Inelastic mechanical behaviour of an additively manufactured titanium alloy: a statistical continuum mechanics theory perspective

Statistical continuum mechanics theory was used to simulate the inelastic stress of polycrystalline materials using two-point statistics. For the experimental part, the Electron beam melting (EBM) technique (Arcam EBM Q10 additive machine) was used to fabricate cylindrical rods of Ti-6Al-4V both in horizontal and vertical directions. Electron backscatter diffraction (EBSD) technique was employed to achieve statistically reliable orientation maps of vertically and horizontally printed samples. In this study, high strain rate compression tests at six different strain rates were performed, and the stress–strain curves were generated. This work is amongst the first attempts to model the microstructure of additively manufactured hexagonal alloys under compressive loadings using the statistical continuum mechanics theory. The model is capable of simulating reasonably large microstructures (statistically representative) with a practical computational cost and accuracy, unlike numerical models that require a high computational cost. It should be noted that in additive manufacturing, due to large grains and high anisotropy, microstructures used in the simulations should be large enough to include sufficient information from the material’s structure. Therefore, using finite element models would be very challenging here. On the other hand, the statistical continuum mechanics theory uses the statistical representation of the material’s characteristics for solving the governing equations with Green’s function that enables this methodology to use more microstructure characteristic information without having a noticeable change to the computational cost. The proposed model in this study uses different microstructure characteristics such as crystal grain orientation, total slip systems, active slip systems, gain morphology, and chemical phases that are obtained from EBSD images for simulating the inelastic mechanical behavior of polycrystalline materials. Although this model simulates polycrystalline materials by considering various crystal and grain information, unlike numerical methods, it doesn’t simulate the grain interactions well and we cannot study local deformation and crack nucleation sites. This model works very well for simulating the overall behavior of material instead of each individual grain and failure analysis. This model has shown a good combination of computational cost and accuracy in which the error between the simulated and experimental strength for vertical and horizontal samples was 6.21% and 8.07%, respectively.


Introduction
ASTM defines additive manufacturing (AM) as 'a process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies' [1].AM methods can be classified based on the energy source (laser power, electron beam, arc, etc) and material feedstock (powder, filament, wire, sheet, liquid, rod), etc [2].Among different AM techniques, powder bed fusion (PBF) technology is considered as one of the primary techniques for printing metallic parts.The PBF technology can mainly be divided according to the energy source into the following techniques: Electron Beam Melting (EBM), selective laser melting (SLM), and selective laser sintering (SLS).In these techniques, the AM process occurs in an inert or vacuum atmosphere, and the deposited layer of powder is scanned and melted using an energy source.In the next step, the build plate is lowered by a distance equal to the powder layer thickness, followed by the deposition of another layer of powder.This procedure continues until the designed geometry is entirely printed [3].
Ti-6Al-4V is a dual-phase material (α+β) and the most extensively used titanium alloy due to its applications in marine, biomedical, aerospace, automotive, and offshore industries.This alloy has a high strength and superior corrosion resistivity [4].Despite the wide usage of this alloy, many challenges can arise in the production of Ti-6Al-4V parts because of its significant reactivity with oxygen and poor thermal conductivity; therefore, AM has attracted attention as the atmosphere is controllable and intense energy sources can overcome the poor conductivity problem [5].
Among the PBF techniques, the EBM technique has shown to be the most proper way to manufacture Ti-6Al-4V parts, as it operates in a vacuum chamber and avoids powder oxidation during printing.Three lenses (astigmatism, focus, and deflection) are utilized to focus and accelerate electron beams.Powder layers have a thickness of 50-150 μm and are deposited on a preheated stainless build plate (650-700 °C), which reduces internal stresses [6].
Understanding the relationship between microstructures and macro-properties (e.g., fracture, toughness, strength, etc) has been a remarkable interest in recent decades in order to control the macroscopic properties of materials by changing their microstructures [7][8][9][10][11][12].Polycrystals are used in many engineering applications, and there has been a huge interest in understanding, simulating, and predicting their mechanical behavior [13,14].The deformation behavior of polycrystals is dependent on the loading direction because of the elastic tensor anisotropy.This anisotropy originates from different crystallographic deformation mechanisms (such as dislocation, twins, martensitic transformation).Crystalline anisotropy leads to the directional dependency of all mechanical phenomena such as strength, strain hardening, and crystallographic texture, where in this case, the most famous mechanical properties are the stress values that are obtained through the uniaxial stress-strain test, which is a one-dimensional yield curve; however, it is a six-dimensional yield surface.For 3D polycrystalline structures, this behavior can be monitored by loading in different directions.
3D polycrystalline microstructures can be achieved by electron backscattering diffraction (EBSD) at different layers of a polycrystalline material [15].The 3D microstructure has local phase, grain size, and crystallographic texture information at the micro-level.To study the mechanical behavior of polycrystalline materials, crystal plasticity (CP) is a useful theory that models the response of each grain under deformation [16,17].Then, polycrystal plasticity correlates the grain behavior to the effective mechanical behavior of a polycrystalline material.The existed CP models have different computational costs and accuracies.The most popular methodologies are Taylor [17], self-consistent [18,19], and other models that homogenize the neighborhood of each grain [20][21][22].
Taylor model: In Taylor model [17], which was then followed by Asaro and Needleman [16], it's assumed that all grains in the medium have identical volumes with a uniform deformation gradient value within each grain.In this methodology, equilibrium exists in each grain but not among grains.The macroscopic effective stress in the polycrystalline material is determined by calculating the mean value of the Cauchy stress in the grains.This methodology does not consider any twinning, diffusion, and grain boundary sliding mechanisms.
Self-consistent models: The interaction of each grain with its surroundings directly affects the mechanical behavior of a polycrystalline material at the macro-level.Kroner [23] is among the first ones who worked on the grain interaction with its surrounding during plastic deformation.Hill [24] proposed an elasto-plastic method to predict the plastic behavior in the material, which Hutchinson [25] used this model to predict the elastovisco-plastic transition in the grains.To solve the problem of large plastic deformations, we can assume that a visco-plastic equation governs the stress equilibrium state by neglecting the elastic effects.This approach was used by Hutchinson [26] to model steady-state creep in polycrystals, where Molinari et al [27] used it to model the texture development of cubic polycrystals.
Crystal plasticity finite element (CPFE) models: These models consist of the different force equilibrium forms and the displacement compatibilities in each finite-volume element.CPFE is capable of simulating the entire sample with any geometry under loading by discretizing the sample into small elements.The main value of CPFE models is that they can deal with complicated internal/external boundary conditions.Also, these models have great flexibility to use different constitutive laws for stress flow.CPFE models can be used for both micro-level (such as inter-grain studies) and macro-level phenomena; however, they are very time-consuming and even depend on the CPFE solver (uniaxial test for a simple geometry with 81 grains takes between 60 h-150 h.) [28].Therefore, it is not feasible to deal with large polycrystalline aggregate.
Among the existing models, the crystal plasticity model works well in considering the grain crystal orientation as well as grain size and distribution in the microstructure for simulating the inelastic behavior.However, the computational cost even for small microstructures is significant as mentioned above.On the other hand, the existing statistical-analytical models are not accurate enough as they don't use all important microstructure characteristics in their algorithm.In this study, we used the Taylor model capability in simulating the mechanical responses at the grain level using crystal orientation, crystal system, and slip systems information.Then, used the mechanical response of each grain and imported them in the self-consistent model by knowing the location of each grain in the media to simulate the mechanical response of entire microstructure.In this way, we could consider the effect of grain orientation and active slip systems in the self-consistent model to have an accurate simulation of mechanical behavior of the polycrystalline materials.
In this study, a former study by Garmestani et al [29] was used in which statistical continuum mechanics theory was employed to simulate the inelastic behavior of a dual-phase medium.Moreover, the two-point statistics were employed to capture the spatial correlation for each phase in the medium.In the next step, the Green's function proposed by Molinari et al [27] was utilized to derive the final closed-form solution.Therefore, in this study, the mentioned formulations are expanded to model additively manufactured Ti64 alloys so that anisotropy in a two-phase material and finally N-phase material can be considered.

Single crystal behavior
The mechanical behavior of a polycrystalline material is influenced by the crystal structure and orientation of each individual grain.The response to mechanical loadings is determined by the slip systems associated with its crystal structure.In a study conducted by Molinary et al [21], a self-consistent approach based on polycrystal viscoplasticity was introduced to predict the behavior of polycrystalline materials under mechanical loadings.They utilized a power law to define the plastic shear rate s  g for slip system s, which was expressed as where s 0 t is the reference stress, 0  g is the corresponding reference shear strain rate, m represents the strain rate sensitivity, and s t denotes the resolved shear stress for a specific slip system s.The resolved shear stress s t is expressed as: where n b , s s are the normal and Burgers vector of slip system s, and S represents the deviatoric Cauchy stress tensor.The deviatoric strain rate is also defined as: where i j , n represents the velocity gradient.The deviatoric strain rate is a function of the microscopic shear strain rates as follows: By substituting equation (4) in equations ( 1) and (2), where elasticity is neglected, we arrive at the following equation that illustrates the relationship between the strain rate and the deviatoric Cauchy stress: It is important to note that in equation (5) s 0 t and r kl s are updated during deformation as a result of hardening and changes in grain orientation.Consequently, they reformulated the nonlinear equation (5) as following: Drawing inspiration from this relationship, they expressed the stress S as a function of the strain rate D in the following manner: Subsequently, they considered a tangent behavior for equation (7).They calculated the tangent behavior by first-order Taylor expansion for an applied strain rate D¢ as: where G S / ¶ ¶ is the Jacobian matrix of G(S) and S D kl 0 ( ) ¢ is expressed as: Finally, they got to the following formulation, which calculates the strain rate of each grain in a polycrystalline material: In this study, the aforementioned formulations were utilized to calculate the strain rate of each individual grain in response to mechanical loading.These calculated strain rates were then employed within the framework of statistical continuum theory to determine the effective mechanical property.

Statistical continuum mechanics methodology
Garmestani et al [29] utilized statistical continuum mechanics formulations to simulate the inelastic properties of a dual-isotropic-phase microstructure.In section 3.4.3,this model was further extended to account for anisotropic microstructures, allowing it to predict the inelastic properties in all directions [30].In this chapter, their model is expanded into a comprehensive framework that is applicable to N-phase polycrystalline materials.This expanded model allows for the efficient prediction of inelastic behavior across various material systems, offering faster results compared to other numerical methods.They employed the power-law steady-state creep equation to establish the relationship between the strain rate and stress for each phase, which can be expressed as follows: where D is strain rate, s is stress, n is the inverse strain rate sensitivity, D* and σ * are the reference configuration for each phase.In the same study by Garmestani et al [29], these calculations were initially performed on a twophase medium.However, to extend this formulation to polycrystalline materials, where there are N different phases, the strain rate needs to be calculated for each phase.As mentioned, equation (11) calculates the strain rate for each specific grain under a given mechanical load, which can then be substituted into equation (12).In the following, Garmestani et al [29] introduced Cauchy stress tensor as: which N denotes inelastic modulus, D is strain rate and p is hydrostatic pressure.The velocity gradient, which is the derivative of velocity L ij i j , ( ) = can be decomposed into two parts: asymmetrical part, D kl (strain rate), and antisymmetrical part, w kl (rotation rate).Also, as N is a symmetric tensor, the following relation is achieved [29]: Then, by substituting equation (15) in equation (13): In the following, incompressibility condition is expressed as: )is a spatially dependent part.In the following, they used Green's function to introduce the relation of velocity gradient and inelastic modulus.We used the same strategy for each grain instead of each phase as follows: ñ shows the velocity gradient for grain h .i Consequently, N L h , ,KN is grain index) is the ensemble average of polarized modulus for the velocity gradient of grain h .
i L ¯denotes macroscopic homogeneous velocity gradient and G r r is the conditional probability and defined as p r h r h V, which p is absolute probability and V i is the volume fraction of grain h .
i The absolute probability p is calculated using two-point statistics, which gives us the spatial information of grains.Finally, they derived the Cauchy stress as follows: By integrating the single crystal plasticity model with statistical continuum mechanics formulations, it becomes possible to simulate the deformation behavior of a polycrystal.This approach offers the advantage of low computational cost while still accommodating inputs with a high level of detail, addressing a key concern in numerical approaches.Furthermore, this model takes into account the hardening effect that occurs during deformation, and it can be easily adapted to incorporate any new hardening law as needed.

EBM process and EBM-Ti64 rods fabrication
As mentioned earlier, the Ti-6Al-4V alloy is categorized among oxygen-sensitive alloys; therefore, the EBM process performs in a vacuum chamber preheated up to 650 °C-700 °C to avoid martensite a ¢ and internal stress formation.The schematic of the EBM process can be seen in [31].
Using the EBM technique (Arcam EBM Q10 additive machine), cylindrical parts (9.5 mm × 120 mm in diameter and length, respectively) were fabricated in two horizontal and vertical directions, where in horizontal and vertical samples, the building direction is perpendicular and parallel to the longitudinal axis of the rods.The process parameters used in fabricating the mentioned rods are the maximum beam power of 3000 W, layer thickness of 50 μm, beam spot size of 450-500 μm, hatch spacing of 0.2 mm, and a scan speed of 800 mm sec −1 and 4500 mm sec −1 in contour and hatch, respectively.In addition, each layer was oriented 46˚relative to the former layer on a heated bed (650-700 ˚C) in a vacuumed chamber (5 × 10 −4 mbar).

Microstructural observation
To examine the texture of the samples, a standard procedure was followed.This procedure involved initially cutting the samples, as illustrated by the hatched surfaces in figure 1.The next step involved polishing using polishing cloths (320, 500, 800, 1200, 2400, and 4000) and then achieving mirror-like surface using a vibratory polishing machine.The results of electron backscatter diffraction (EBSD) characterization for the undeformed samples are shown in figure 2.
Based on the phase maps provided in the supplementary document, the α-phase volume fraction is determined to be 96% for the vertically printed sample and 100% for the horizontally printed sample.Since the β-phase volume fraction is negligible compared to the α-phase, all subsequent interpretations will focus on the α-phase.
The textures of both the vertically and horizontally printed samples exhibit elongated prior β-grain boundaries along the building direction.This directional growth of prior β-grain boundaries aligns with the directional thermal gradient present during the printing process.In Electron Beam Melting (EBM), the parts are surrounded by deposited powders that act as an isolation layer, allowing heat to dissipate either from the preheated build plate or the top layer.
Additionally, the vertical sample displays a finer microstructure and thinner prior β-grains compared to the horizontal sample.This can be attributed to the higher cooling rate experienced by the vertical sample.The inverse pole figure (IPF) maps for both sets of samples do not exhibit a clear preferential orientation.However, it appears that a basal texture is slightly more favored in the samples.

High strain rate tests
The compression tests were conducted on samples that were machined from the original printed rods, with dimensions of 9.5 mm × 10.4 mm.The Split-Hopkinson Pressure Bar (SHPB) equipment was employed for these tests.Initial data was obtained using a differential amplifier and a high-speed oscilloscope, allowing for the calculation of nominal strain, strain rate, and stress values.
To account for any necessary corrections, the nominal values obtained from the experiment were converted to true values using the equations provided by Ramesh et al [32].These equations enable the adjustment of the nominal values to their corresponding true values, ensuring accuracy in the data analysis.
According to figure 3 it can be observed that both the horizontal and vertical samples exhibit an increase in yield strength (YS), ultimate compression strength (UCS), and total strain as the strain rate is increased.The flow curves demonstrate different behaviors at different strain rates.At higher strain rates, there is a slight decrease in stress, indicating the dominance of thermal softening over strain hardening due to the rise in temperature during deformation.However, as the deformation progresses, strain hardening becomes the predominant mechanism, leading to an increase in stress.Table 1 provides a detailed overview of the yield strength (YS) and ultimate compression strength (UCS) values obtained from the stress-strain curves in figure 3.This table presents specific data points that capture the strength characteristics of the samples under varying strain rates.
Figure 4 represents the EBSD images of the samples used in this study.In the first step, we needed to characterize these EBSD microstructures.In this regard, as each pixel carries three Euler angles and one chemical phase information, if all these four values were identical between two pixels, then they are assumed as one phase.In this way, due to the broad range of Euler angles, specific bin sizes are defined, and pixels that fall into them are recognized as the same phases.The larger the bin size, the less distinct phases are defined.Therefore, there is a trade-off between bin size (which affects the number of distinct phases) and computational cost.For example, figure 4 illustrates the distinct phases for a bin size of 15°and 30 o in vertical and horizontal samples.According to figures 4(b), (c), the number of distinct phases with the bin size of 15 o and 30 o for the vertical sample is 1800 and  300, respectively.Also, for the horizontal sample, the number of distinct phases with the bin size of 15 o and 30 o is 1000 and 180, respectively (figures 4(e), (f)).
It should be noted that the computational cost only for calculating the two-point statistics for an N-phase It also adds additional computational cost of calculating strain rate and effective dissolved stress in each phase.In this study, both bin sizes of 15 o and 30 o are used to see how the model's accuracy is affected by the bin size.After the distinct phases are recognized in the next step, the two-point statistics are calculated for the microstructures b, c, e, and f in figure 4. The two-point statistics are expressed as [33]: where s S 1, 2, , = ¼ denotes the total pixels in the media, h represents the distinct local phases in the microstructure, m s h is the volume fraction of local phase h in pixel s, and f r hh¢ represents the probability of finding the local state h at the tail and the local state of h¢ at the head of a vector, r placed randomly in the microstructure.Two-point statistics carry a comprehensive set of statistical information of the microstructure [34,35].For instance, (r =2) vector in the autocorrelation of local phase h, represents the volume fraction of the local phase h.Moreover, two-point statistics enclose all size and shape distributions of each local phase of the microstructure [34].Another important feature of the two-point statistics is capturing anisotropy patterns of the microstructure.
As an example, figure 5 represents the autocorrelation of phase 1 as well as cross-correlation of phases 1 and 2 with the bin size of 30 o .
As previously mentioned, two-point statistics provide valuable information about the microstructure.In figures 5(a) and (b), the center of the two-point statistics map represents the volume fraction of phase 1 in both the vertical and horizontal samples.Specifically, the volume fractions are measured as 0.18 10 4  ´for the vertical sample and 0.07 for the horizontal sample.Furthermore, two-point statistics offer insights into feature sizes and their distribution within the microstructure.By comparing figures 5(a) and (c), it can be concluded that the feature size in the horizontal sample is approximately 40 μ, while in the vertical sample, it is around 5 μ (the axis values in the autocorrelation maps indicate the feature sizes).It's important to note that these maps only represent phase one, and the vertical sample contains 300 phases, while the horizontal sample contains 180 phases.The two-point statistics analysis provides a detailed understanding of feature sizes and their distribution within the specific phases of the microstructure.
In the subsequent analysis, the single crystal formulations described in section 2.1 were employed and applied to the microstructures depicted in figure 4. The goal was to calculate the resulting strain rate for each microstructure under different mechanical loadings.As equation (4) indicates, the resolved shear stress for a specific slip system s is determined in order to determine the local strain rate within each grain.
For the EBM-Ti64 samples studied in this research, the microstructures consisted of over 96% volume fraction of the α-phase, which possesses a hexagonal close-packed crystal system.This information regarding the predominant phase and its crystal system is crucial for the subsequent calculations of the strain rate within the grains.
Based on the two main Burgers vectors in HCP materials, which are along the 〈a〉 (〈0001〉) and 〈c+a〉 (〈1123 ¯¯〉) directions, slip systems can be specified.As can be seen in figure 6 from the author's previous study on the HCP materials slip systems, 〈a〉 Burgers vector is on the basal (0001), prismatic (11 ¯00), and 1st order pyramidal (11 ¯01) and (11 ¯02) slip planes, and 〈c+a〉 Burger vector is on the prismatic (11 ¯00), 1st order pyramidal (101 ¯1) and (211 ¯¯1), and 2nd order pyramidal (112 ¯2) slip planes [36].According to table 2, which shows the number of active slip systems at room temperature, the basal and 〈a〉-type prismatic slip planes provide four slip planes, while in order to obtain a plastic deformation, five independent slip planes are required; therefore, the importance of 〈a〉and 〈c+a〉-type pyramidal slip systems with four and five active slip planes at room temperature becomes clear [36].
In addition to slip deformation, twin deformation is another important mechanism that needs to be taken into account.Twins play a significant role in the overall deformation behavior, primarily due to their capability to accumulate deformation along the c-axis.
Twin deformation can be classified into three categories: extension twins, contraction twins, and double twins.The specific planes and directions associated with each type of twin deformation are provided in table 2 [37].Depending on the stress conditions, different types of twins may be activated.Table 2. Available slip system in HCP system at room temperature [36].

Slip plane
Plane/Direction Number of active planes at room temperature It is worth noting that up to 5 twin systems can be activated simultaneously, working in conjunction with slip systems to satisfy the Von-Mises criteria for plastic deformation [36].The activation of twin systems in conjunction with slip systems allows for a more comprehensive understanding of the deformation behavior observed in the material.
In the following, equation ( 11) is utilized to calculate a 3×3 strain rate tensor for each distinct phase present in the material.This equation provides a comprehensive representation of the strain rate within the polycrystalline structure.
As an illustrative example, figure 7 displays the resulting components of the strain rate tensor in the horizontal sample subjected to a mechanical loading with a strain rate of 157 s −1 .This figure highlights the anisotropy present in polycrystals and demonstrates how the texture of the material influences the distribution of the load within the microstructure.The visualization of the strain rate components provides valuable insights into the local deformation behavior and the role of texture in influencing the mechanical response of the material.
Subsequently, all the collected strain rate information for each phase was utilized in section 2.2 to calculate the velocity gradient, inelastic modulus, and, ultimately, the Cauchy stress.In equation (21), the Cauchy stress is obtained by summing over all the phases present in the microstructure.
An essential parameter in this study is the inverse strain rate sensitivity, denoted as n, which plays a crucial role in equation (12).The strain rate sensitivity is a material-specific parameter and is determined based on the stress-strain data obtained at different strain rates.In this study, different values of inverse strain rate sensitivity, namely n = 4, 6, 8, and 10, were employed.The resulting strength values were then plotted and compared to experimental data, as depicted in figure 8.The bin size used for this comparison was set to 30 o .By comparing the simulated strength values with experimental results, the accuracy and effectiveness of the chosen inverse strain rate sensitivities can be assessed.This analysis aids in validating the material model and provides insights into the appropriate selection of the strain rate sensitivity parameter for future simulations.
Based on the analysis of figure 8, it can be observed that the inverse strain rate sensitivity value of n=6 yielded the lowest error when compared to experimental data.This indicates that the model's predictions aligned most closely with the observed behavior when using this particular value.
In the literature, it has been reported that the strain rate sensitivity of the α phase in Ti-6Al-4V varies within the range of 0.1 to 0.3, depending on the temperature and strain rate conditions.These values reflect the sensitivity of the material's response to changes in the applied strain rate.Considering this information, the selection of n = 6 as the inverse strain rate sensitivity in this study appears to be reasonable, as it falls within the reported range for Ti-6Al-4V [38].In this study, it was found that the inverse strain rate sensitivity value of 6 yielded the lowest error when compared to experimental data.This implies that the corresponding strain rate sensitivity is 1/6, which is approximately equal to 0.166.
To further investigate and determine the actual strain rate sensitivity in this study, the following equation was utilized [39]: where s is the flow stress at a constant strain and deformation temperature (MPa) and D is the strain rate (s −1 ).By calculating the logarithmic ratios of stress rates and strain rates at different strain rate conditions, the strain rate sensitivity value specific to this study can be determined.This approach allows for a more precise characterization of the material's response to changes in strain rate, facilitating accurate modeling and prediction of the mechanical behavior of the material.According to figure 9, the strain rate sensitivity results obtained using equation (12) are m=0.1537(corresponding to an inverse strain rate sensitivity of n=6.5) for the vertical sample and m=0.1584 (corresponding to an inverse strain rate sensitivity of n=6.31) for the horizontal sample.These values are consistent with the stress results shown in figure 8, where the inverse strain rate sensitivity of n=6 (corresponding to m=0.1667) resulted in the lowest error.
It is worth noting that the strain rate sensitivity for the horizontal sample is slightly higher than that of the vertical sample.This difference can be attributed to the slight variation in the volume fraction of the α-phase between these two samples.The composition and microstructure variations can influence the material's response to strain rate, resulting in subtle differences in the strain rate sensitivity values.
Overall, the strain rate sensitivity values obtained align well with the stress results and provide further insight into the material's behavior under different loading conditions.These findings contribute to a better understanding of the mechanical response of the material and enhance the accuracy of the predictive models used in this study.
In order to assess the impact of using an average inverse strain rate sensitivity value, the value of n=(6.31+6.5)/2=6.4 was employed.This value was used to calculate the resulting Cauchy stress for different strain rates in both the vertical and horizontal samples.The comparison between the experimental ultimate compression strength (UCS) and the resulting Cauchy stress for different strain rates is presented in table 3.
The results in table 3 are shown for two different bin sizes, 30 degrees, and 15 degrees, and demonstrate the effect of bin size on the accuracy of the model.For the vertical sample, the average error for a bin size of 30 degrees is 7.87%, while for a bin size of 15 degrees, it reduces to 5.67%.Similarly, for the horizontal sample, the average error decreases from 12.57% to 6.85% when the bin size is reduced from 30 degrees to 15 degrees.
These results highlight the significance of bin size in achieving accurate predictions.The reduction in the bin size from 30 degrees to 15 degrees leads to a decrease in the average error, indicating improved agreement between the model predictions and experimental data.Moreover, table 3 illustrates that even with a bin size of 30 degrees, the model can yield low errors when appropriate material properties are employed.
Overall, these findings emphasize the importance of selecting an appropriate bin size and utilizing suitable material properties to achieve accurate results in modeling the mechanical behavior of the material.

Conclusion
In this study, the mechanical behaviour of additively manufactured polycrystalline materials was investigated.Cylindrical specimens were fabricated using the EBM technique, and their microstructures were characterized using EBSD.Elevated and high strain rate compression tests were conducted at different strain rates to obtain stress-strain data.To simulate the mechanical behaviour of the material, a visco-plastic model combining single crystal behaviour and statistical continuum mechanics theory was employed.Different inverse strain rate sensitivity parameters were tested, and it was found that an inverse strain rate sensitivity of 6 resulted in the lowest error compared to other values tested (4, 8, and 10).Experimental data revealed an inverse strain rate sensitivity of 6.32 for the vertical sample and 6.5 for the horizontal sample.Based on these findings, an inverse strain rate sensitivity of 6.4 was chosen for further simulations.The resulting stress-strain predictions using this parameter showed an average error of 6.21% for the vertical sample and 8.07% for the horizontal sample, when using a bin size of 15 degrees.This modeling approach demonstrates the capability to simulate large microstructures with over 2000 individual grains, providing a balance between computational cost and accuracy.It offers a unique method for studying the mechanical behaviour of different materials systems in a cost-effective and efficient manner.

Figure 1 .
Figure 1.Schematic of the samples showing the loading and building directions and the surfaces used for texture studies (hatched surfaces).

Figure 2 .
Figure 2. EBSD results for the initial texture of the (a) vertical and (b) horizontal sample.(c) Represents the IPF color map for the EBSD microstructures in (a) and (b).

Figure 3 .
Figure 3. Compression test results of the samples studied in this paper.

Figure 4 .
Figure 4. EBSD microstructure representation for (a) vertical sample and (d) horizontal sample; Distinct phase representation with a bin size of 15 o for (b) vertical sample and (e) horizontal sample; Distinct phase representation with a bin size of 30 o for (c) vertical sample and (f) horizontal sample.

Figure 5 .
Figure 5. Examples of two-point statistics for the bin size of 30 o ; autocorrelation of phase 1 in the vertical sample (a) and horizontal sample (c); cross-correlation of phase 1 and 2 in the vertical sample (b) and horizontal sample (d).

Figure 6 .
Figure 6.Schematic of the slip systems in the HCP structure [40].

Figure 7 .
Figure 7. Representation of strain rate tensor components in the microstructure for the horizontal sample under loading with a strain rate of 157 s −1 with the bin size of 30 o ; (a) D , 11 (b) D , 12 (c) D , 13 (d) D , 21 (e) D , 22 (f) D , 23 (g) D , 31 (h) D , 32 (i) D .

Figure 8 .
Figure 8. Resulted stress values for both samples at different strain rates.

Figure 9 .
Figure 9. Strain rate sensitivity calculations for the vertical and horizontal samples.

Table 1 .
Compression test results extracted from figure 3.