the φ2=45° section of the ODF after annealing. As mentioned above, the orientations of newly nucleated grains were determined based on calculations from the GSERM model before they underwent grain growth. The calculated recrystallization texture displays the γ-fiber centered on the {111}<112> component, aligning with the texture reported in experimental studies for ultra-low carbon steel. During the annealing process, the minor rolling component {111}<112> was fairly stationary, whereas other texture components moved towards {111}<112> following a rotational relationship of ±54.7°<110>. In particular, the proposed model was capable of predicting the transition from the γ-fiber {111}<110> to the {111}<112> component.

Prediction of Anisotropy and Formability of Ultra-low Carbon Steel. Utilizing the microstructural information of annealed (or fully recrystallized) ultra-low carbon steel obtained from the proposed integrated crystal plasticity and phase field model, virtual mechanical tests was executed. The information included recrystallization texture, grain size, and its distribution. The RVE model for virtual experiments was generated using the same procedure as the cold rolling simulation, except that it utilized the predicted microstructural information of annealed ultra-low carbon steel. The RVE model consisted of 941 orientations and 27000 elements.

Based on the RVE models, constructed with the predicted microstructural information, uniaxial tensile tests along material orientations differing from the RD and balanced biaxial tensile test were virtually performed. For the CPFEM parameters of the recrystallized grains, the initial dislocation densities of the recrystallized grain were reduced to 18.2% of their value, a reduction derived by fitting the yield stress of annealed ultra-low carbon steel in the RD. And the average grain size was derived from the predicted value. However, all other hardening-related parameters remained identical to those of undeformed parent grains. Subsequently, the Lankford coefficients
<figure><img src="image_2071.png" /><figcaption>Fig. 5. Predicted microstructure evolution during static recrystallization of ultra-low carbon steel.</figcaption></figure>

The recrystallization texture derived from the PFM simulation is depicted in Fig. 6, represented by the φ2=45° section of the ODF after annealing. As mentioned above, the orientations of newly nucleated grains were determined based on calculations from the GSERM model before they underwent grain growth. The calculated recrystallization texture displays the γ-fiber centered on the {111}<112> component, aligning with the texture reported in experimental studies for ultra-low carbon steel. During the annealing process, the minor rolling component {111}<112> was fairly stationary, whereas other texture components moved towards {111}<112> following a rotational relationship of ±54.7°<110>. In particular, the proposed model was capable of predicting the transition from the γ-fiber {111}<110> to the {111}<112> component.

Prediction of Anisotropy and Formability of Ultra-low Carbon Steel. Utilizing the microstructural information of annealed (or fully recrystallized) ultra-low carbon steel obtained from the proposed integrated crystal plasticity and phase field model, virtual mechanical tests was executed. The information included recrystallization texture, grain size, and its distribution. The RVE model for virtual experiments was generated using the same procedure as the cold rolling simulation, except that it utilized the predicted microstructural information of annealed ultra-low carbon steel. The RVE model consisted of 941 orientations and 27000 elements.

Based on the RVE models, constructed with the predicted microstructural information, uniaxial tensile tests along material orientations differing from the RD and balanced biaxial tensile test were virtually performed. For the CPFEM parameters of the recrystallized grains, the initial dislocation densities of the recrystallized grain were reduced to 18.2% of their value, a reduction derived by fitting the yield stress of annealed ultra-low carbon steel in the RD. And the average grain size was derived from the predicted value. However, all other hardening-related parameters remained identical to those of undeformed parent grains. Subsequently, the Lankford coefficients
<figure><img src="image_2080.png" /><figcaption>Fig. 6. Predicted recrystallization texture of ultra-low carbon steel, represented by $\phi_2=45^\circ$ orientation distribution function section.</figcaption></figure>

(or r-values) and yield stresses in different material directions were predicted as measures for the plastic anisotropy.

The boundary conditions for the uniaxial tension and balanced biaxial tension were as follows: PBCs were applied to all simulations, and the prescribed displacement was controlled to maintain a constant strain rate of $10^{-3}$ s$^{-1}$, suitable for the quasi-static test conditions.

Fig. 7 depicts the predicted plastic anisotropy obtained from the proposed virtual mechanical tests, which includes the r-values and yield stresses (normalized by the yield stress at RD) along different angles form the RD. These predicted results were compared with experimental data for validation. The r-values were measured at an engineering strain of 20%, taking into account a uniform elongation of

approximately 25%. The results showed that the r-value at 45° from the RD was lower than those along the RD and TD, which is in line with experimental observations. Moreover, the yield stresses along different loading directions matched well with experimental values.

A more detailed investigation of plastic anisotropy was carried out, as presented in Fig. 8, using various anisotropic yield functions. For modeling plastic anisotropy of annealed ultra-low carbon steel, the Hill 1948 yield function (Hill'48) [16] and the Yld2000-2d non-quadratic yield function [17] were considered. For validation purposes, both measured and predicted plastic anisotropies, including r-values and yield stresses, were utilized to identify the anisotropic yield functions. Fig. 8(a) displays the variations of r-value and normalized yield stress based on experimental values, while Fig. 8(b) presents the corresponding variations based on predicted values obtained from virtual experiments.

<figure><img src="image_2081.png" /><figcaption>Fig. 7. Comparison of the measured and predicted plastic anisotropy in recrystallized ultra-low carbon steel, featuring the Lankford coefficients and normalized yield stresses along various material orientations.</figcaption></figure>
<figure><img src="image_2072.png" /><figcaption>Fig. 6. Predicted recrystallization texture of ultra-low carbon steel, represented by $\phi_2=45^\circ$ orientation distribution function section.</figcaption></figure>

(or r-values) and yield stresses in different material directions were predicted as measures for the plastic anisotropy.

The boundary conditions for the uniaxial tension and balanced biaxial tension were as follows: PBCs were applied to all simulations, and the prescribed displacement was controlled to maintain a constant strain rate of $10^{-3}$ s$^{-1}$, suitable for the quasi-static test conditions.

Fig. 7 depicts the predicted plastic anisotropy obtained from the proposed virtual mechanical tests, which includes the r-values and yield stresses (normalized by the yield stress at RD) along different angles form the RD. These predicted results were compared with experimental data for validation. The r-values were measured at an engineering strain of 20%, taking into account a uniform elongation of

approximately 25%. The results showed that the r-value at 45° from the RD was lower than those along the RD and TD, which is in line with experimental observations. Moreover, the yield stresses along different loading directions matched well with experimental values.

A more detailed investigation of plastic anisotropy was carried out, as presented in Fig. 8, using various anisotropic yield functions. For modeling plastic anisotropy of annealed ultra-low carbon steel, the Hill 1948 yield function (Hill'48) [16] and the Yld2000-2d non-quadratic yield function [17] were considered. For validation purposes, both measured and predicted plastic anisotropies, including r-values and yield stresses, were utilized to identify the anisotropic yield functions. Fig. 8(a) displays the variations of r-value and normalized yield stress based on experimental values, while Fig. 8(b) presents the corresponding variations based on predicted values obtained from virtual experiments.

<figure><img src="image_2073.png" /><figcaption>Fig. 7. Comparison of the measured and predicted plastic anisotropy in recrystallized ultra-low carbon steel, featuring the Lankford coefficients and normalized yield stresses along various material orientations.</figcaption></figure>
Sheet formability of annealed ultra-low carbon steel, represented by FLD, was evaluated. Based on the M-K model [5], FLDs were calculated using various anisotropic yield functions, as identified in Fig. 8. In addition to this yield function-based approach, a hybrid FLD prediction scheme [6], which combines the CPFEM and M-K model to account for the microstructure evolution at grain-scale and macro-scale, respectively, was utilized. Unlike the yield function-based approach, which uses fixed r-values as parameters, this approach was designed to consider the evolution of crystallographic texture during deformation, which means the evolution of r-values during deformation. Since drawability, one of the perspectives for evaluating sheet formability, is primarily characterized by sheet anisotropy, accurate predictions of anisotropic mechanical properties can be beneficial for the predictions of FLD.

<figure><img src="image_2082.png" /><figcaption>Fig. 8. Variations of r-value and normalized yield stress for annealed ultra-low carbon steel along various material orientations, obtained using various anisotropic yield functions based on (a) experimental and (b) predicted values.</figcaption></figure>

<figure><img src="image_2083.png" /><figcaption>Fig. 9. Comparison of forming limit diagrams for recrystallized ultra-low carbon steel, obtained using various yield functions and crystal plasticity-based approach (CPFLD), with experimental values.</figcaption></figure>

Fig. 9 displays the FLDs for annealed ultra-low carbon steel obtained using both yield function-based and crystal plasticity-based (referred to as CPFLD in Fig. 9) approaches. These predicted results were compared with experimental data, which was obtained from the well-known Nakajima-type ISO standard test [18]. The initial imperfection factor was determined to be 0.995 for both approaches through an iterative fitting process of the forming limit strains under the plane strain tension condition or $FLD_0$. In the yield function-based approach, a Swift-type strain hardening law was employed.

The predicted sheet anisotropy and formability confirmed that the proposed integrated crystal plasticity and phase field model can effectively reproduce the microstructural characteristics of
Sheet formability of annealed ultra-low carbon steel, represented by FLD, was evaluated. Based on the M-K model [5], FLDs were calculated using various anisotropic yield functions, as identified in Fig. 8. In addition to this yield function-based approach, a hybrid FLD prediction scheme [6], which combines the CPFEM and M-K model to account for the microstructure evolution at grain-scale and macro-scale, respectively, was utilized. Unlike the yield function-based approach, which uses fixed r-values as parameters, this approach was designed to consider the evolution of crystallographic texture during deformation, which means the evolution of r-values during deformation. Since drawability, one of the perspectives for evaluating sheet formability, is primarily characterized by sheet anisotropy, accurate predictions of anisotropic mechanical properties can be beneficial for the predictions of FLD.

<figure><img src="image_2074.png" /><figcaption>Fig. 8. Variations of r-value and normalized yield stress for annealed ultra-low carbon steel along various material orientations, obtained using various anisotropic yield functions based on (a) experimental and (b) predicted values.</figcaption></figure>

<figure><img src="image_2075.png" /><figcaption>Fig. 9. Comparison of forming limit diagrams for recrystallized ultra-low carbon steel, obtained using various yield functions and crystal plasticity-based approach (CPFLD), with experimental values.</figcaption></figure>

Fig. 9 displays the FLDs for annealed ultra-low carbon steel obtained using both yield function-based and crystal plasticity-based (referred to as CPFLD in Fig. 9) approaches. These predicted results were compared with experimental data, which was obtained from the well-known Nakajima-type ISO standard test [18]. The initial imperfection factor was determined to be 0.995 for both approaches through an iterative fitting process of the forming limit strains under the plane strain tension condition or $FLD_0$. In the yield function-based approach, a Swift-type strain hardening law was employed.

The predicted sheet anisotropy and formability confirmed that the proposed integrated crystal plasticity and phase field model can effectively reproduce the microstructural characteristics of
thermo-mechanically processed ultra-low carbon steel, while so taking into account the effect of microstructure evolution on mechanical properties. This reinforced the validity and accuracy of the proposed model as a robust tool for comprehending the intricate relationship between process, structure, property, and performance.

## Summary

In this work, the numerical reproduction of static recrystallization in ultra-low carbon steel was presented using the proposed integrated crystal plasticity and phase field model. The CPFEM successfully simulated microstructural heterogeneities induced by cold rolling, while the PFM was employed to simulate microstructure evolution during the annealing process. Subsequently, the effects of microstructure evolution on mechanical properties, such as sheet anisotropy and formability, were assessed through virtual mechanical tests. The accuracy and validity of the proposed model were confirmed by comparing its predicted results with experimental observations.

## Acknowledgements

This work was supported by the Fundamental Research Program of the Korea Institute of Materials Science (KIMS, PNK9800) and the National Research Council of Science and Technology (NST) grant by the Korea government (MSIT) (CRC23011-210).

## References

[1] M.E. McHenry, D.E. Laughlin, Magnetic Properties of Metals and Alloys, in: Physical Metallurgy, Elsevier, 2014, pp. 1881–2008. https://doi.org/10.1016/B978-0-444-53770-6.00019-8

[2] T. Urabe, High Formability Steel, in: Encyclopedia of Materials: Metals and Alloys, Elsevier, 2022, pp. 3–11. https://doi.org/10.1016/B978-0-12-819726-4.00131-9

[3] S.K. Paul, S. Mishra, On the formability of sheet steels, Bull. Mater. Sci. 19 (1996) 963–984. https://doi.org/10.1007/BF02744631

[4] K.S. Chan, Effects of plastic anisotropy and yield surface shape on sheet metal stretchability, Metall. Trans. A 16 (1985) 629–639. https://doi.org/10.1007/BF02814237

[5] Z. Marciniak, K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609–620. https://doi.org/10.1016/0020-7403(67)90066-5

[6] H.J. Bong, J. Lee, X. Hu, X. Sun, M.G. Lee, Predicting forming limit diagrams for magnesium alloys using crystal plasticity finite elements, Int. J. Plast. 126 (2020) 102630. https://doi.org/10.1016/j.ijplas.2019.11.009

[7] C.P. Kohar, J.L. Bassani, A. Brahme, W. Muhammad, R.K. Mishra, K. Inal, A new multi-scale framework to incorporate microstructure evolution in phenomenological plasticity: Theory, explicit finite element formulation, implementation and validation, Int. J. Plast. 117 (2019) 122–156. https://doi.org/10.1016/j.ijplas.2017.08.006

[8] I. Steinbach, F. Pezzolla, A generalized field method for multiphase transformations using interface fields, Phys. D 134 (1999) 385–393. https://doi.org/10.1016/S0167-2789(99)00129-3

[9] S.G. Kim, D.I. Kim, W.T. Kim, Y.B. Park, Computer simulations of two-dimensional and three-dimensional ideal grain growth, Phys. Rev. E 74 (2006) 061605. https://doi.org/10.1103/PhysRevE.74.061605

[10] D.N. Lee, The evolution of recrystallization textures from deformation textures, Scr. Metall. Mater. 32 (1995) 1689–1694. https://doi.org/10.1016/0956-716X(95)00256-U
thermo-mechanically processed ultra-low carbon steel, while so taking into account the effect of microstructure evolution on mechanical properties. This reinforced the validity and accuracy of the proposed model as a robust tool for comprehending the intricate relationship between process, structure, property, and performance.

## Summary

In this work, the numerical reproduction of static recrystallization in ultra-low carbon steel was presented using the proposed integrated crystal plasticity and phase field model. The CPFEM successfully simulated microstructural heterogeneities induced by cold rolling, while the PFM was employed to simulate microstructure evolution during the annealing process. Subsequently, the effects of microstructure evolution on mechanical properties, such as sheet anisotropy and formability, were assessed through virtual mechanical tests. The accuracy and validity of the proposed model were confirmed by comparing its predicted results with experimental observations.

## Acknowledgements

This work was supported by the Fundamental Research Program of the Korea Institute of Materials Science (KIMS, PNK9800) and the National Research Council of Science and Technology (NST) grant by the Korea government (MSIT) (CRC23011-210).

## References

[1] M.E. McHenry, D.E. Laughlin, Magnetic Properties of Metals and Alloys, in: Physical Metallurgy, Elsevier, 2014, pp. 1881–2008. https://doi.org/10.1016/B978-0-444-53770-6.00019-8

[2] T. Urabe, High Formability Steel, in: Encyclopedia of Materials: Metals and Alloys, Elsevier, 2022, pp. 3–11. https://doi.org/10.1016/B978-0-12-819726-4.00131-9

[3] S.K. Paul, S. Mishra, On the formability of sheet steels, Bull. Mater. Sci. 19 (1996) 963–984. https://doi.org/10.1007/BF02744631

[4] K.S. Chan, Effects of plastic anisotropy and yield surface shape on sheet metal stretchability, Metall. Trans. A 16 (1985) 629–639. https://doi.org/10.1007/BF02814237

[5] Z. Marciniak, K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609–620. https://doi.org/10.1016/0020-7403(67)90066-5

[6] H.J. Bong, J. Lee, X. Hu, X. Sun, M.G. Lee, Predicting forming limit diagrams for magnesium alloys using crystal plasticity finite elements, Int. J. Plast. 126 (2020) 102630. https://doi.org/10.1016/j.ijplas.2019.11.009

[7] C.P. Kohar, J.L. Bassani, A. Brahme, W. Muhammad, R.K. Mishra, K. Inal, A new multi-scale framework to incorporate microstructure evolution in phenomenological plasticity: Theory, explicit finite element formulation, implementation and validation, Int. J. Plast. 117 (2019) 122–156. https://doi.org/10.1016/j.ijplas.2017.08.006

[8] I. Steinbach, F. Pezzolla, A generalized field method for multiphase transformations using interface fields, Phys. D 134 (1999) 385–393. https://doi.org/10.1016/S0167-2789(99)00129-3

[9] S.G. Kim, D.I. Kim, W.T. Kim, Y.B. Park, Computer simulations of two-dimensional and three-dimensional ideal grain growth, Phys. Rev. E 74 (2006) 061605. https://doi.org/10.1103/PhysRevE.74.061605

[10] D.N. Lee, The evolution of recrystallization textures from deformation textures, Scr. Metall. Mater. 32 (1995) 1689–1694. https://doi.org/10.1016/0956-716X(95)00256-U
[11] D.N. Lee, Strain energy release maximization model for evolution of recrystallization textures, Int. J. Mech. Sci. 42 (2000) 1645-1678. doi.org/10.1016/S0020-7403(99)00095-8

[12] D.N. Lee, Texture and related phenomena, second ed., The Korean Institute of Metals and Materials, Seoul, 2014.

[13] K.M. Min, W. Jeong, S.H. Hong, C.A. Lee, P.-R. Cha, H.N. Han, M.-G. Lee, Integrated crystal plasticity and phase field model for prediction of recrystallization texture and anisotropic mechanical properties of cold-rolled ultra-low carbon steels, Int. J. Plast. 127 (2020) 102644. https://doi.org/10.1016/j.ijplas.2019.102644

[14] F.J. Humphreys, M. Hatherly, Recrystallization and Related Annealing Phenomena, second ed., Elsevier, 2004. https://doi.org/10.1016/B978-0-08-044164-1.X5000-2

[15] H. Zurob, Y. Brechet, J. Dunlop, Quantitative criterion for recrystallization nucleation in single-phase alloys: Prediction of critical strains and incubation times, Acta Mater. 54 (2006) 3983-3990. https://doi.org/10.1016/j.actamat.2006.04.028

[16] H. Rodney, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A Math. Phys. Sci. 193 (1948) 281-297. https://doi.org/10.1098/rspa.1948.0045

[17] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheets—part 1: theory, Int. J. Plast. 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[18] Metallic materials-Sheet and strip-Determination of forming-limit curves-Part 2: Determination of forming-limit curves in the laboratory, International Organization for Standardization, 2008.
[11] D.N. Lee, Strain energy release maximization model for evolution of recrystallization textures, Int. J. Mech. Sci. 42 (2000) 1645-1678. doi.org/10.1016/S0020-7403(99)00095-8

[12] D.N. Lee, Texture and related phenomena, second ed., The Korean Institute of Metals and Materials, Seoul, 2014.

[13] K.M. Min, W. Jeong, S.H. Hong, C.A. Lee, P.-R. Cha, H.N. Han, M.-G. Lee, Integrated crystal plasticity and phase field model for prediction of recrystallization texture and anisotropic mechanical properties of cold-rolled ultra-low carbon steels, Int. J. Plast. 127 (2020) 102644. https://doi.org/10.1016/j.ijplas.2019.102644

[14] F.J. Humphreys, M. Hatherly, Recrystallization and Related Annealing Phenomena, second ed., Elsevier, 2004. https://doi.org/10.1016/B978-0-08-044164-1.X5000-2

[15] H. Zurob, Y. Brechet, J. Dunlop, Quantitative criterion for recrystallization nucleation in single-phase alloys: Prediction of critical strains and incubation times, Acta Mater. 54 (2006) 3983-3990. https://doi.org/10.1016/j.actamat.2006.04.028

[16] H. Rodney, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A Math. Phys. Sci. 193 (1948) 281-297. https://doi.org/10.1098/rspa.1948.0045

[17] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheets—part 1: theory, Int. J. Plast. 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[18] Metallic materials-Sheet and strip-Determination of forming-limit curves-Part 2: Determination of forming-limit curves in the laboratory, International Organization for Standardization, 2008.
# A robust identification protocol of flow curve adjusting parameters using uniaxial tensile curve

LEMOINE Xavier¹, a*, MUNIER Rémi¹, b and BELLUT Xavier², c

¹ Global R&D ArcelorMittal Maizières, voie Romaine, BP30320, F-57283 Maizières-lès-Metz, France

² Global R&D ArcelorMittal Montataire, 1 Route de Saint-Leu, F-60160 Montataire, France

a xavier.lemoine@arcelormittal.com, b remi.munier@arcelormittal.com,

c xavier.bellut@arcelormittal.com

**Keywords:** Large Strain, Steel, Hardening Curve, Formability

**Abstract.** ArcelorMittal is constantly developing new steel grades to enable the automotive industry to offer safer, lighter, and more environmentally friendly vehicles. These new grades include advanced high-strength steels (AHSS) and Ultra High Strength steels (UHSS) having for some of them lower uniform elongation (UE) than conventional drawing steels. This particularity needs to be considered for an accurate formability prediction in sheet forming numerical simulations. One of these difficulties is the effect of the relatively low uniform elongation on the identification of the parameters of the isotropic hardening model. Various experimental tests can be used to reach the large plastic deformation (hydraulic bulge test, stack compression test, shear test, torsion test or plane strain compression test). The identification protocol of ArcelorMittal for hardening models is based solely on stress-strain curves determined in uniaxial tension. The Exp_S hardening law (TU experimental values before UE%, Swift extension above) was validated by comparison with the stress-strain curves obtained from measurements of experimental tests reaching large strains.

**Introduction**

The new advanced high-strength steels (AHSS) and Ultra High Strength steels (UHSS) for the automotive industry to improve weight reduction and passive safety result in lower uniform elongation than conventional drawing steels. In general, strain hardening laws are identified from the stress-plastic strain curve obtained from the uniaxial tensile test. Five measurements points on a B-pillar stamping part for seven steels with uniform elongation ranging from 2% to 14% (Figure 1) exceed the strain of the uniform elongation, sometimes to more than 12 times this strain where β is:

$$ \beta = \frac{\text{Measured strain}}{\text{Strain of Uniform Elongation}} \qquad (1) $$

<figure><img src="image_2084.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.
# A robust identification protocol of flow curve adjusting parameters using uniaxial tensile curve

LEMOINE Xavier¹, a*, MUNIER Rémi¹, b and BELLUT Xavier², c

¹ Global R&D ArcelorMittal Maizières, voie Romaine, BP30320, F-57283 Maizières-lès-Metz, France

² Global R&D ArcelorMittal Montataire, 1 Route de Saint-Leu, F-60160 Montataire, France

a xavier.lemoine@arcelormittal.com, b remi.munier@arcelormittal.com,

c xavier.bellut@arcelormittal.com

**Keywords:** Large Strain, Steel, Hardening Curve, Formability

**Abstract.** ArcelorMittal is constantly developing new steel grades to enable the automotive industry to offer safer, lighter, and more environmentally friendly vehicles. These new grades include advanced high-strength steels (AHSS) and Ultra High Strength steels (UHSS) having for some of them lower uniform elongation (UE) than conventional drawing steels. This particularity needs to be considered for an accurate formability prediction in sheet forming numerical simulations. One of these difficulties is the effect of the relatively low uniform elongation on the identification of the parameters of the isotropic hardening model. Various experimental tests can be used to reach the large plastic deformation (hydraulic bulge test, stack compression test, shear test, torsion test or plane strain compression test). The identification protocol of ArcelorMittal for hardening models is based solely on stress-strain curves determined in uniaxial tension. The Exp_S hardening law (TU experimental values before UE%, Swift extension above) was validated by comparison with the stress-strain curves obtained from measurements of experimental tests reaching large strains.

**Introduction**

The new advanced high-strength steels (AHSS) and Ultra High Strength steels (UHSS) for the automotive industry to improve weight reduction and passive safety result in lower uniform elongation than conventional drawing steels. In general, strain hardening laws are identified from the stress-plastic strain curve obtained from the uniaxial tensile test. Five measurements points on a B-pillar stamping part for seven steels with uniform elongation ranging from 2% to 14% (Figure 1) exceed the strain of the uniform elongation, sometimes to more than 12 times this strain where β is:

$$ \beta = \frac{\text{Measured strain}}{\text{Strain of Uniform Elongation}} \qquad (1) $$

<figure><img src="image_2076.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
<figure><img src="image_2085.png" /><figcaption>Figure 1 – Five measurements points on a B-pillar stamping part for seven steels with uniform elongation ranging from 2% to 14% exceed the strain of the uniform elongation.</figcaption></figure>

There are 2 possible ways for a hardening model to correctly represent the stress-strain curve in large deformation [1], using tests to reach such strain levels:

* First way: to identify isotropic hardening parameters with the true stress-plastic strain curve obtained from these experimental tests allowing to reach large deformations [2].

* Second way: to identify isotropic hardening parameters with uniaxial tensile test and to choose the better prediction after uniform elongation with another rheological tests enabling to reach larger deformations [3].

## Different large plastic strain experimental tests

Different experimental tests (cf. figure 2) allow to reach large plastic strain levels. Figure 3 shows the β-values (eq. 1) achieved for these various experimental tests on different steel grades (Uniform elongation between 3% to 27% with thickness between 0.5 to 3 mm). Each point corresponds to a test, the majority of which comes from a publication [3 to 13]. Each of these tests has its advantages, disadvantages, and limitations (cf. table 1):

a) Uniaxial Tensile Test with local measure (UTT + DIC) [1, 4]: Based on a standard uniaxial tensile test coupled with digital image correlation for strain measurement, it is possible to decrease the measured length of 80 mm to 1 mm or less. As long as the deformation is constant over the width of the specimen, the local stress remains uniform and can be deduced from the tensile stress. Strains range between 0.2 and 0.4 can be reached depending on the material.

b) Shear Test (ST) [1]: The planar simple shear test device consists of two rigid bodies subjected to a parallel movement. With a high-resolution optical technique, the shear strain γ is deduced from the change in slope of an initially straight line or with digital image correlation. The shear stress is deduced by the shearing force F divided by the length l and by the sheet thickness t. The comparison of the shear flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Depending on sheet thickness and material, sometimes the test is limited by fracture or buckling. Strains range between 0.1 and 0.5 can be reached.

c) Stack Compression Test (SCT) [1, 4]: The test consists of stacking a number of sheet specimens and compressing the stack using platens. The procedure adopted by An and
<figure><img src="image_2077.png" /><figcaption>Figure 1 – Five measurements points on a B-pillar stamping part for seven steels with uniform elongation ranging from 2% to 14% exceed the strain of the uniform elongation.</figcaption></figure>

There are 2 possible ways for a hardening model to correctly represent the stress-strain curve in large deformation [1], using tests to reach such strain levels:

* First way: to identify isotropic hardening parameters with the true stress-plastic strain curve obtained from these experimental tests allowing to reach large deformations [2].

* Second way: to identify isotropic hardening parameters with uniaxial tensile test and to choose the better prediction after uniform elongation with another rheological tests enabling to reach larger deformations [3].

## Different large plastic strain experimental tests

Different experimental tests (cf. figure 2) allow to reach large plastic strain levels. Figure 3 shows the β-values (eq. 1) achieved for these various experimental tests on different steel grades (Uniform elongation between 3% to 27% with thickness between 0.5 to 3 mm). Each point corresponds to a test, the majority of which comes from a publication [3 to 13]. Each of these tests has its advantages, disadvantages, and limitations (cf. table 1):

a) Uniaxial Tensile Test with local measure (UTT + DIC) [1, 4]: Based on a standard uniaxial tensile test coupled with digital image correlation for strain measurement, it is possible to decrease the measured length of 80 mm to 1 mm or less. As long as the deformation is constant over the width of the specimen, the local stress remains uniform and can be deduced from the tensile stress. Strains range between 0.2 and 0.4 can be reached depending on the material.

b) Shear Test (ST) [1]: The planar simple shear test device consists of two rigid bodies subjected to a parallel movement. With a high-resolution optical technique, the shear strain γ is deduced from the change in slope of an initially straight line or with digital image correlation. The shear stress is deduced by the shearing force F divided by the length l and by the sheet thickness t. The comparison of the shear flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Depending on sheet thickness and material, sometimes the test is limited by fracture or buckling. Strains range between 0.1 and 0.5 can be reached.

c) Stack Compression Test (SCT) [1, 4]: The test consists of stacking a number of sheet specimens and compressing the stack using platens. The procedure adopted by An and
Vegter [14] reduces friction between each platen. The comparison of the stacked compression flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Strains range between 0.2 and 0.3 can be reached before "barreling" effect. Using two video systems to characterize the anisotropy of a stacked compression test, M. Merklein [15] obtains a value of 0.5 despite the "barreling" effect.

d) Hydraulic Bulge Test (HBT) [1, 4]: This test deforms sheet specimen under action of hydraulic pressure through circular die. The state of stress can be determined analytically according to membrane theory using pressure, radius of curvature and thickness at the top of the dome. The bi-axial strains are measured with digital image correlation. The comparison of the equi-biaxial flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Sometimes the test is limited by fracture or maximal pressure. Strains range between 0.5 and 0.7 can be reached.

e) In-plane Torsion Test (ITT) [4]: The in-plane torsion test is a shear test for sheet metals. A circular specimen is clamped at the center as well as on the outer rim. By imposing a different rotational movement between outer and inner clamping, the annular free area between the clamping devices is loaded under simple shear in the sheet plane. The shear stress can be deduced from the measured external torque. Shear strain can be calculated from the change in slope of an initially straight line or with digital image correlation. The comparison of the shear flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. According to the geometry of specimen groove, strains reach more than 0.5. Sometimes the test is limited by wrinkling instability or fracture.

f) Plane Strain Compression Test (PSCT) [4]: A rectangular sample of the sheet is compressed through the sheet thickness between two dies. The stress and strain fields determined analytically are based on the load and die displacement measurements, taking into account the friction influence. The comparison of the plane strain compression flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Strains reach more than 0.3. Sometimes the test is limited by fracture.

g) Uniaxial Tensile Test on pre-rolled flat (UT on pre-rolled) [1]: Sheets (cold rolled and annealed) of several material types are rolled on an experimental mill several times up to specified thicknesses. Afterwards standard tensile tests are carried out on these pre-rolled specimens. The stress strain response of pre-strained test samples is obtained. The amount of pre-strain is assumed by the thickness reduction of the rolled samples - the deformation induced by rolling is not uniaxial compression in the thickness direction, but it is assumed to be a compressive deformation in plane strain with the plastic strain being zero in the width direction of the samples. Strains reach more than 0.7.
Vegter [14] reduces friction between each platen. The comparison of the stacked compression flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Strains range between 0.2 and 0.3 can be reached before "barreling" effect. Using two video systems to characterize the anisotropy of a stacked compression test, M. Merklein [15] obtains a value of 0.5 despite the "barreling" effect.

d) Hydraulic Bulge Test (HBT) [1, 4]: This test deforms sheet specimen under action of hydraulic pressure through circular die. The state of stress can be determined analytically according to membrane theory using pressure, radius of curvature and thickness at the top of the dome. The bi-axial strains are measured with digital image correlation. The comparison of the equi-biaxial flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Sometimes the test is limited by fracture or maximal pressure. Strains range between 0.5 and 0.7 can be reached.

e) In-plane Torsion Test (ITT) [4]: The in-plane torsion test is a shear test for sheet metals. A circular specimen is clamped at the center as well as on the outer rim. By imposing a different rotational movement between outer and inner clamping, the annular free area between the clamping devices is loaded under simple shear in the sheet plane. The shear stress can be deduced from the measured external torque. Shear strain can be calculated from the change in slope of an initially straight line or with digital image correlation. The comparison of the shear flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. According to the geometry of specimen groove, strains reach more than 0.5. Sometimes the test is limited by wrinkling instability or fracture.

f) Plane Strain Compression Test (PSCT) [4]: A rectangular sample of the sheet is compressed through the sheet thickness between two dies. The stress and strain fields determined analytically are based on the load and die displacement measurements, taking into account the friction influence. The comparison of the plane strain compression flow curve with the tensile flow curve needs to generate an equivalent stress and equivalent strain. Strains reach more than 0.3. Sometimes the test is limited by fracture.

g) Uniaxial Tensile Test on pre-rolled flat (UT on pre-rolled) [1]: Sheets (cold rolled and annealed) of several material types are rolled on an experimental mill several times up to specified thicknesses. Afterwards standard tensile tests are carried out on these pre-rolled specimens. The stress strain response of pre-strained test samples is obtained. The amount of pre-strain is assumed by the thickness reduction of the rolled samples - the deformation induced by rolling is not uniaxial compression in the thickness direction, but it is assumed to be a compressive deformation in plane strain with the plastic strain being zero in the width direction of the samples. Strains reach more than 0.7.
<figure><img src="image_2086.png" /><figcaption>Figure 2 – Different experimental tests allowing to reach large plastic strain levels: a) Uniaxial Tensile Test with Local Measure [1], b) Shear Test [1], c) Stacked Compression Test [1], d) Bulge Test [1], e) In-plane Torsion Test [4], f) Plane Strain Compression Test [4] and g) Uniaxial Tensile Test on pre-rolled Flat [1].</figcaption></figure>

<figure><img src="image_2087.png" /><figcaption>Figure 3 – β-values (eq. 1) achieved for various experimental tests on different steel grades (Uniform elongation between 3% to 27% with thickness between 0.5 to 3 mm).</figcaption></figure>
<figure><img src="image_2078.png" /><figcaption>Figure 2 – Different experimental tests allowing to reach large plastic strain levels: a) Uniaxial Tensile Test with Local Measure [1], b) Shear Test [1], c) Stacked Compression Test [1], d) Bulge Test [1], e) In-plane Torsion Test [4], f) Plane Strain Compression Test [4] and g) Uniaxial Tensile Test on pre-rolled Flat [1].</figcaption></figure>

<figure><img src="image_2079.png" /><figcaption>Figure 3 – β-values (eq. 1) achieved for various experimental tests on different steel grades (Uniform elongation between 3% to 27% with thickness between 0.5 to 3 mm).</figcaption></figure>
<table><caption>Table 1 – Different Experimental Tests regarding Standard Uniaxial Tensile Test (Standard UTT)</caption><thead><tr><th>Experimental tests</th><th>β (eq. 1 and figure 3)</th><th>Sheet Surface Used Ratio</th><th>Complexity of Specimen Geometry</th><th>Test Duration Ratio</th></tr></thead><tbody><tr><td>Standard UTT</td><td>1</td><td>1</td><td>-</td><td>1</td></tr><tr><td>UTT + DIC</td><td>1.5 to 2.5</td><td>1</td><td>Identic</td><td>1.5</td></tr><tr><td>ST</td><td>1.5 to 5</td><td>0.75</td><td>Simpler</td><td>1.5</td></tr><tr><td>SCT</td><td>2 to 4</td><td>0.5</td><td>Simpler</td><td>1</td></tr><tr><td>HBT</td><td>More 2.5</td><td>15</td><td>Simpler</td><td>3</td></tr><tr><td>ITT</td><td>More 4.5</td><td>1.5</td><td>More complex</td><td>1.5</td></tr><tr><td>PSCT</td><td>More 3</td><td>0.5</td><td>Simpler</td><td>1</td></tr><tr><td>UT on pre-rolled (10 pts)</td><td>More 3</td><td>15</td><td>Identic</td><td>10 + tooling</td></tr></tbody></table>

Flow curves comparison for different experimental tests

<figure><img src="image_2088.png" /><figcaption>Figure 4 – True stress-plastic strain curves comparison for different steel grades and experimental tests</figcaption></figure>

Figure 4 shows the true stress-plastic strain curves comparison for 2 different steel grades. As noted, beforehand, for several tests, the comparison of the flow curve with tensile flow curve needs to generate an equivalent stress-plastic strain curve. In this study, the generation of the equivalent stress-plasticity curve is based on a scalar multiplier parameter $\chi$ (Table 2) derived from the associated plasticity theory where the equivalent stress and equivalent strain are determined with plastic work per volume and the normality rule:

$$dW = d\epsilon_{eq}^p \sigma_{eq} = d\epsilon_{ij}^p \sigma_{ij} \quad \text{with} \quad d\epsilon_{ij}^p = \lambda \frac{\partial \sigma_{eq}}{\partial \sigma_{ij}} = \lambda V_{ij} \quad (2)$$
<table><caption>Table 1 – Different Experimental Tests regarding Standard Uniaxial Tensile Test (Standard UTT)</caption><thead><tr><th>Experimental tests</th><th>β (eq. 1 and figure 3)</th><th>Sheet Surface Used Ratio</th><th>Complexity of Specimen Geometry</th><th>Test Duration Ratio</th></tr></thead><tbody><tr><td>Standard UTT</td><td>1</td><td>1</td><td>-</td><td>1</td></tr><tr><td>UTT + DIC</td><td>1.5 to 2.5</td><td>1</td><td>Identic</td><td>1.5</td></tr><tr><td>ST</td><td>1.5 to 5</td><td>0.75</td><td>Simpler</td><td>1.5</td></tr><tr><td>SCT</td><td>2 to 4</td><td>0.5</td><td>Simpler</td><td>1</td></tr><tr><td>HBT</td><td>More 2.5</td><td>15</td><td>Simpler</td><td>3</td></tr><tr><td>ITT</td><td>More 4.5</td><td>1.5</td><td>More complex</td><td>1.5</td></tr><tr><td>PSCT</td><td>More 3</td><td>0.5</td><td>Simpler</td><td>1</td></tr><tr><td>UT on pre-rolled (10 pts)</td><td>More 3</td><td>15</td><td>Identic</td><td>10 + tooling</td></tr></tbody></table>

Flow curves comparison for different experimental tests

<figure><img src="image_2080.png" /><figcaption>Figure 4 – True stress-plastic strain curves comparison for different steel grades and experimental tests</figcaption></figure>

Figure 4 shows the true stress-plastic strain curves comparison for 2 different steel grades. As noted, beforehand, for several tests, the comparison of the flow curve with tensile flow curve needs to generate an equivalent stress-plastic strain curve. In this study, the generation of the equivalent stress-plasticity curve is based on a scalar multiplier parameter $\chi$ (Table 2) derived from the associated plasticity theory where the equivalent stress and equivalent strain are determined with plastic work per volume and the normality rule:

$$dW = d\epsilon_{eq}^p \sigma_{eq} = d\epsilon_{ij}^p \sigma_{ij} \quad \text{with} \quad d\epsilon_{ij}^p = \lambda \frac{\partial \sigma_{eq}}{\partial \sigma_{ij}} = \lambda V_{ij} \quad (2)$$
The $\chi$-parameter value is not imposed with yield criterion, but it is defined by an inverse method to coincide the loading path flow curve with the uniaxial tensile flow curve like the HBT standardized procedure given by ISO 16808. For Von Mises material, the uniaxial tensile, the shear, the biaxial and plane strain compression $\chi$-parameters are respectively equal to $\chi^{UT} = 1$, $\chi^{SH} = \sqrt{3}$, $\chi^{EEB} = 1$ and $\chi^{PSC} = -\sqrt{3}/2$.

<table><caption>Table 2 – Different Experimental Tests regarding Standard Uniaxial Tensile Test (Standard UTT)</caption><thead><tr><th>Experimental tests</th><th>ε<sub>eq</sub><sup>p</sup></th><th>σ<sub>eq</sub></th></tr></thead><tbody><tr><td>Standard UTT, UTT + DIC, UT on pre-rolled</td><td>ε<sub>11</sub><sup>p</sup>/χ<sup>UT</sup></td><td>χ<sup>UT</sup>σ<sub>11</sub></td></tr><tr><td>ST, ITT</td><td>2ε<sub>12</sub><sup>p</sup>/χ<sup>SH</sup></td><td>χ<sup>SH</sup>σ<sub>12</sub></td></tr><tr><td>SCT, HBT</td><td>ε<sub>11</sub><sup>p</sup>/χ<sup>EEB</sup></td><td>χ<sup>EEB</sup>σ<sub>11</sub></td></tr><tr><td>PSCT</td><td>ε<sub>11</sub><sup>p</sup>/χ<sup>PSC</sup></td><td>χ<sup>PSC</sup>σ<sub>11</sub></td></tr></tbody></table>

## Isotropic hardening laws

Six isotropic hardening laws (Swift, Voce, SV, Exp_S, Exp_V and Exp_SV) are studied here based on three mathematical equations (Table 3). The isotropic hardening laws Exp_SV, Exp_S, Exp_V consist of considering the experimental points before the uniform elongation (UE) followed by the predictions of the mathematical equation at larger strain levels (Figure 5). The selection of experimental points before UE allows the Lüders plateau to be taken into account when it exists. For the selection of experimental points, Autoform® software imposes that the stress to be continuously increasing. To identify the adjusting parameters, 3 different constraints are satisfied (Table 4):

* Constraint 1: the best approximation between the Yield Strength (YS) (or after Lüders plateau) and Ultimate Tensile Strength (UTS) points on the experimental flow curve issue of uniaxial tensile test (true stress-plastic strain curve).

* Constraint 2: the law passes through YS point (or low yield strength (LYS) if Lüders plateau).

* Constraint 3: Considère's criterion. At the point (UTS-UE), the Considère's criterion imposes that the slope of the true stress/true plastic strain curve is equal to the true stress at this point.

<table><caption>Table 3 – Different isotropic hardening laws</caption><thead><tr><th>Name</th><th>Equation</th><th>Material Parameters</th></tr></thead><tbody><tr><td>Swift (S)</td><td>σ<sub>Swift</sub> = K(ε<sub>0</sub> + ε)<sup>n</sup> (3)</td><td>K; n; ε<sub>0</sub></td></tr><tr><td>Voce (V)</td><td>σ<sub>Voce</sub> = σ<sub>0</sub> + R<sub>Sat</sub>(1 - exp(-C<sub>r</sub>ε)) (4)</td><td>σ<sub>0</sub>; R<sub>Sat</sub>; C<sub>r</sub></td></tr><tr><td>Swift and Voce (SV)</td><td>σ<sub>SV</sub> = (1 - α)σ<sub>Swift</sub> + ασ<sub>Voce</sub> (5)</td><td>α; K; n; ε<sub>0</sub>; σ<sub>0</sub>; R<sub>Sat</sub>; C<sub>r</sub></td></tr></tbody></table>
The $\chi$-parameter value is not imposed with yield criterion, but it is defined by an inverse method to coincide the loading path flow curve with the uniaxial tensile flow curve like the HBT standardized procedure given by ISO 16808. For Von Mises material, the uniaxial tensile, the shear, the biaxial and plane strain compression $\chi$-parameters are respectively equal to $\chi^{UT} = 1$, $\chi^{SH} = \sqrt{3}$, $\chi^{EEB} = 1$ and $\chi^{PSC} = -\sqrt{3}/2$.

<table><caption>Table 2 – Different Experimental Tests regarding Standard Uniaxial Tensile Test (Standard UTT)</caption><thead><tr><th>Experimental tests</th><th>ε<sub>eq</sub><sup>p</sup></th><th>σ<sub>eq</sub></th></tr></thead><tbody><tr><td>Standard UTT, UTT + DIC, UT on pre-rolled</td><td>ε<sub>11</sub><sup>p</sup>/χ<sup>UT</sup></td><td>χ<sup>UT</sup>σ<sub>11</sub></td></tr><tr><td>ST, ITT</td><td>2ε<sub>12</sub><sup>p</sup>/χ<sup>SH</sup></td><td>χ<sup>SH</sup>σ<sub>12</sub></td></tr><tr><td>SCT, HBT</td><td>ε<sub>11</sub><sup>p</sup>/χ<sup>EEB</sup></td><td>χ<sup>EEB</sup>σ<sub>11</sub></td></tr><tr><td>PSCT</td><td>ε<sub>11</sub><sup>p</sup>/χ<sup>PSC</sup></td><td>χ<sup>PSC</sup>σ<sub>11</sub></td></tr></tbody></table>

## Isotropic hardening laws

Six isotropic hardening laws (Swift, Voce, SV, Exp_S, Exp_V and Exp_SV) are studied here based on three mathematical equations (Table 3). The isotropic hardening laws Exp_SV, Exp_S, Exp_V consist of considering the experimental points before the uniform elongation (UE) followed by the predictions of the mathematical equation at larger strain levels (Figure 5). The selection of experimental points before UE allows the Lüders plateau to be taken into account when it exists. For the selection of experimental points, Autoform® software imposes that the stress to be continuously increasing. To identify the adjusting parameters, 3 different constraints are satisfied (Table 4):

* Constraint 1: the best approximation between the Yield Strength (YS) (or after Lüders plateau) and Ultimate Tensile Strength (UTS) points on the experimental flow curve issue of uniaxial tensile test (true stress-plastic strain curve).

* Constraint 2: the law passes through YS point (or low yield strength (LYS) if Lüders plateau).

* Constraint 3: Considère's criterion. At the point (UTS-UE), the Considère's criterion imposes that the slope of the true stress/true plastic strain curve is equal to the true stress at this point.

<table><caption>Table 3 – Different isotropic hardening laws</caption><thead><tr><th>Name</th><th>Equation</th><th>Material Parameters</th></tr></thead><tbody><tr><td>Swift (S)</td><td>σ<sub>Swift</sub> = K(ε<sub>0</sub> + ε)<sup>n</sup> (3)</td><td>K; n; ε<sub>0</sub></td></tr><tr><td>Voce (V)</td><td>σ<sub>Voce</sub> = σ<sub>0</sub> + R<sub>Sat</sub>(1 - exp(-C<sub>r</sub>ε)) (4)</td><td>σ<sub>0</sub>; R<sub>Sat</sub>; C<sub>r</sub></td></tr><tr><td>Swift and Voce (SV)</td><td>σ<sub>SV</sub> = (1 - α)σ<sub>Swift</sub> + ασ<sub>Voce</sub> (5)</td><td>α; K; n; ε<sub>0</sub>; σ<sub>0</sub>; R<sub>Sat</sub>; C<sub>r</sub></td></tr></tbody></table>
<figure><img src="image_2089.png" /><figcaption>Figure 5 – The isotropic hardening laws Exp_SV, Exp_S, Exp_V consist of experimental points before UE and the mathematical equation after UE.</figcaption></figure>

<table><caption>Table 4 – Constraints satisfied according to isotropic hardening laws.</caption><thead><tr><td>Name</td><td>Constraint 1</td><td>Constraint 2</td><td>Constraint 3</td></tr></thead><tbody><tr><td>Swift</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>Voce</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>SV</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Exp_S</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Exp_V</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Exp_SV</td><td>Yes</td><td>Yes</td><td>Yes</td></tr></tbody></table>

### Selection of the best isotropic hardening law.

For several comparison cases with experimental tests reaching large deformations, Figure 6 shows the ranking of these six isotropic hardening laws from first to sixth place. The ranking was based on the squared sum of stress deviations (Figure 7) between the mathematical law and experimental test measurements for several strain values after the uniform elongation point. The first place is the curve closest to the experimental curve (minimal squared sum) and the sixth place the furthest away (maximal squared sum) (Figure 8). According to our identification protocols, the best isotropic hardening law is Exp_S.
<figure><img src="image_2081.png" /><figcaption>Figure 5 – The isotropic hardening laws Exp_SV, Exp_S, Exp_V consist of experimental points before UE and the mathematical equation after UE.</figcaption></figure>

<table><caption>Table 4 – Constraints satisfied according to isotropic hardening laws.</caption><thead><tr><td>Name</td><td>Constraint 1</td><td>Constraint 2</td><td>Constraint 3</td></tr></thead><tbody><tr><td>Swift</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>Voce</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>SV</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Exp_S</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Exp_V</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Exp_SV</td><td>Yes</td><td>Yes</td><td>Yes</td></tr></tbody></table>

### Selection of the best isotropic hardening law.

For several comparison cases with experimental tests reaching large deformations, Figure 6 shows the ranking of these six isotropic hardening laws from first to sixth place. The ranking was based on the squared sum of stress deviations (Figure 7) between the mathematical law and experimental test measurements for several strain values after the uniform elongation point. The first place is the curve closest to the experimental curve (minimal squared sum) and the sixth place the furthest away (maximal squared sum) (Figure 8). According to our identification protocols, the best isotropic hardening law is Exp_S.
<figure><img src="image_2090.png" /><figcaption>Figure 6 – Out of 32 cases of comparison, number of times the isotropic hardening law is in first, second ... or sixth place.</figcaption></figure>

<figure><img src="image_2091.png" /><figcaption>Figure 7 – For DP450 steel, Examples of stress deviations between the mathematical law and experimental test measurements for several strain values after the uniform elongation point.</figcaption></figure>
<figure><img src="image_2082.png" /><figcaption>Figure 6 – Out of 32 cases of comparison, number of times the isotropic hardening law is in first, second ... or sixth place.</figcaption></figure>

<figure><img src="image_2083.png" /><figcaption>Figure 7 – For DP450 steel, Examples of stress deviations between the mathematical law and experimental test measurements for several strain values after the uniform elongation point.</figcaption></figure>
<figure><img src="image_2092.png" /><figcaption>Figure 8 – Prediction/experience comparison of extended curves after uniform elongation defining the choice of isotropic hardening model.</figcaption></figure>

## Conclusions

Different experimental tests allowing to reach large strain have been presented: uniaxial tensile test with local measure, shear test, stack compression test, bulge test, in-plane torsion test, plane strain compression test and uniaxial tensile test on pre-rolled flat. The “Exp_S” isotropic hardening law is recommended for an accurate formability prediction in sheet forming numerical simulations, as soon as an adequate identification protocol, as proposed here, is used to determine the adjusting parameters.

## References

[1] X. Lemoine, S. Sriram, R. Kergen, Flow Curve determination at large plastic Strain levels to accurately constitutive equations of AHSS in forming simulation, ESAFORM Conference 2011, Belfast, AIP publishing, https://doi.org/10.1063/1.3589715

[2] M. Jain, D.J. Lloyd, S.R. Macewen, Hardening laws, surface roughness and biaxial tensile limit strains of sheet aluminium alloys, Int. F. Mech. Sci. 38 (1996) 219-232.

[3] S. Dziallach, W. Bleck, M. Blumbach, T. Hallfeldt, Sheet metal testing and flow curve determination under Multiaxial conditions, Advanced Engineering Materials, 9 (2007) 987-994, https://doi.org/10.1002/adem.200700129

[4] S. Coppieters, H. Traphöner, F. Stiebert, T. Balan, T. Kuwabara, A.E. Tekkaya, Large strain flow curve identification for sheet metal, J. Mat. Proc. Techn. 308 (2022) 117725, https://doi.org/10.1016/j.jmatprotec.2022.117725

[5] M. Ben Tahar, Contribution à l'étude et la simulation du procédé d'hydroformage, PhD Report, Ecole des Mines de Paris, 2005

[6] X. Lemoine, E. Till, L. Kessler, T. Balan, Improved constitutive equations for forming processes including ductility criteria, Technical Report EUR 22836 EN, Office for Official Publications of the European Communities, 2006.
<figure><img src="image_2084.png" /><figcaption>Figure 8 – Prediction/experience comparison of extended curves after uniform elongation defining the choice of isotropic hardening model.</figcaption></figure>

**Conclusions**

Different experimental tests allowing to reach large strain have been presented: uniaxial tensile test with local measure, shear test, stack compression test, bulge test, in-plane torsion test, plane strain compression test and uniaxial tensile test on pre-rolled flat. The “Exp_S” isotropic hardening law is recommended for an accurate formability prediction in sheet forming numerical simulations, as soon as an adequate identification protocol, as proposed here, is used to determine the adjusting parameters.

**References**

[1] X. Lemoine, S. Sriram, R. Kergen, Flow Curve determination at large plastic Strain levels to accurately constitutive equations of AHSS in forming simulation, ESAFORM Conference 2011, Belfast, AIP publishing, https://doi.org/10.1063/1.3589715

[2] M. Jain, D.J. Lloyd, S.R. Macewen, Hardening laws, surface roughness and biaxial tensile limit strains of sheet aluminium alloys, Int. F. Mech. Sci. 38 (1996) 219-232.

[3] S. Dziallach, W. Bleck, M. Blumbach, T. Hallfeldt, Sheet metal testing and flow curve determination under Multiaxial conditions, Advanced Engineering Materials, 9 (2007) 987-994, https://doi.org/10.1002/adem.200700129

[4] S. Coppieters, H. Traphöner, F. Stiebert, T. Balan, T. Kuwabara, A.E. Tekkaya, Large strain flow curve identification for sheet metal, J. Mat. Proc. Techn. 308 (2022) 117725, https://doi.org/10.1016/j.jmatprotec.2022.117725

[5] M. Ben Tahar, Contribution à l'étude et la simulation du procédé d'hydroformage, PhD Report, Ecole des Mines de Paris, 2005

[6] X. Lemoine, E. Till, L. Kessler, T. Balan Improved constitutive equations for forming processes including ductility criteria, Technical Report EUR 22836 EN, Office for Official Publications of the European Communities, 2006.
[7] W. Bleck, S. Brühl, T. Gerber, A. Katsamas, R. Ottaviani, L. Samek, and S. Traint. Control and exploitation of the bake-hardening effect in multi-phase high-strength steels, Technical Report EUR 22448 EN, Office for Official Publications of the European Communities, 2006.

[8] M. Kleiner, V. Hellinger, M. Schikorra, L. Kessler, H. Vegter, D. Cornette, M. Kulik and T. Brenne. Material parameters for sheet metal forming simulations by means of optimisation algorithms, Technical Report EUR 20907 EN, Office for Official Publications of the European Communities, 2003.

[9] C. Chermette, K. Unruh, I. Peshekhodov, J. Chottin and T. Balan. A new analytical method for determination of the flow curve for high-strength sheet steels using the plane strain compression test, Int. J. Mater Form, 13 (2019) 269-292. https://doi.org/10.1007/s12289-019-01485-4.

[10] H. Traphöner, T. Clausmeyer, and A.E. Tekkaya. Methods for measuring large shear strains in in-plane torsion tests, Journal of Materials Processing Tech. 287 (2021) 116516. https://doi.org/10.1016/j.jmatprotec.2019.116516

[11] H. Traphöner, T. Clausmeyer, S. Heibel and A.E. Tekkaya. Influence of manufacturing processes on material characterization with the grooved in-plane torsion test, Int. J. Mech. Sci. 146-147 (2018) 544-555. https://doi.org/10.1016/j.ijmecsci.2017.12.052

[12] V. Grolleau, C. Roth and D. Mohr. Design of in-plane torsion experiment to characterize anisotropic plasticity and fracture under simple shear, Int. J. Solids Struct. 236-237 (2022) 111314. https://doi.org/10.1016/j.ijsolstr.2021.111341

[13] M. Merklein, V. Gödel, Characterization of the flow behavior of deep drawing steel grades in dependency of stress state and its impact on FEA, ESAFORM Conference 2009, University of Twente, Netherland, Int J Mater Form 2 (Suppl 1), 415 (2009). https://doi.org/10.1007/s12289-009-0506-9

[14] Y.G. An, H. Vegter, Analytical and experimental study of frictional behavior in through-thickness compression test, Journal of Materials Processing Technology 160 (2005) 148-155. https://doi.org/10.1016/j.jmatprotec.2004.05.026

[15] M. Merklein, A. Kuppert, A method for the layer compression test considering the anisotropic material behavior, ESAFORM Conference 2009, University of Twente, Netherland, Int J Mater Form 2 (Suppl 1), 483 (2009). https://doi.org/10.1007/s12289-009-0592-8
[7] W. Bleck, S. Brühl, T. Gerber, A. Katsamas, R. Ottaviani, L. Samek, and S. Traint. Control and exploitation of the bake-hardening effect in multi-phase high-strength steels, Technical Report EUR 22448 EN, Office for Official Publications of the European Communities, 2006.

[8] M. Kleiner, V. Hellinger, M. Schikorra, L. Kessler, H. Vegter, D. Cornette, M. Kulik and T. Brenne. Material parameters for sheet metal forming simulations by means of optimisation algorithms, Technical Report EUR 20907 EN, Office for Official Publications of the European Communities, 2003.

[9] C. Chermette, K. Unruh, I. Peshekhodov, J. Chottin and T. Balan. A new analytical method for determination of the flow curve for high-strength sheet steels using the plane strain compression test, Int. J. Mater Form, 13 (2019) 269-292. https://doi.org/10.1007/s12289-019-01485-4.

[10] H. Traphöner, T. Clausmeyer, and A.E. Tekkaya. Methods for measuring large shear strains in in-plane torsion tests, Journal of Materials Processing Tech. 287 (2021) 116516. https://doi.org/10.1016/j.jmatprotec.2019.116516

[11] H. Traphöner, T. Clausmeyer, S. Heibel and A.E. Tekkaya. Influence of manufacturing processes on material characterization with the grooved in-plane torsion test, Int. J. Mech. Sci. 146-147 (2018) 544-555. https://doi.org/10.1016/j.ijmecsci.2017.12.052

[12] V. Grolleau, C. Roth and D. Mohr. Design of in-plane torsion experiment to characterize anisotropic plasticity and fracture under simple shear, Int. J. Solids Struct. 236-237 (2022) 111314. https://doi.org/10.1016/j.ijsolstr.2021.111341

[13] M. Merklein, V. Gödel, Characterization of the flow behavior of deep drawing steel grades in dependency of stress state and its impact on FEA, ESAFORM Conference 2009, University of Twente, Netherland, Int J Mater Form 2 (Suppl 1), 415 (2009). https://doi.org/10.1007/s12289-009-0506-9

[14] Y.G. An, H. Vegter, Analytical and experimental study of frictional behavior in through-thickness compression test, Journal of Materials Processing Technology 160 (2005) 148-155. https://doi.org/10.1016/j.jmatprotec.2004.05.026

[15] M. Merklein, A. Kuppert, A method for the layer compression test considering the anisotropic material behavior, ESAFORM Conference 2009, University of Twente, Netherland, Int J Mater Form 2 (Suppl 1), 483 (2009). https://doi.org/10.1007/s12289-009-0592-8
# High strain rate investigation on the mechanical anisotropy induced by SLM technology on a 3D printed steel

MANCINI Edoardo<fnref n="1" />,ª*, UTZERI Mattia<fnref n="2" />,ᵇ, CORTIS Gabriele<fnref n="3" />,ᶜ, SASSO Marco<fnref n="2" />,ᵈ, CORTIS Daniele<fnref n="4" />,ᵉ, ORLANDI Donato<fnref n="4" />,ᵉ, CORTESE Luca<fnref n="3" />,<fnref n="9" /> and DI ANGELO Luca<fnref n="1" />,ʰ

**Keywords:** SLM, High Strain Rate, Material Anisotropy

**Abstract.** Selective Laser Melting (SLM) technology is an additive manufacturing (AM) technique that involves selective or localized melting of metallic powder bed by a high-intensity laser beam source. This technology is responsible for an induced anisotropy in the printed components: in fact, depending on the orientation of the part within the powder bed, different mechanical properties are usually obtained. It is also known that mechanical properties may depend on the rate of load application. This work studies the mechanical anisotropy induced by SLM technology on a 16MnCr5 case hardening steel subjected to high strain rate loading. A direct Split Hopkinson Bar was employed for testing. The Cowper-Symonds model was used to describe the material behavior at high strain rates for samples loaded in the building and in the normal to building directions.

## Introduction

Additive Manufacturing (AM) has gained popularity in recent decades due to the peculiar process and material properties [1], and it is today widely used in a variety of specific applications. Compared to traditional techniques [2], this technology allows for a greater design customization and leads to cost and time savings during design and production. AM enables the construction of sophisticated 3D structures with specifically designed features [3]. Rapid development in the previous 20 years has been driven by increased availability of materials, cheap machines, and always novel applications [4]. AM is considered also a straightforward manufacturing technology that allows fabricating parts from mixed materials composed of metals, polymers, ceramics, and composites with complex features, reducing material consumption [5].

Among the AM technologies, in recent years, a particular development has involved Selective Laser Melting (SLM), which is the most common technology for fabricating objects from metallic powders. The SLM employs a high-intensity laser as an energy source to melt and fuse particular powder regions layer by layer. Regarding the mechanical properties of objects manufactured by the SLM process of steels, the literature mainly concentrates on the combination of parameters that optimize hardness, tensile and compression yield, and high cycle fatigue properties.

It is known that this technology may be responsible for an induced mechanical anisotropy as a consequence of the orientation of the part that is realized within the powder bed [8]. It is also known that mechanical properties may depend on the rate of load application [9][10]. Many AM-

<figure><img src="image_2093.png" /></figure>

<footnote n="1">¹ Università degli Studi dell'Aquila, DIIIE, Piazzale Ernesto Pontieri I, 67100 L'Aquila, Italy</footnote>

<footnote n="2">² Università Politecnica delle Marche, DIISM, Via Brecce Bianche, 60121 Ancona, Italy</footnote>

<footnote n="3">³ Sapienza University of Rome, DIMA, 00184 Rome, Italy</footnote>

<footnote n="4">⁴ Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Gran Sasso, via G. Acitelli 22, 67100 L'Aquila, Italy</footnote>

<footnote>ªedoardo.mancini@univaq.it, ᵇm.utzeri@staff.univpm.it, ᵉgabriele.cortis@uniroma1.it,</footnote>

<footnote>ᵈm.sasso@staff.univpm.it, ᵉdaniele.cortis@lngs.infn.it, ᵇdonato.orlandi@lngs.infn.it,</footnote>

<footnote n="9">⁹luca.cortese@uniroma1.it, ᵇluca.diangelo@univaq.it</footnote>
# High strain rate investigation on the mechanical anisotropy induced by SLM technology on a 3D printed steel

MANCINI Edoardo<fnref n="1" />, a*, UTZERI Mattia<fnref n="2" />, b, CORTIS Gabriele<fnref n="3" />, c, SASSO Marco<fnref n="2" />, d, CORTIS Daniele<fnref n="4" />, e, ORLANDI Donato<fnref n="4" />, f, CORTESE Luca<fnref n="3" />, g and DI ANGELO Luca<fnref n="1" />, h

**Keywords:** SLM, High Strain Rate, Material Anisotropy

**Abstract.** Selective Laser Melting (SLM) technology is an additive manufacturing (AM) technique that involves selective or localized melting of metallic powder bed by a high-intensity laser beam source. This technology is responsible for an induced anisotropy in the printed components: in fact, depending on the orientation of the part within the powder bed, different mechanical properties are usually obtained. It is also known that mechanical properties may depend on the rate of load application. This work studies the mechanical anisotropy induced by SLM technology on a 16MnCr5 case hardening steel subjected to high strain rate loading. A direct Split Hopkinson Bar was employed for testing. The Cowper-Symonds model was used to describe the material behavior at high strain rates for samples loaded in the building and in the normal to building directions.

**Introduction**

Additive Manufacturing (AM) has gained popularity in recent decades due to the peculiar process and material properties [1], and it is today widely used in a variety of specific applications. Compared to traditional techniques [2], this technology allows for a greater design customization and leads to cost and time savings during design and production. AM enables the construction of sophisticated 3D structures with specifically designed features [3]. Rapid development in the previous 20 years has been driven by increased availability of materials, cheap machines, and always novel applications [4]. AM is considered also a straightforward manufacturing technology that allows fabricating parts from mixed materials composed of metals, polymers, ceramics, and composites with complex features, reducing material consumption [5].

Among the AM technologies, in recent years, a particular development has involved Selective Laser Melting (SLM), which is the most common technology for fabricating objects from metallic powders. The SLM employs a high-intensity laser as an energy source to melt and fuse particular powder regions layer by layer. Regarding the mechanical properties of objects manufactured by the SLM process of steels, the literature mainly concentrates on the combination of parameters that optimize hardness, tensile and compression yield, and high cycle fatigue properties.

It is known that this technology may be responsible for an induced mechanical anisotropy as a consequence of the orientation of the part that is realized within the powder bed [8]. It is also known that mechanical properties may depend on the rate of load application [9][10]. Many AM-

<figure><img src="image_2085.png" /></figure>

<footnote n="1">¹ Università degli Studi dell'Aquila, DIIIE, Piazzale Ernesto Pontieri I, 67100 L'Aquila, Italy</footnote>

<footnote n="2">² Università Politecnica delle Marche, DIISM, Via Brecce Bianche, 60121 Ancona, Italy</footnote>

<footnote n="3">³ Sapienza University of Rome, DIMA, 00184 Rome, Italy</footnote>

<footnote n="4">⁴ Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Gran Sasso, via G. Acitelli 22, 67100 L'Aquila, Italy</footnote>

<footnote>ªedoardo.mancini@univaq.it, ᵇm.utzeri@staff.univpm.it, ᵇgabriele.cortis@uniroma1.it,</footnote>

<footnote>d⁵m.sasso@staff.univpm.it, e⁶daniele.cortis@lngs.infn.it, f⁷donato.orlandi@lngs.infn.it,</footnote>

<footnote>g⁹luca.cortese@uniroma1.it, h⁹luca.diangelo@univaq.it</footnote>
built metal parts, such as engineered components, are subjected to dynamic loads under operating conditions. So, high-strain dynamic conditions must be investigated for an efficient design and manufacturing of such parts. Although the study of the dynamic behavior of additively manufactured materials is increasing, literature referring to high strain rate properties of steel alloys is limited ([6] and [7]).

The aim of this work is to study the mechanical anisotropy induced by SLM technology on a case hardening steel (16MnCr5) subjected to high strain rate loading.

## Materials and methods

Material. The material tested is a 16MnCr5 whose typical characteristics resulting from tensile test in as built conditions, for a relative density > 99.9%, are reported in Table 1.

<table><caption>Table 1. Tensile test material characteristics</caption><thead><tr><th></th><th>Tensile strength Rm<br>[N/mm²]</th><th>Yield strength Re<br>[N/mm²]</th><th>Elongation at break A<br>[%]</th></tr></thead><tbody><tr><td>as-built</td><td>1095</td><td>1040</td><td>10</td></tr></tbody></table>

Specimens have been produced considering three different orientations with respect to the z-building direction: printing direction 0°, also indicated with Z, 30° with respect to the z-direction, and normal direction 90°, indicated with X. A SISMA MySint100 PM/RM machine equipped with a single InfraRed (IR) laser was used. Fig. 1 shows the samples, the support structures and the samples orientations. The ids of the samples tested in compression in quasi-static and dynamic condition are reported in Table 2.

<figure><img src="image_2094.png" /><figcaption>Fig. 1. Produced compression specimens: 0°(Z), 30°, 90°(X) orientation respect to z-direction</figcaption></figure>

<table><caption>Table 2. Specimens for compression tests: printing and loading directions</caption><thead><tr><th>Test type</th><th>Specimen id</th><th>Printing direction</th><th>Loading direction</th></tr></thead><tbody><tr><td rowspan="2">Quasi-Static</td><td>Ci X</td><td>Z</td><td>X</td></tr><tr><td>Ci Z</td><td>Z</td><td>Z</td></tr><tr><td rowspan="2">Dynamic</td><td>Ti X</td><td>Z</td><td>X</td></tr><tr><td>Ti Z</td><td>Z</td><td>Z</td></tr></tbody></table>

An hydrostatic balance (Fig. 2a) was used to measure the samples relative density. The results of the measurements are shown in Fig. 2b.
built metal parts, such as engineered components, are subjected to dynamic loads under operating conditions. So, high-strain dynamic conditions must be investigated for an efficient design and manufacturing of such parts. Although the study of the dynamic behavior of additively manufactured materials is increasing, literature referring to high strain rate properties of steel alloys is limited ([6] and [7]).

The aim of this work is to study the mechanical anisotropy induced by SLM technology on a case hardening steel (16MnCr5) subjected to high strain rate loading.

## Materials and methods

Material. The material tested is a 16MnCr5 whose typical characteristics resulting from tensile test in as built conditions, for a relative density > 99.9%, are reported in Table 1.

<table><caption>Table 1. Tensile test material characteristics</caption><thead><tr><th></th><th>Tensile strength Rm<br>[N/mm²]</th><th>Yield strength Re<br>[N/mm²]</th><th>Elongation at break A<br>[%]</th></tr></thead><tbody><tr><td>as-built</td><td>1095</td><td>1040</td><td>10</td></tr></tbody></table>

Specimens have been produced considering three different orientations with respect to the z-building direction: printing direction 0°, also indicated with Z, 30° with respect to the z-direction, and normal direction 90°, indicated with X. A SISMA MySint100 PM/RM machine equipped with a single InfraRed (IR) laser was used. Fig. 1 shows the samples, the support structures and the samples orientations. The ids of the samples tested in compression in quasi-static and dynamic condition are reported in Table 2.

<figure><img src="image_2086.png" /><figcaption>Fig. 1. Produced compression specimens: 0°(Z), 30°, 90°(X) orientation respect to z-direction</figcaption></figure>

<table><caption>Table 2. Specimens for compression tests: printing and loading directions</caption><thead><tr><th>Test type</th><th>Specimen id</th><th>Printing direction</th><th>Loading direction</th></tr></thead><tbody><tr><td rowspan="2">Quasi-Static</td><td>Ci X</td><td>Z</td><td>X</td></tr><tr><td>Ci Z</td><td>Z</td><td>Z</td></tr><tr><td rowspan="2">Dynamic</td><td>Ti X</td><td>Z</td><td>X</td></tr><tr><td>Ti Z</td><td>Z</td><td>Z</td></tr></tbody></table>

An hydrostatic balance (Fig. 2a) was used to measure the samples relative density. The results of the measurements are shown in Fig. 2b.
<figure><img src="image_2095.png" /><figcaption>Fig. 2. Relative density measurements: (a) hydrostatic balance and (b) relative density as a function of printing direction</figcaption></figure>

Equipment. Quasi static compression tests were performed by a universal servo-hydraulic MTS with a maximum axial load of 250 kN available at “Sapienza” University of Rome (Fig. 3a). A direct Split Hopkinson bar, available at Politecnica University of Marche, was used for the compression dynamic tests (Fig. 3b) [9][11]. Figure 4 shows a schematic representation of the sample with respect to printing and loading direction. The support structure is also indicated.

<figure><img src="image_2096.png" /><figcaption>Fig. 3. Equipment: (a) Servo-hydraulic MTS machine and (b) Split Hopkinson Bar with a detailed view of the sample</figcaption></figure>

<figure><img src="image_2097.png" /><figcaption>Fig. 4. Printing and loading directions; the latter identified by the red arrows</figcaption></figure>
<figure><img src="image_2087.png" /><figcaption>Fig. 2. Relative density measurements: (a) hydrostatic balance and (b) relative density as a function of printing direction</figcaption></figure>

Equipment. Quasi static compression tests were performed by a universal servo-hydraulic MTS with a maximum axial load of 250 kN available at “Sapienza” University of Rome (Fig. 3a). A direct Split Hopkinson bar, available at Politecnica University of Marche, was used for the compression dynamic tests (Fig. 3b) [9][11]. Figure 4 shows a schematic representation of the sample with respect to printing and loading direction. The support structure is also indicated.

<figure><img src="image_2088.png" /><figcaption>Fig. 3. Equipment: (a) Servo-hydraulic MTS machine and (b) Split Hopkinson Bar with a detailed view of the sample</figcaption></figure>

<figure><img src="image_2089.png" /><figcaption>Fig. 4. Printing and loading directions; the latter identified by the red arrows</figcaption></figure>
## Results and discussion

Results. True stress-strain for quasi-static compression tests are reported in Fig. 5. The difference in terms of Young's modulus in different direction is $\Delta_{QS}(E_Z E_X) \approx 11000$ MPa.

<figure><img src="image_2098.png" /><figcaption>Fig. 5. Experimental quasi-static compression stress-strain results.</figcaption></figure>

Engineering stress and strain were calculated for dynamic tests using the formulas and procedure described in [9], and then converted in true values. The obtained true stress-strain curves, for different strain rates and loading directions, are shown in Fig. 6a and b. For all tests the strain rate was fairly constant (Fig. 6c), within a range from 800 to 1800 1/s.

<figure><img src="image_2099.png" /><figcaption>Fig. 6. Experimental dynamic compression results: (a), (b) true stress-strain ~@1500 /s and ~@1000 /s respectively; (c) strain rate-strain curves</figcaption></figure>
## Results and discussion

Results. True stress-strain for quasi-static compression tests are reported in Fig. 5. The difference in terms of Young's modulus in different direction is $\Delta_{QS}(E_Z E_X) \approx 11000$ MPa.

<figure><img src="image_2090.png" /><figcaption>Fig. 5. Experimental quasi-static compression stress-strain results.</figcaption></figure>

Engineering stress and strain were calculated for dynamic tests using the formulas and procedure described in [9], and then converted in true values. The obtained true stress-strain curves, for different strain rates and loading directions, are shown in Fig. 6a and b. For all tests the strain rate was fairly constant (Fig. 6c), within a range from 800 to 1800 1/s.

<figure><img src="image_2091.png" /><figcaption>Fig. 6. Experimental dynamic compression results: (a), (b) true stress-strain ~@1500 /s and ~@1000 /s respectively; (c) strain rate-strain curves</figcaption></figure>
In Fig. 7 the loads acting on the left and right side of the sample during the test are also shown. Apart the oscillations, which are expected on the input bar side (Load Sx Side) in dynamic SHB tests, the test occurred in dynamic equilibrium.

<figure><img src="image_2100.png" /><figcaption>Fig. 7. Load on sample sides; loading direction (a) Z and (b) X</figcaption></figure>

Experimental quasi-static and dynamic compression tests have been used to identify the material strain rate sensitivity at different building orientations and strain rates by the dynamic increase factors (DIFs). The DIFs is a scalar computed as the ratio of dynamic over quasi-static strength of the material at a given value of strain ε. A DIF is larger than one indicates an increase in stress under dynamic conditions. The Cowper-Symonds model (Eq. 1) was used to describe the material behavior at strain rates for the X and Z loading directions. The results of the calibration are shown in Fig. 8.

$$ \frac{\sigma}{\sigma_y} = \left[ 1 + \left( \frac{\dot{\epsilon}}{C} \right)^{\frac{1}{q}} \right] \quad (1) $$

<figure><img src="image_2101.png" /><figcaption>Fig. 8. DIFs and Cowper-Symonds interpolation: a) sample printed and loaded in Z direction, b) sample printed in Z direction and loaded in X direction</figcaption></figure>

Discussion. Quasi static tests revealed an anisotropic behavior both in terms of Young modulus ($\Delta_{QS}(E_Z E_X) \approx 11000$ MPa) and plastic flow curve (Fig. 5); The effect of printing direction on Young modulus seems to disappear under dynamic conditions (Fig. 6a and b; $\Delta_{Dyn}(E_Z E_X) \sim 0$ MPa). A DIFs larger than one, indicating an increase in stress for the dynamic conditions when compared with the quasi-static ones, revealed a strain rate sensitivity for both
In Fig. 7 the loads acting on the left and right side of the sample during the test are also shown. Apart the oscillations, which are expected on the input bar side (Load Sx Side) in dynamic SHB tests, the test occurred in dynamic equilibrium.

<figure><img src="image_2092.png" /><figcaption>Fig. 7. Load on sample sides; loading direction (a) Z and (b) X</figcaption></figure>

Experimental quasi-static and dynamic compression tests have been used to identify the material strain rate sensitivity at different building orientations and strain rates by the dynamic increase factors (DIFs). The DIFs is a scalar computed as the ratio of dynamic over quasi-static strength of the material at a given value of strain ε. A DIF is larger than one indicates an increase in stress under dynamic conditions. The Cowper-Symonds model (Eq. 1) was used to describe the material behavior at strain rates for the X and Z loading directions. The results of the calibration are shown in Fig. 8.

$$ \frac{\sigma}{\sigma_y} = \left[ 1 + \left( \frac{\dot{\epsilon}}{C} \right)^{\frac{1}{q}} \right] \quad (1) $$

<figure><img src="image_2093.png" /><figcaption>Fig. 8. DIFs and Cowper-Symonds interpolation: a) sample printed and loaded in Z direction, b) sample printed in Z direction and loaded in X direction</figcaption></figure>

Discussion. Quasi static tests revealed an anisotropic behavior both in terms of Young modulus ($\Delta_{QS}(E_Z E_X) \approx 11000$ MPa) and plastic flow curve (Fig. 5); The effect of printing direction on Young modulus seems to disappear under dynamic conditions (Fig. 6a and b; $\Delta_{Dyn}(E_Z E_X) \sim 0$ MPa). A DIFs larger than one, indicating an increase in stress for the dynamic conditions when compared with the quasi-static ones, revealed a strain rate sensitivity for both
loading directions (Fig. 8) pointing out a higher effect for the samples loaded in the X direction normal to the printing one (Fig. 8b).

## Conclusion

A preliminary investigation of the mechanical response of a steel 16MnCr5 printed with SLM technology was made. The study was carried out under both quasi-static and dynamic conditions. The DIFs were used to evaluate the strain rate sensitivity. Anisotropy due to the printing process and a dependence on the strain rate of the printed material emerged. An anisotropic behavior in the elastic regime under quasi-static conditions has been found, while at a high strain rate no significant difference with orientation was observed. However, the result must be assumed with caution, as further investigations with an accuracy assessment at small strains are needed. Eventually, the calibration of Cowper-Symonds model in different loading directions was done.

## References

[1] T. DebRoy, H.L. Wei, J.S. Zuback, T. Mukherjee, J.W. Elmer, J.O. Milewski, et al. Additive manufacturing of metallic components – process, structure and properties, Prog. Mater. Sci 92 (2018) 112–224. https://doi.org/10.1016/j.pmatsci.2017.10.001

[2] W.E. Frazier, Metal additive manufacturing: a review, J. Mater. Eng. Perform. 2014;23: 1917–28. https://doi.org/10.1007/s11665-014-0958-z

[3] S. Cooke, K. Ahmadi, S. Willerth, R. Herring, Metal additive manufacturing: technology, metallurgy and modelling, J. Manuf. Process. 57 (2020) 978–1003 https://doi.org/10.1016/j.jmapro.2020.07.025

[4] A. Gisario, M. Kazarian, F. Martina, M. Mehrpouya, Metal additive manufacturing in the commercial aviation industry: a review. J. Manuf. Syst. 2019;53:124–49. https://doi.org/10.1016/j.jmsy.2019.08.005

[5] S. R. Narasimharaju, W. Zeng, T. See, Z. Zhu, P. Scott, X. Jiang & S. Lou, A comprehensive review on laser powder bed fusion of steels: Processing, microstructure, defects and control methods, mechanical properties, current challenges and future trends, J. Manuf. Processes. 75 (2022) 375-414.

[6] B. Song, E. Nishida, B. Sanborn, M. Maguire, D. Adams, J. Carroll, & T. Palmer. Compressive and tensile stress–strain responses of additively manufactured (AM) 304L stainless steel at high strain rates. J. Dyn. Behav. Mater. 3 (2017) 412-425. https://doi.org/10.1007/s40870-017-0122-6

[7] S. H. Riza, S. H. Masood, C. Wen, D. Ruan, & S. Xu, Dynamic behaviour of high strength steel parts developed through laser assisted direct metal deposition. Mater. Des. 64 (2014) 650-659. https://doi.org/10.1016/j.matdes.2014.08.026

[8] A. Charmi et al., Mechanical anisotropy of additively manufactured stainless steel 316L: An experimental and numerical study", Mater. Sci. Engineering: A. 799 (2021) 140-154

[9] E. Mancini, M. Sasso, M. Rossi, G. Chiappini, G. Newaz, D. Amodio, Design of an Innovative System for Wave Generation in Direct Tension–Compression Split Hopkinson Bar, J. Dyn. Behav. Mater. 1 (2015) 201–213. https://doi.org/10.1007/s40870-015-0019-1

[10] B. Song, W. Chen (2010) Split Hopkinson Kolsky bar: design testing and applications. Springer, New York. ISBN 978-1-4419-7981-0

[11] M. Martarelli, E. Mancini, B. Lonzi, M. Sasso, Sensor calibration of polymeric Hopkinson bars for dynamic testing of soft materials, Meas. Sci. Technol. 29 (2018) 025601. https://doi.org/10.1088/1361-6501/aa9136
loading directions (Fig. 8) pointing out a higher effect for the samples loaded in the X direction normal to the printing one (Fig. 8b).

## Conclusion

A preliminary investigation of the mechanical response of a steel 16MnCr5 printed with SLM technology was made. The study was carried out under both quasi-static and dynamic conditions. The DIFs were used to evaluate the strain rate sensitivity. Anisotropy due to the printing process and a dependence on the strain rate of the printed material emerged. An anisotropic behavior in the elastic regime under quasi-static conditions has been found, while at a high strain rate no significant difference with orientation was observed. However, the result must be assumed with caution, as further investigations with an accuracy assessment at small strains are needed. Eventually, the calibration of Cowper-Symonds model in different loading directions was done.

## References

[1] T. DebRoy, H.L. Wei, J.S. Zuback, T. Mukherjee, J.W. Elmer, J.O. Milewski, et al. Additive manufacturing of metallic components – process, structure and properties, Prog. Mater. Sci 92 (2018) 112–224. https://doi.org/10.1016/j.pmatsci.2017.10.001

[2] W.E. Frazier, Metal additive manufacturing: a review, J. Mater. Eng. Perform. 2014;23: 1917–28. https://doi.org/10.1007/s11665-014-0958-z

[3] S. Cooke, K. Ahmadi, S. Willerth, R. Herring, Metal additive manufacturing: technology, metallurgy and modelling, J. Manuf. Process. 57 (2020) 978–1003 https://doi.org/10.1016/j.jmapro.2020.07.025

[4] A. Gisario, M. Kazarian, F. Martina, M. Mehrpouya, Metal additive manufacturing in the commercial aviation industry: a review. J. Manuf. Syst. 2019;53:124–49. https://doi.org/10.1016/j.jmsy.2019.08.005

[5] S. R. Narasimharaju, W. Zeng, T. See, Z. Zhu, P. Scott, X. Jiang & S. Lou, A comprehensive review on laser powder bed fusion of steels: Processing, microstructure, defects and control methods, mechanical properties, current challenges and future trends, J. Manuf. Processes. 75 (2022) 375-414.

[6] B. Song, E. Nishida, B. Sanborn, M. Maguire, D. Adams, J. Carroll, & T. Palmer. Compressive and tensile stress–strain responses of additively manufactured (AM) 304L stainless steel at high strain rates. J. Dyn. Behav. Mater. 3 (2017) 412-425. https://doi.org/10.1007/s40870-017-0122-6

[7] S. H. Riza, S. H. Masood, C. Wen, D. Ruan, & S. Xu, Dynamic behaviour of high strength steel parts developed through laser assisted direct metal deposition. Mater. Des. 64 (2014) 650-659. https://doi.org/10.1016/j.matdes.2014.08.026

[8] A. Charmi et al., Mechanical anisotropy of additively manufactured stainless steel 316L: An experimental and numerical study", Mater. Sci. Engineering: A. 799 (2021) 140-154

[9] E. Mancini, M. Sasso, M. Rossi, G. Chiappini, G. Newaz, D. Amodio, Design of an Innovative System for Wave Generation in Direct Tension–Compression Split Hopkinson Bar, J. Dyn. Behav. Mater. 1 (2015) 201–213. https://doi.org/10.1007/s40870-015-0019-1

[10] B. Song, W. Chen (2010) Split Hopkinson Kolsky bar: design testing and applications. Springer, New York. ISBN 978-1-4419-7981-0

[11] M. Martarelli, E. Mancini, B. Lonzi, M. Sasso, Sensor calibration of polymeric Hopkinson bars for dynamic testing of soft materials, Meas. Sci. Technol. 29 (2018) 025601. https://doi.org/10.1088/1361-6501/aa9136
# Mechanical characterization and behavior modelling of Ti-6Al-4V alloy in hot forming conditions

D'ARCHIVIO Lucas¹,ᵃ*, PENAZZI Luc¹,ᵇ, VELAY Vincent¹,ᶜ, VIDAL Vanessa¹,ᵈ

¹Institut Clément Ader (ICA), Université de Toulouse, CNRS, IMT Mines Albi, INSA, UPS, ISAE-SUPAERO, UT3, Toulouse, France

ᵃlucas.d_archivio@mines-albi.fr, ᵇluc.penzazzi@mines-albi.fr, ᶜvincent.velay@mines-albi.fr, ᵈvanessa.vidal@mines-albi.fr

**Keywords:** Behavior Modelling, Hot Forming, Ti-6Al-4V, Mechanical Testing, Digital Image Correlation (DIC)

**Abstract.** Ti-6Al-4V is a widely used titanium alloy in superplastic forming process which requires high temperatures (T ≈ 900 °C) and low strain rates (ε̇ ≈ 10⁻³s⁻¹). One way to reduce the costs of the process is to use a lower forming temperature and/or higher strain rate, up to 10⁻²s⁻¹. However, the behavior of Ti-6Al-4V alloy still need to be modeled at such forming conditions and for different initial microstructure. In order to characterize the mechanical behavior of Ti-6Al-4V at temperature between 400 °C and 700 °C, relaxation and tensile tests performed under small and large deformation conditions were conducted. Depending on the test, the deformations were evaluated through two kinds of measurements, respectively with an extensometer and with a digital image correlation technique (DIC). Similar experimental results were obtained, validating the use of DIC at high temperature to evaluate high strain levels. Two different microstructures of Ti-6Al-4V alloy were tested to study the impact of the initial microstructure on the mechanical behavior. For similar conditions of strain rate and temperature, representative of the forming process, a fine-grained microstructure exhibits an enhancement of mechanical behavior in comparison with the classical coarse-grained microstructure used in the current industrial process. Finally, an elasto-viscoplastic has been identified for each microstructure.

**Introduction**

Titanium alloys are widely used in the aerospace industry thanks to their high strength to weight ratio [1]. Titanium sheets are manufactured into parts for industrial applications by thermo-mechanical processes (heat treatment, machining, drilling, hot forming, ...). These processes, along with the chemical composition of the alloy, can produce various microstructures which in turn influences the mechanical behavior. Therefore, a wide range of mechanical properties can be achieved for titanium alloys. One particular alloy stands out for aerospace application, the Ti-6Al-4V titanium alloy, an α + β alloy, used in critical parts of an airplane such as the engine. To shape it, a specific high temperature process is required to unlock its superplastic properties, the Superplastic Forming process (SPF). This process combines very high temperature (≈ 900 °C) and low deformation rates (ε̇ ≈ 10⁻³s⁻¹) using inert gas punch to form parts with high level of deep-drawing. Its behavior has been well studied at room temperature [2] and at SPF temperatures [3]–[5]. To ensure lower energy consumption and faster processing time, some parts could be manufactured at lower temperature and higher strain rates in Hot Forming conditions (HF). Instead of a gas punch, this process makes use of a metallic punch to form the part. Some research have been made in that direction to explore the feasibility of forming Ti-6Al-4V alloy in such conditions [6], [7], by using numerical models to simulate the material behavior and the forming process, showing promising results.

<figure><img src="image_2102.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.
# Mechanical characterization and behavior modelling of Ti-6Al-4V alloy in hot forming conditions

D'ARCHIVIO Lucas¹,ᵃ*, PENAZZI Luc¹,ᵇ, VELAY Vincent¹,ᶜ, VIDAL Vanessa¹,ᵈ

¹Institut Clément Ader (ICA), Université de Toulouse, CNRS, IMT Mines Albi, INSA, UPS, ISAE-SUPAERO, UT3, Toulouse, France

ᵃlucas.d_archivio@mines-albi.fr, ᵇluc.penzazzi@mines-albi.fr, ᶜvincent.velay@mines-albi.fr, ᵈvanessa.vidal@mines-albi.fr

**Keywords:** Behavior Modelling, Hot Forming, Ti-6Al-4V, Mechanical Testing, Digital Image Correlation (DIC)

**Abstract.** Ti-6Al-4V is a widely used titanium alloy in superplastic forming process which requires high temperatures (T ≈ 900 °C) and low strain rates (ε̇ ≈ 10⁻³s⁻¹). One way to reduce the costs of the process is to use a lower forming temperature and/or higher strain rate, up to 10⁻²s⁻¹. However, the behavior of Ti-6Al-4V alloy still need to be modeled at such forming conditions and for different initial microstructure. In order to characterize the mechanical behavior of Ti-6Al-4V at temperature between 400 °C and 700 °C, relaxation and tensile tests performed under small and large deformation conditions were conducted. Depending on the test, the deformations were evaluated through two kinds of measurements, respectively with an extensometer and with a digital image correlation technique (DIC). Similar experimental results were obtained, validating the use of DIC at high temperature to evaluate high strain levels. Two different microstructures of Ti-6Al-4V alloy were tested to study the impact of the initial microstructure on the mechanical behavior. For similar conditions of strain rate and temperature, representative of the forming process, a fine-grained microstructure exhibits an enhancement of mechanical behavior in comparison with the classical coarse-grained microstructure used in the current industrial process. Finally, an elasto-viscoplastic has been identified for each microstructure.

**Introduction**

Titanium alloys are widely used in the aerospace industry thanks to their high strength to weight ratio [1]. Titanium sheets are manufactured into parts for industrial applications by thermo-mechanical processes (heat treatment, machining, drilling, hot forming, ...). These processes, along with the chemical composition of the alloy, can produce various microstructures which in turn influences the mechanical behavior. Therefore, a wide range of mechanical properties can be achieved for titanium alloys. One particular alloy stands out for aerospace application, the Ti-6Al-4V titanium alloy, an α + β alloy, used in critical parts of an airplane such as the engine. To shape it, a specific high temperature process is required to unlock its superplastic properties, the Superplastic Forming process (SPF). This process combines very high temperature (≈ 900 °C) and low deformation rates (ε̇ ≈ 10⁻³s⁻¹) using inert gas punch to form parts with high level of deep-drawing. Its behavior has been well studied at room temperature [2] and at SPF temperatures [3]–[5]. To ensure lower energy consumption and faster processing time, some parts could be manufactured at lower temperature and higher strain rates in Hot Forming conditions (HF). Instead of a gas punch, this process makes use of a metallic punch to form the part. Some research have been made in that direction to explore the feasibility of forming Ti-6Al-4V alloy in such conditions [6], [7], by using numerical models to simulate the material behavior and the forming process, showing promising results.

<figure><img src="image_2094.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.
Different types of model exist to describe the behavior of the Ti-6Al-4V alloy. They can be classified in a range from phenomenological to physical models. Johnson and Cook proposed a simple phenomenological model taking into account strain, strain-rates and temperature for several metals [8]. This model gives low stress discrepancy (~20 MPa) for room temperature and 400 °C [9], whereas Gao et al. results showed that the values of the parameters identified for 650 °C, 700 °C and 750 °C produces higher stress differences (~50 MPa in the best case) [10]. Physical models tries to integrates information about the microstructure, such as phase volume fraction, grain sizes [11], but also deformation mechanisms [5].

This study aims at characterizing and modelling the behavior of a Ti-6Al-4V titanium alloy at temperatures and strain rates for the HF process, i.e. from 400 °C to 700 °C and for strain rates up to $10^{-2}s^{-1}$. The behavior will be tested and modeled to high deformation levels using non-contact optical measurement of the strain by using Digital Image Correlation (DIC) in high temperature conditions. It will also explore the influence of the initial microstructure on the behavior by comparing two different microstructures.

## Material and methods

Materials description: two different microstructures of Ti-6Al-4V titanium alloy are studied in this paper. SEM micrographs showing the two microstructures are presented in Figure 1. The first microstructure is a Coarse Grained (CG) microstructure having equiaxed alpha nodules with an average grain size of 6 µm (Figure 1a) [6]. The second microstructure is a Fine Grained (FG) microstructure which is not fully equiaxed: the alpha phase is slightly elongated along RD and has an average grain size of 2 µm (Figure 1b) [12].

The CG and the FG microstructures are sampled from Ti-6Al-4V sheets having a thickness of respectively 1.7 mm and 3.2 mm. Their respective chemical composition is given in Table 1.

<figure><img src="image_2103.png" /><figcaption>a) Coarse Grain microstructure (CG)</figcaption></figure>

<figure><img src="image_2104.png" /><figcaption>b) Fine Grain microstructure (FG)</figcaption></figure>

<caption>Figure 1: SEM observations of the two initial microstructures (α phase in dark and β phase in white)</caption>

<table><caption>Table 1: Ti-6Al-4V alloy chemical composition for both microstructure</caption><thead><tr><th>Microstructure</th><th>Ti</th><th>Al</th><th>V</th><th>Fe</th><th>O</th><th>C</th><th>N</th></tr></thead><tbody><tr><td>FG [6]</td><td>Bal.</td><td>6.26</td><td>3.92</td><td>0.24</td><td>0.33</td><td>0.01</td><td>0.01</td></tr><tr><td>CG [12]</td><td>Bal.</td><td>6.46</td><td>4.23</td><td>0.21</td><td>0.174</td><td>0.05</td><td>0.01</td></tr></tbody></table>

Procedure description: to characterize the mechanical behavior of the materials at several temperatures, two types of tests are performed on the Ti-6Al-4V alloy: a tensile relaxation test in small deformation to evaluate the elasto-viscoplastic behavior, and a tensile test to failure to upscale the model to large deformations calculated from a displacement field obtained using a DIC method.
Different types of model exist to describe the behavior of the Ti-6Al-4V alloy. They can be classified in a range from phenomenological to physical models. Johnson and Cook proposed a simple phenomenological model taking into account strain, strain-rates and temperature for several metals [8]. This model gives low stress discrepancy (~20 MPa) for room temperature and 400 °C [9], whereas Gao et al. results showed that the values of the parameters identified for 650 °C, 700 °C and 750 °C produces higher stress differences (~50 MPa in the best case) [10]. Physical models tries to integrates information about the microstructure, such as phase volume fraction, grain sizes [11], but also deformation mechanisms [5].

This study aims at characterizing and modelling the behavior of a Ti-6Al-4V titanium alloy at temperatures and strain rates for the HF process, i.e. from 400 °C to 700 °C and for strain rates up to $10^{-2}s^{-1}$. The behavior will be tested and modeled to high deformation levels using non-contact optical measurement of the strain by using Digital Image Correlation (DIC) in high temperature conditions. It will also explore the influence of the initial microstructure on the behavior by comparing two different microstructures.

## Material and methods

Materials description: two different microstructures of Ti-6Al-4V titanium alloy are studied in this paper. SEM micrographs showing the two microstructures are presented in Figure 1. The first microstructure is a Coarse Grained (CG) microstructure having equiaxed alpha nodules with an average grain size of 6 µm (Figure 1a) [6]. The second microstructure is a Fine Grained (FG) microstructure which is not fully equiaxed: the alpha phase is slightly elongated along RD and has an average grain size of 2 µm (Figure 1b) [12].

The CG and the FG microstructures are sampled from Ti-6Al-4V sheets having a thickness of respectively 1.7 mm and 3.2 mm. Their respective chemical composition is given in Table 1.

<figure><img src="image_2095.png" /><figcaption>Figure 1: SEM observations of the two initial microstructures (α phase in dark and β phase in white)</figcaption></figure>

<table><caption>Table 1: Ti-6Al-4V alloy chemical composition for both microstructure</caption><thead><tr><th>Microstructure</th><th>Ti</th><th>Al</th><th>V</th><th>Fe</th><th>O</th><th>C</th><th>N</th></tr></thead><tbody><tr><td>FG [6]</td><td>Bal.</td><td>6.26</td><td>3.92</td><td>0.24</td><td>0.33</td><td>0.01</td><td>0.01</td></tr><tr><td>CG [12]</td><td>Bal.</td><td>6.46</td><td>4.23</td><td>0.21</td><td>0.174</td><td>0.05</td><td>0.01</td></tr></tbody></table>

Procedure description: to characterize the mechanical behavior of the materials at several temperatures, two types of tests are performed on the Ti-6Al-4V alloy: a tensile relaxation test in small deformation to evaluate the elasto-viscoplastic behavior, and a tensile test to failure to upscale the model to large deformations calculated from a displacement field obtained using a DIC method.
For the relaxation test, the specimen is deformed to 0.02 deformation and relaxed for 30 min, deformation rate is set at $10^{-2}$ s$^{-1}$. The test is controlled by deformation using a contact extensometer, allowing a precise control over the deformation and deformation rate. 6 temperatures are tested, from 400 °C to 700 °C with the use of an AET furnace.

The second test is a tensile test under the same conditions of temperature where the specimen is deformed to rupture. Therefore, an extensometer cannot be used to measure the deformation in this case. Instead, a DIC method is implemented to acquire the displacement field between two images taken during the tests. In this study, logarithmic (Hencky) deformation is considered, and the software VIC-2D® is used for the DIC. In order to evaluate the displacement field, a speckle is applied on the surface of the specimen. A wide range of speckles specially developed for DIC can be found in the literature, from painting and laser marking to chemical etching and anodization [13]. For our test conditions, high temperature paints are used to ensure a resistant speckle even at 700 °C. To take account of the risk of the paint layer tearing due to the expected high deformation, no such layers are applied on the specimen. The paint is directly pulverized on the raw surface of the sample.

The samples are obtained by Water Abrasive Jet Cutting (WAJC) in the rolling direction, this study doesn't consider the anisotropy of the Ti-6Al-4V anisotropy.

## Experimental results

Relaxation tests: the load and relaxation curves obtained for the FG microstructure are shown in Figure 2 for temperature levels between 400°C and 700°C. At 400 °C, the coarse-grained microstructure exhibits a stress 55 MPa higher than the fine-grained microstructure, due to a more pronounced hardening. However, at 600 °C, this gap falls to 10 MPa, and at 700 °C no difference in the stresses has been observed between the two microstructures at 0.02 strain level. It could indicate a similar mechanical response for both microstructures beyond 650 °C. At 650 °C and 700 °C, the stresses are fully relaxed before the 30 min holding time, which is not the case at lower temperatures.

<figure><img src="image_2105.png" /><figcaption>Figure 2: Strain-Stress (a) and relaxation (b) curves for the FG microstructure</figcaption></figure>

Table 2 presents the elastic properties of the two microstructures, the Young's moduli E and the elasticity limits k. The latter are close to those reported by Luong et al. [14] who performed a measurement using a virtual extensometer based on DIC at high temperature. Other authors, such as Odenberger et al. [15], measured higher Young's modulus through an acoustic resonance analysis method (around 80 GPa at 700 °C). The elasticity limits presented Table 2 are estimated based on the method by Rollin et al. using the final stress during relaxation, minus the hardening [16]. For the elasticity limit, a Hall-Petch effect is present at the lower temperature, the fine-grained
For the relaxation test, the specimen is deformed to 0.02 deformation and relaxed for 30 min, deformation rate is set at $10^{-2}$ s$^{-1}$. The test is controlled by deformation using a contact extensometer, allowing a precise control over the deformation and deformation rate. 6 temperatures are tested, from 400 °C to 700 °C with the use of an AET furnace.

The second test is a tensile test under the same conditions of temperature where the specimen is deformed to rupture. Therefore, an extensometer cannot be used to measure the deformation in this case. Instead, a DIC method is implemented to acquire the displacement field between two images taken during the tests. In this study, logarithmic (Hencky) deformation is considered, and the software VIC-2D® is used for the DIC. In order to evaluate the displacement field, a speckle is applied on the surface of the specimen. A wide range of speckles specially developed for DIC can be found in the literature, from painting and laser marking to chemical etching and anodization [13]. For our test conditions, high temperature paints are used to ensure a resistant speckle even at 700 °C. To take account of the risk of the paint layer tearing due to the expected high deformation, no such layers are applied on the specimen. The paint is directly pulverized on the raw surface of the sample.

The samples are obtained by Water Abrasive Jet Cutting (WAJC) in the rolling direction, this study doesn't consider the anisotropy of the Ti-6Al-4V anisotropy.

## Experimental results

Relaxation tests: the load and relaxation curves obtained for the FG microstructure are shown in Figure 2 for temperature levels between 400°C and 700°C. At 400 °C, the coarse-grained microstructure exhibits a stress 55 MPa higher than the fine-grained microstructure, due to a more pronounced hardening. However, at 600 °C, this gap falls to 10 MPa, and at 700 °C no difference in the stresses has been observed between the two microstructures at 0.02 strain level. It could indicate a similar mechanical response for both microstructures beyond 650 °C. At 650 °C and 700 °C, the stresses are fully relaxed before the 30 min holding time, which is not the case at lower temperatures.

<figure><img src="image_2096.png" /><figcaption>Figure 2: Strain-Stress (a) and relaxation (b) curves for the FG microstructure</figcaption></figure>

Table 2 presents the elastic properties of the two microstructures, the Young's moduli E and the elasticity limits k. The latter are close to those reported by Luong et al. [14] who performed a measurement using a virtual extensometer based on DIC at high temperature. Other authors, such as Odenberger et al. [15], measured higher Young's modulus through an acoustic resonance analysis method (around 80 GPa at 700 °C). The elasticity limits presented Table 2 are estimated based on the method by Rollin et al. using the final stress during relaxation, minus the hardening [16]. For the elasticity limit, a Hall-Petch effect is present at the lower temperature, the fine-grained
microstructure having a higher elasticity limit than the coarse-grained one. Rollin reported an elasticity limit near zero for temperature higher than 650 °C with an initial 3 µm mean grain size Ti-6Al-4V alloy [16].

<table><caption>Table 2: Young's modulus and elasticity limit evolution with temperature for the two microstructures</caption><thead><tr><th rowspan="2">Temperature [°C]</th><th colspan="2">CG</th><th colspan="2">FG</th></tr><tr><th>k [MPa]</th><th>E [GPa]</th><th>k [MPa]</th><th>E [GPa]</th></tr></thead><tbody><tr><td>400</td><td>250</td><td>94.1</td><td>300</td><td>98.2</td></tr><tr><td>500</td><td>80</td><td>87.1</td><td>100</td><td>89.1</td></tr><tr><td>550</td><td>30</td><td>70.8</td><td>40</td><td>69.7</td></tr><tr><td>600</td><td>5</td><td>64.9</td><td>10</td><td>57.6</td></tr><tr><td>650</td><td>0</td><td>50.8</td><td>0</td><td>54.1</td></tr><tr><td>700</td><td>0</td><td>39.9</td><td>0</td><td>33.6</td></tr></tbody></table>

Tensile tests: The strain-stress curves are presented in Figure 3. They were obtained by extracting the maximal deformation value from the deformation fields at each time increment. Each experimental point on the graphs is related to one image taken with the camera, and one deformation field. As the speckle deforms with the specimen, high deformation induces a severe decrease in the contrast of the speckle, almost deleting any speckle for the highest deformation value. When deformation reaches a value of about 2, errors in the DIC become too high and decorrelation occurs at the center of the necking area so the measurement cannot be done up to the rupture. Stress-strain values beyond that point should be carefully considered.

<figure><img src="image_2106.png" /><figcaption>Figure 3: Tensile strain-stress curves for all temperatures at 1.10⁻⁴ s⁻¹ (a, c) and 1.10² s⁻¹ (b, d) for the FG and CG microstructure</figcaption></figure>
microstructure having a higher elasticity limit than the coarse-grained one. Rollin reported an elasticity limit near zero for temperature higher than 650 °C with an initial 3 µm mean grain size Ti-6Al-4V alloy [16].

<table><caption>Table 2: Young's modulus and elasticity limit evolution with temperature for the two microstructures</caption><thead><tr><th rowspan="2">Temperature [°C]</th><th colspan="2">CG</th><th colspan="2">FG</th></tr><tr><th>k [MPa]</th><th>E [GPa]</th><th>k [MPa]</th><th>E [GPa]</th></tr></thead><tbody><tr><td>400</td><td>250</td><td>94.1</td><td>300</td><td>98.2</td></tr><tr><td>500</td><td>80</td><td>87.1</td><td>100</td><td>89.1</td></tr><tr><td>550</td><td>30</td><td>70.8</td><td>40</td><td>69.7</td></tr><tr><td>600</td><td>5</td><td>64.9</td><td>10</td><td>57.6</td></tr><tr><td>650</td><td>0</td><td>50.8</td><td>0</td><td>54.1</td></tr><tr><td>700</td><td>0</td><td>39.9</td><td>0</td><td>33.6</td></tr></tbody></table>

Tensile tests: The strain-stress curves are presented in Figure 3. They were obtained by extracting the maximal deformation value from the deformation fields at each time increment. Each experimental point on the graphs is related to one image taken with the camera, and one deformation field. As the speckle deforms with the specimen, high deformation induces a severe decrease in the contrast of the speckle, almost deleting any speckle for the highest deformation value. When deformation reaches a value of about 2, errors in the DIC become too high and decorrelation occurs at the center of the necking area so the measurement cannot be done up to the rupture. Stress-strain values beyond that point should be carefully considered.

<figure><img src="image_2097.png" /><figcaption>Figure 3: Tensile strain-stress curves for all temperatures at 1.10⁻⁴ s⁻¹ (a, c) and 1.10² s⁻¹ (b, d) for the FG and CG microstructure</figcaption></figure>
For each combination of microstructure and deformation rate, an increase in temperature induces an increase in the strain at rupture and a decrease in true stress. The strain drops observed are not related to necking, but more likely to material damage. FG microstructure exhibits a higher deformation to rupture (Figure 4) and higher maximum stress in almost all cases compared to the coarse-grained microstructure. The viscoplastic effect activates for temperature starting at 500 °C for each microstructure, with no major impact at 400 °C. Strains at rupture between 0.92 and 1 were measured for the FG microstructure at 400 °C, whereas at 600 °C, a 0.8 strain difference between $1.10^{-2}$ s$^{-1}$ and $1.10^{-4}$ s$^{-1}$ has been evaluated. This difference is even higher considering that at $1.10^{-4}$ s$^{-1}$ a loss of correlation prevented the full acquisition of the strain level up to rupture.

<figure><img src="image_2107.png" /><figcaption>Figure 4: Strain to rupture for all temperature, strain rate and both microstructure (* last strain measured before loss of correlation)</figcaption></figure>

Tensile-relaxation tests dialogue: the two campaigns generate experimental data in two different ways: through an extensometer for the relaxation tests and through DIC for the tensile tests. The concordance between the two campaigns has to be pointed out, particularly in the low deformation part. Figure 5 presents a comparison between the two kinds of tests, defined by the same loading path up to 0.02 deformation. The tensile curves measured by DIC shown here are slightly different from the ones shown in Figure 3b, as the deformation used here is a mean deformation in the whole region of interest instead of the maximum deformation value measured. This allows to be less sensitive on the noise and therefore to have smoother curves. This is further supported as there is no localization of the deformation at this strain. This method was not used in the Figure 5 as it underestimates the strain levels; For the FG microstructure, it lowers the strain at rupture by 0.21 at 400 °C and by 0.39 at 700 °C for a strain rate of $1.10^{-2}$ s$^{-1}$.

The two campaigns present close similarities even with two completely different means of measurement. This further validate DIC as a robust method to acquire deformation, at least for low strain.
For each combination of microstructure and deformation rate, an increase in temperature induces an increase in the strain at rupture and a decrease in true stress. The strain drops observed are not related to necking, but more likely to material damage. FG microstructure exhibits a higher deformation to rupture (Figure 4) and higher maximum stress in almost all cases compared to the coarse-grained microstructure. The viscoplastic effect activates for temperature starting at 500 °C for each microstructure, with no major impact at 400 °C. Strains at rupture between 0.92 and 1 were measured for the FG microstructure at 400 °C, whereas at 600 °C, a 0.8 strain difference between $1.10^{-2}$ s$^{-1}$ and $1.10^{-4}$ s$^{-1}$ has been evaluated. This difference is even higher considering that at $1.10^{-4}$ s$^{-1}$ a loss of correlation prevented the full acquisition of the strain level up to rupture.

<figure><img src="image_2098.png" /><figcaption>Figure 4: Strain to rupture for all temperature, strain rate and both microstructure (* last strain measured before loss of correlation)</figcaption></figure>

Tensile-relaxation tests dialogue: the two campaigns generate experimental data in two different ways: through an extensometer for the relaxation tests and through DIC for the tensile tests. The concordance between the two campaigns has to be pointed out, particularly in the low deformation part. Figure 5 presents a comparison between the two kinds of tests, defined by the same loading path up to 0.02 deformation. The tensile curves measured by DIC shown here are slightly different from the ones shown in Figure 3b, as the deformation used here is a mean deformation in the whole region of interest instead of the maximum deformation value measured. This allows to be less sensitive on the noise and therefore to have smoother curves. This is further supported as there is no localization of the deformation at this strain. This method was not used in the Figure 5 as it underestimates the strain levels; For the FG microstructure, it lowers the strain at rupture by 0.21 at 400 °C and by 0.39 at 700 °C for a strain rate of $1.10^{-2}$ s$^{-1}$.

The two campaigns present close similarities even with two completely different means of measurement. This further validate DIC as a robust method to acquire deformation, at least for low strain.