Cyclic mechanical response of LPBF Hastelloy X over a wide temperature and strain range: Experiments and modelling

Additive manufacturing (AM) of high-temperature alloys through processes such as laser powder bed fusion (LPBF) has gained significant interest and is rapidly expanding due to its exceptional design freedom, which enables the fabrication of complex parts that contribute to the increased efficiency of aerospace and energy systems. The materials produced through this process exhibit unique microstructures and mechanical properties, which necessitate dedicated study and characterization. In this context, our research focuses on the experimental characterization of the isothermal cyclic viscoplastic mechanical response of Hastelloy X (HX) over the temperature range of 22 to 1000 ° C and at various strain rates, addressing a current gap in the literature. Recognizing the need for material models that can accurately represent the cyclic mechanical response of LPBF HX across a broad temperature range, we developed a robust extension of the viscoplastic isotropic-kinematic hardening Chaboche model, intended for applications in the thermomechanical simulation of the LPBF process for the analysis of residual stress and distortion, as well as for assessing the mechanical integrity of LPBF components. The extension involves expressing the entire set of model parameters explicitly with analytical functions to account for their temperature dependence. Consequently, the model includes a relatively large number of parameters to represent the isotropic-kinematic hardening viscoplastic response of the alloy over a wide temperature range, and hence to overcome the endeavor of its systematic calibration, a dedicated calibration approach was introduced. The model ultimately demonstrated its capability to precisely represent the isothermal response of the alloy over the examined temperatures and strain rates. To evaluate the model’s predictiveness for non-isothermal conditions, out-of-phase thermomechanical cyclic experiments were also conducted as independent benchmark tests, where the model’s predictions were fairly consistent with the experimental results. As a part of this study, the derived material model has been integrated into the UMAT subroutine, complete with an analytical derivation of the consistent Jacobian matrix.


Introduction
Hastelloy X (HX) is a solid-solution-strengthened nickel-based superalloy with application in high-temperature components of gas turbines, jet engines, and chemical reactors.Its balanced contents of chromium, molybdenum, and iron result in outstanding hightemperature strength and excellent resistance against oxidation, nitriding, and carburizing (Mertens, 2018).
Additive manufacturing processes such as laser powder bed fusion (LPBF) enable the fabrication of complex geometries with internal features that cannot be realized through conventional manufacturing.This technology is particularly valuable for the high-temperature sector, e.g., for the fabrication of parts with embedded internal cooling channels having complex geometries.For this reason, its application using high-temperature alloys such as HX has gained increasing attention (Harrison et al., 2015;Han et al., 2019;Pilgar et al., 2022).
Importantly, the microstructure of LPBF alloys is substantially different from that of the conventionally manufactured variants (see Fig. 1(a)).Rapid solidification and cooling during LPBF (10 3 -10 6 K/s) result in a very fine microstructure (Gu et al., 2012;Herzog et al., 2016).For LPBF HX, Montero-Sistiaga et al. (2019) reported cellular microscale structures which are embedded in macroscale columnar grains, grown epitaxially in the build direction throughout several deposition layers, as shown in Fig. 1(a).Also, a much higher dislocation density, as shown in Fig. 1(c), and finer precipitates have been observed for LPBF HX (Ni et al., 2019).The combination of these features consequently leads to a different mechanical behaviour for LPBF HX compared with the conventionally manufactured HX, for example, significantly higher yield strength in the LPBF variant as reported by Li et al. (2023).
A reliable representation of the stress-strain response of such materials through robust constitutive models is essential for assessing the mechanical integrity of high temperature components.Moreover, such constitutive models are crucially required for LPBF alloys in the context of thermomechanical simulation of the LPBF manufacturing process, where the reliable representation of the stress-strain response plays a key role in accurately predicting residual stress development in the parts.
Rapid thermal cycles during the LPBF process induce high residual stresses and distortions, which may lead to cracking or failure of the printing process (Ming et al., 2019;Fang et al., 2020).To develop strategies for mitigating residual stress and increasing the likelihood of high-quality manufacturing, numerous studies have focused on thermomechanical modelling of the LPBF process, e.g., Chen et al. (2019), Walker et al. (2019), Tan et al. (2019), Xiao et al. (2020), Song et al. (2020).In the absence of better alternatives, most of these studies resort to highly simplified constitutive models.For instance, while the material experiences rapid cyclic strains, the adopted constitutive models are often based on rate-independent plasticity with or without an isotropic hardening consideration.This ignores important aspects such as kinematic hardening and rate dependence arising from the viscoplastic nature (Promoppatum and Rollett, 2021) which is relevant for LPBF HX at temperatures above 600 °C (Li et al., 2023).These aspects should be accounted for in models, underpinned by appropriate and relevant experiments, i.e. across various temperatures and strain rates under cyclic loading conditions.
The existing knowledge on the mechanical response of LPBF HX for such purposes is however limited, as previous studies primarily focused on the alloy's room temperature properties (Mertens, 2018;Montero-Sistiaga et al., 2019;Ni et al., 2019;Sanchez-mata et al., 2021;Esmaeilizadeh et al., 2020).Exceptions are the studies of Montero Sistiaga (2019), Sanchez-Mata et al. (2020), Pilgar et al. (2022), and Li et al. (2023) which additionally explored the tensile response of LPBF HX at elevated temperatures with ranges/values of 650-850, 750, 200-750, and 200-800 °C, respectively. Furthermore, Lindström et al. (2020) investigated the cyclic response of a number of LPBF nickel-based superalloys, including LPBF HX, for a temperature range of 200-600 °C.However, considering the alloy's application field and its exposure to extreme thermomechanical histories during the LPBF process, there remains a notable absence of systematic investigation into its iso-and non-isothermal mechanical behaviour over a broader range of temperatures in the existing literature, with particular emphasis on examining its cyclic viscoplastic response.To address this need, an experimental campaign was conducted, including a set of dedicated isothermal, strain-controlled tests at temperatures of 22,200,400,600,800,900, and 1000 °C, and strain rates of 0.02, 0.1, and 0.5% s −1 , with and without strain-controlled holds at peak/valley strains, as well as a few out-of-phase, non-isothermal cyclic tests over the temperature range of 200 to 1000 °C.
The analysis of the generated experimental data is followed by the development of an advanced constitutive material model, applicable both in the mechanical integrity assessment of high-temperature components and the thermomechanical simulation of the LPBF process.The model is based on the well-accepted Chaboche type constitutive models (Lemaitre and Chaboche, 1990), which, while inspired by physical perspectives on the evolution of the dislocation microstructure during deformation, do not specifically account for detailed deformation mechanisms such as dislocation glide, climb, or cross-slip to maintain a simpler formulation and improve computational efficiency, i.e., in comparison to dislocation-based crystal plasticity models.Variants of the Chaboche model allow for the consideration of isotropic/kinematic hardening in a thermodynamically consistent manner (Chaboche, 1989(Chaboche, , 1993(Chaboche, , 2008)).Isotropic hardening or softening describes an expansion or contraction of the yield surface, while kinematic hardening describes its translation within the stress space.Chaboche (1986) employed a non-linear hardening rule, first introduced by Armstrong and Frederick (1966), for describing the hardening responses.The Chaboche model has then gone through several modifications and extensions by others to improve for representing more complex loading scenarios, e.g., by Bari and Hassan (2002), Yaguchi et al. (2002a), Ahmed et al. (2016), Ahmed and Hassan (2017), Zhang and Xuan (2017), Barrett and Hassan (2020), Bartošák et al. (2020), and Bartošák and Horváth (2024).Particularly relevant for this study, viscoplastic variants of the Chaboche model address rate-independent and rate-dependent plasticity in a unified manner and inherently account for plasticity-creep interactions.
For the application of such constitutive models to non-isothermal conditions, originally, Chaboche (1989) introduced an additional term into the Armstrong-Frederick rule to incorporate the effect of temperature variation into the hardening response.Additionally, a critical aspect of applying the model to non-isothermal conditions involves accounting for temperature effects on the model parameters.Traditionally, parameters are identified for several temperatures using isothermal experimental data, followed by linear interpolation to estimate the model parameters at intermediate temperatures.This method, however, results in an extensive list of model parameters, introduces uncertainties due to the application of linear interpolation in inherently nonlinear contexts, and can cause convergence issues in finite element simulations of non-isothermal loadings -issues that stem from the non-monotonic and abrupt changes in model parameters with temperature variations, as highlighted by Morch et al. (2021).In response, previous works suggested a more reliable approach that involves considering predefined trends and formulation for temperature-dependence of model parameters, e.g.polynomial functions (Yaguchi et al., 2002b), exponential (Hosseini et al., 2015) and double-exponential functions (Morch et al., 2021).This approach necessitates taking an inverse analysis for model calibration through the concurrent analysis of experimental data across a range of temperatures for determining all model parameters, a task that is often found to be cumbersome.

Table 1
Nominal chemical composition of HX in comparison with that of the supplied powder by Oerlikon AM, Germany (weight %) (Haynes, 1997;Oerlikon Metco, 2022 In this study, we introduce an extension of the viscoplastic Chaboche material model suited to represent the cyclic mechanical behaviour of LPBF HX across the temperature range of 22 to 1000 °C and strain rate range of 0.02 to 0.5% s −1 , aiming at establishing a robust constitutive model with applications for thermomechanical simulations of the LPBF process as well as for the mechanical integrity assessment of LPBF HX components.The main contributions of our study are: • experimentally assessing the cyclic elastic-viscoplastic response of LPBF HX over a wide range of temperatures and strain rates, • extending the viscoplastic Chaboche model by explicitly incorporating temperature effects into the entire set of model parameters (using a combination of exponential and sigmoid functions).This eliminates the need for separate parameter sets for each temperature, allowing a single set of parameters to be applicable across the entire temperature range, • introducing a dedicated calibration strategy for the model to systematically determine its 39 model parameters, • demonstrating the representativeness and predictive capability of the model by comparing its outcomes with the results of 28 isothermal tests and three independent non-isothermal benchmark tests, • implementing the constitutive model into the Fortran-based UMAT subroutine for integration into the finite element solver of ABAQUS.

Material and LPBF process
A commercially available, gas-atomized HX powder was used in this study.The powder is provided by Oerlikon AM and has a chemical composition that is presented in Table 1.Cylindrical bars with diameters and lengths of 8 mm and 86.3 mm, respectively, were fabricated using a Sisma MySint 100 (Sisma S.p.A., Italy).The machine is equipped with a 200 W 1070 nm-fiber-laser operating in continuous wave mode and featuring a Gaussian intensity distribution with a 55 μm spot size.The cylinders were built vertically on a 100 mm diameter stainless steel build plate with the following LPBF parameters: 150 W laser power, 1000 mm s −1 scanning speed, 100 μm hatch spacing, and 20 μm layer thickness.No build plate preheating was applied.A bidirectional scan strategy was employed, featuring a 90°rotation between layers.Flowing argon shielding gas with 99.996% purity ensured that the oxygen content in the build chamber did not exceed 0.1% O 2 .All samples were intended to be printed in a single build job to reduce variability among them.Therefore, the diameters of the bars had to be limited to 8 mm to fit them on the build plate, which therefore allowed extraction of M7 testpieces.To maintain a high ratio between the cross-section of the testpieces in the thread and the gauge section, the gauge diameter was reduced to 4 mm.Considering the 15 mm gauge length of the extensometer of the testing machine, the testpieces gauge length was set at 20 mm.Therefore, cylindrical dog-bone testpieces, according to the drawing provided in Fig. 2, were designed to be machined from the manufactured bars for mechanical testing.No post-process heat treatment was applied to the testpieces.

Isothermal mechanical experiments
Isothermal mechanical experiments were conducted using an MTS servohydraulic universal testing machine with a 100 kN load capacity.For the isothermal experiments, the machine was equipped with a closed-loop, two-zone resistance furnace.Two K-type thermocouples were spot-welded to monitor the temperature along the parallel length of the testpiece, each positioned 8 mm from either side of the midpoint.The signals from the thermocouples were used to control the sample gauge temperature within ±0.5 °C of the target temperature throughout the entire test.A Class 0.5 side-entry extensometer, with a datum leg spacing of 15 mm, was used for the control and measurement of axial strain.A calibrated integral load cell was employed for measuring the axial load acting on the specimen.Fig. 3(a) illustrates the testing setup for the isothermal mechanical experiments.
A total of 28 isothermal experiments were carried out to characterize the cyclic mechanical response of LPBF HX at temperature and strain rate ranges of 22-1000 °C and 0.02-0.50%∕s, respectively (Table 2).Each experiment comprised five strain-controlled cycles up to a strain value of 0.75%, with or without a 15 min constant-strain dwell at peak and valley strains, as depicted in Fig. 3(b).Incorporating straincontrolled dwell times into the loading cycles facilitates the direct characterization of the alloy's viscoplastic response.By opting for five cycles with a strain amplitude of 0.75%, our objective was to characterize the alloy's behaviour under relatively large strain amplitudes akin to those experienced during the LPBF fabrication process, and enabling the analysis of various stages of strain hardening without a risk of testpiece buckling.In principle, conducting mechanical experiments at even higher temperatures and strain rates was desirable, aiming to more accurately capture the constitutive behaviour of the alloy under conditions mirroring those encountered during the LPBF process, i.e., temperatures close to the melting point and strain rates up to 400% s −1 (Promoppatum and Rollett, 2021).Unfortunately, the limitations of our experimental setup precluded such experiments.
It is worth mentioning that some degree of anisotropy in the mechanical behaviour of LPBF HX is anticipated and important to consider.Keshavarzkermani et al. (2019), under relatively similar LPBF conditions to ours, observed 9-10% higher yield strength for orientations perpendicular to the build direction compared to those parallel.A systematic exploration of this anisotropic behaviour across the specified temperature and strain rate spectrum in this study would necessitate the fabrication and examination of specimens oriented at various angles relative to the build direction.However, such an exploration would have significantly broadened the scope of our experimental endeavors and hence was not pursued.

Non-isothermal mechanical experiments
The experimental setup used for non-isothermal tests was an extension of the one used for isothermal experiments.To accelerate the  heating and cooling of the samples, the resistive heating system was replaced with an induction heating system, combined with a compressed air blowing mechanism (see Fig. 4(a)).To ensure precise temperature control, three K-type thermocouples were spot-welded along the parallel length of the samples-one in the middle and two positioned 8 mm from the centre on each side.The central thermocouple controlled the temperature with an accuracy of ±0.5 °C throughout the test.Before initiating each test, signals from the two side thermocouples were used to monitor the temperature gradients within the parallel length of the sample during thermal cycles.The arrangement of the induction heating and air cooling systems was optimized for each individual sample prior to the initiation of the actual test, ensuring a temperature uniformity of at least ±3 °C along the sample's parallel length.No corrective measures were taken during the course of the test when the temperature uniformity occasionally exceeded the ±3 °C limit.
The designed temperature and mechanical strain cycle for the nonisothermal experiments are depicted in Fig. 4(b).In all three conducted tests, the temperature cycled between 200 °C and 1000 °C at a rate of approximately 4 °C s −1 .The mechanical strain ranges applied for the three tests varied between 0.75%, 1.125%, and 1.5%, with a 90 • phase-shift relative to the temperature cycle (i.e., an out-of-phase thermomechanical cycle).
Control of the mechanical strain during the non-isothermal cycles requires consideration of a compensation function for the thermal strain.In adherence to the ISO12111 (2011) standard, a pre-test free thermal expansion measurement was performed prior to the experiments to calibrate a second-order polynomial temperature-based thermal strain compensation function.The mechanical strain was then calculated as the difference between the total strain measured using the side-entry extensometer and the thermal strain calculated from the compensation function.The accuracy of this compensation strategy was validated before the start of each test, through ensuring minimal stress development during a pre-test temperature cycle when the mechanical strain was kept at zero.

Experimental results
The following presents an overview of the experimental results generated in this study, while complete records of all the isothermal and non-isothermal experiments are made available as a MATLAB database in the Supplementary Materials.

Isothermal mechanical experiments
The isothermal mechanical experiments aimed to characterize the cyclic elastic-viscoplastic response of LPBF HX at different temperatures and strain rates.Figs. 5 to 8 and Fig. 11 in Section 4 provide examples of the experimental results.In general, the observed stress response during the imposed deformation profiles in this experimental campaign can be explained by considering dislocation micromechanisms.Upon initial loading, the material's response is elastic until the stress exceeds the yield strength, inducing dislocation movement and thus (visco)plastic deformation.It should be noted that, in comparison with the response of conventionally manufactured HX reported by Tsuji and Nakajima (1987), the strength of LPBF HX is substantially higher, mainly due to its higher dislocation density (see Fig. 1(c)), consistent with the findings of Karapuzha et al. (2021).Further straining, particularly at low temperatures, increases the alloy's dislocation density due to the generation and multiplication of dislocations.This rise in dislocation population and the resulting dislocation interactions impede the movement of mobile dislocations, necessitating higher stress for further straining, i.e., strain hardening.At larger strain values and particularly higher temperatures, dislocation annihilation mechanisms slow down the increase in dislocation density during straining, thereby reducing the strain-hardening slope.With the start of strain-controlled dwell, if the temperature is sufficiently high, viscoplasticity accompanied by dislocation annihilation results in time-dependent deformation and stress reduction and relaxation.The viscoplastic response at high temperatures is driven by thermally activated creep deformation mechanisms.Wu et al. (2023) reported three main mechanisms for LPBF HX: dislocation motion by glide and climb, grain boundary sliding, and diffusional creep.After the dwell period and upon sufficient reverse loading, the stress state returns to the elastic regime, theoretically halting dislocation movement and hence their generation and dynamic annihilation (although annihilation due to static recovery cannot be excluded).Ultimately, further reverse loading causes dislocation (back) movement and the initiation of (reverse) viscoplastic deformation.The prior loading, however, has a significant influence on the material's response during reverse loading.Although some of the dislocations generated during forward loading might interact with back-moving dislocations and impede their movement (isotropic hardening), a greater portion of the previously generated dislocations might even facilitate dislocation movement during reverse loading, resulting in early yielding (kinematic hardening and the Bauschinger effect).With further reverse loading, the kinematic hardening effect decays, and dislocation generation and multiplication again retard dislocation movement, causing the observation of strain hardening.Similar to forward loading, dislocation annihilation due to static and dynamic softening mechanisms reduces the rate of hardening during larger extents of reverse loading, depending on the imposed temperature, thereby reducing the strain-hardening slope.A similar interpretation can be applied to explain the material response during subsequent stress relaxation and loading ramps.In the absence of a detailed experimental assessment of dislocation structure during cyclic deformations, the main focus of this study is on the macroscopic mechanical response of the material and the development of a robust phenomenological constitutive model.Fig. 5 presents examples of the observed stress evolutions for experiments with ε = 0.10% s −1 and   = 15 min.As expected, stresses were higher at lower testing temperatures.Interestingly, the maximum extent of stress-relaxation during the 15 min dwell time was observed at intermediate testing temperature of 800 °C.Indeed, the extent of the stress relaxation during the conducted experiments depends on the stresses generated after 0.75% straining (i.e., the start of the relaxation period) and the significance of the activity of viscoplasticity and creep at the test temperature.An increase in temperature, on one hand, decreases the stress at the beginning of the relaxation period, and on the other hand, eases the viscoplasticity.Therefore, the extent of stress relaxation at both extreme high and low temperatures is minimal, either due to the small stress at the beginning of the stress relaxation period or due to the low activity of viscoplastic mechanisms.In intermediate temperatures, however, it is possible to have a combination of relatively high stresses after 0.75% straining and sufficient activity of viscoplasticity, leading to a higher level of stress relaxation.Further examples of the alloy's stress relaxation response are illustrated in Fig. S5 in Supplementary Materials.
Figs. 6 and 7 illustrate the temperature and strain rate sensitivity of the mechanical response of LPBF HX during the initial loading transients in the experiments.A higher level of strain rate sensitivity was observed at higher temperatures, indicating the viscoplastic nature of the deformation response.The observed stress level significantly dropped at temperatures above 800 °C.Additionally, indications of dynamic strain aging (DSA) were seen at temperatures of 400 and 600 °C for faster strain rates, consistent with observation reported in our previous study (Li et al., 2023).
The hysteresis response of the alloy under all examined loading conditions is presented in Section 4, Fig. 11.From these figures and Figures S1 to S3, a transition from cyclic hardening (at T ≤ 600 °C) to cyclic softening (at  ≥ 900 °C) can be identified.At 800 °C, lower strain rates resulted in a cyclic softening response in the alloy while the highest strain rate led to the observation of cyclic hardening.Due to the high initial dislocation density in LPBF HX and hence a high driving force for (dynamic) recovery, cyclic deformation at high temperature causes a drop in dislocation density and hence cyclic softening.Meanwhile, due to its low initial dislocation density, conventionally manufactured HX exhibits intense cyclic hardening during early cycles at both low and high temperature conditions (Miner and Castelli, 1992;Lu et al., 2004).
Finally, Fig. 8 indicates only a slight sample-to-sample variability in the mechanical response of LPBF HX when the deformation responses of different samples tested at the same temperature under the strain rate of 0.1% s −1 were compared (from tests with and without a dwell period).The extent of tension-compression asymmetry in the mechanical response of LPBF HX is also examined in Fig. S6 and S7 in Supplementary Materials which did not indicate any systematic or significant asymmetry, consistent with the findings of Esmaeilizadeh et al. (2020).

Non-isothermal mechanical experiments
Fig. 9 presents the observed stress evolutions for the three conducted non-isothermal experiments.It can be observed that at the beginning, with a decrease in temperature from 1000 to 200 °C, a tensile stress develops during the straining of the sample.Within the second transient, as the temperature increases and strain decreases, tensile stress reduces and compressive stress begins to develop.The development of compressive stress is due to the accumulation of nonelastic deformation within the sample during the first transient.Interestingly, the increase in the compressive stress level does not sustain during the second transition, and after reaching a peak compressive stress at a temperature of about 650-730 °C, the compressive stress drops and reaches 59-69 MPa at 1000 °C.The drop in the compressive stress value is led by a reduction of material strength and enhanced viscoplasticity due to temperature increase.The evolution of the stress in subsequent cycles follows a similar logic and is not discussed for the sake of brevity.
Indications of DSA were observed for the non-isothermal experiment at a strain range of   = 0.75%.Interestingly, the experimental observation reveals overall cyclic softening behaviour despite the fact that a significant portion of the applied temperature range is associated with cyclic hardening in the isothermal tests.When comparing the maximum stress levels during the non-isothermal experiments with those of the isothermal test at 200 °C, lower values are obtained during the nonisothermal tests.This is expected as straining at elevated temperatures, up to 1000 °C, during non-isothermal testing does not induce the same extent of material hardening as at a constant temperature of 200 °C during isothermal testing.Importantly, exposure of the material to 1000 °C during non-isothermal testing can induce substantial changes in the microstructure and dislocation configuration of the material, causing enhanced softening and hence leading to the development of lower stress levels during subsequent loading.It should be noted that the complex nature of these experiments makes it challenging to extract fundamental information about the material's mechanical response and indeed, the non-isothermal mechanical experiments are primarily aimed at generating independent benchmarking data for assessing the predictiveness of the developed model, as discussed in Section 5.

Constitutive model for LPBF HX
The viscoplastic Chaboche model formulation decomposes the strain rate tensor into elastic and viscoplastic components (Chaboche, 2008): Hooke's law is considered to represent elastic behaviour, whereas the viscoplastic part of the model follows the normality rule and is governed by the viscosity function and the strain hardening equations: where  and  are the applied stress and back stress tensors, respectively,  ′ and  ′ are their corresponding deviatoric parts, ε  = International Journal of Solids and Structures 305 (2024) 113047 is the von Mises equivalent of the relative stress.The yield function  = ‖ − ‖ −  governs the viscoplastic deformation behaviour of the material such that the state remains elastic for  ≤ 0 and enters the viscoplastic regime for  > 0. In contrast to classical (rate-independent) plasticity, where the stress state is confined within the yield surface  = 0, in viscoplasticity, the stress state can exceed the yield surface, which is known as viscous stress or overstress.Norton's law is employed in this study as the viscosity function for defining the equivalent viscoplastic strain rate: where  and  are temperature-dependent material parameters, and ⟨⟩ are Macaulay brackets, employed here to limit viscoplastic deformation to the condition of  > 0.
Quantities  and  in Eq. (3) represent the contributions from isotropic and kinematic hardening/softening, respectively, both of which are essential for describing the cyclic response of alloys.The isotropic hardening variable  represents the size of the elastic domain, and the kinematic hardening tensor  determines the center of the elastic domain in the stress space.Thus, during the course of viscoplastic deformation, the kinematic term describes a translation of the yield surface within the stress space, and the isotropic term characterizes an isotropic expansion or contraction of the yield surface.
An extension of the Armstrong and Frederick (1966) nonlinear hardening/softening equation by Chaboche (1989) is used to describe the evolution of  and  in this study.The evolution of the isotropic hardening is given as: where   ,   ,   are temperature-dependent material parameters, and  0 is the initial size of the yield surface.Similarly, considering a twoterm back stress formulation, (i.e.,  = ∑ 2 =1   ), the evolution of each back stress tensor is defined as: are temperature-dependent material parameters.The first, second, and third terms in Eqs. ( 4) and ( 5) represent contributions from strain hardening, dynamic and static recovery, respectively.The last term in each equation accounts for changes in  and  with alterations in temperature and becomes relevant for nonisothermal loading conditions.It should be noted that the selection of the above model formulation resulted from a sensitivity analysis aimed at finding the simplest possible formulation with the minimum number of model parameters capable of representing the cyclic viscoplastic response of LPBF HX.It was found that further extension of the model, for example, by increasing the number of terms for kinematic or isotropic hardening description, does not bring significant advantages for enhancing the representativeness of the model.
Consequently, for a known set of model parameters , ,  and a given deformation profile, a solution of the presented equations describes the elastic-viscoplastic response of materials.As discussed by Lemaitre and Chaboche (1990), these 13 model parameters can be calibrated based on experimental cyclic mechanical experiments for a given temperature through inverse analysis.
For the application of the model over a temperature range, our previous study (Hosseini et al., 2015) has recommended avoiding independent determination of the model parameters for individual temperatures, as that not only causes a long list of model parameters for the material, but also typically leads to a non-monotonous evolution of each parameter value with temperature.This brings uncertainty about the validity of the model representation for temperatures in between two testing temperatures and additionally may result in convergence issues in non-isothermal computations, as demonstrated by Morch et al. (2021).Therefore, we explicitly consider a set of functions for describing the temperature dependence of the model parameters, as summarized in Table 3.The selection of function forms is based on existing knowledge about the nature of the dependence of different model parameters on temperature, complemented with some trial and error exercises.
The calibration of the model for LPBF HX involved determining  1−39 by simultaneously considering all cyclic isothermal mechanical testing data through an inverse analysis.This process is particularly challenging and required the development of a dedicated strategy, as described below.
The inverse analysis begins with an initial guess for coefficients  1−39 and involves calculating the stress response for all 28 strain-time profiles of the isothermal experiments.The calculated stresses are then compared with experimentally recorded stress profiles to determine a discrepancy factor, referred to as the 'loss'.MATLAB's ''surrogateopt '' optimizer is utilized to search globally for the best set of   to minimize this loss (the inverse analysis code is provided in Supplementary Materials).
One of the main challenges in our optimization the inhomogeneity of the problem, characterized by significant variations in the sensitivity of the predicted profiles, and consequently the loss function, to various   coefficients.To address this, instead of optimizing the   coefficients directly, we redefine and initiate with   = 1, searching for the optimal set of   values to minimize the loss.The optimization process is structured into multiple stages, each involving over 2000 optimization iterations, required ca.24 h on Table 3 Adopted functions for representing the temperature dependence of constitutive model parameters.
is in Kelvin.15 CPU cores.For each stage, the   values calculated in the previous stage are used as the new    .Prior to each sub-optimization, the parameters of   are set such that a 10% change in any   results in only a 1% change in the loss value.This 'normalization' effectively enhances the convergence rate of the optimization effort.In summary, simultaneous consideration of all experimental data during the calibration process, along with explicit integration of temperature-dependence of model parameters and the adoption of the above described optimization strategy, enhances the model calibration's resistance to noise and its effectiveness in identifying trends in the data.This approach facilitates the rational identification of model parameters, particularly benefiting the determination of parameters related to aspects where only limited information exists in individual tests.For example, this method enables the identification of trends in the isotropic response of LPBF HX across different temperatures and strain rates, and thus the determination of the corresponding parameters by considering the results of only five deformation cycles.It is acknowledged that including deformation responses from a greater number of cycles would certainly enhance the model's capability, albeit at the cost of longer testing durations and greater computational efforts for model calibration.
Due to the complexity of simultaneously determining 39   parameters, the optimization process initially focuses on a simplified version of the model that includes a reduced number of model parameters.This version omits isotropic hardening and static recovery, and requires the two kinematic hardening terms to evolve in a similar manner, mathematically expressed as . This reduces the number of unknowns for calibration from 39 to 19, thereby easing the optimization task initially.Furthermore, the optimization initially considers only the first cycle of the experiments to reduce computational costs.The removed complexities are then reintroduced into the model step-by-step until its complete formulation is restored.The final step involves the simultaneous fine-tuning of all coefficients  1−39 , taking into account stress-strain data from all five cycles.Table 4 reports the calibrated values of  1−39 , and Fig. 10 presents the variation of ,  0 , , ,  with temperature.Fig. 11 (as well as Fig. 5) compares the model and experimental stress-strain responses of LPBF HX under isothermal testing conditions, demonstrating excellent consistency between model representation and the experimental data (for a more detailed comparison of the experimental data and model representation, readers are referred to Supplementary Materials).The calibrated model accurately captures the temperature and rate sensitivity of the material response.Particularly, in agreement with the experimental observations, the model illustrates the transition from cyclic hardening to softening and the largest amount of stress relaxation during hold period at 800 °C as shown in Figs. 12 and 5, respectively.In the worst example, an overprediction of the stress state, with deviations of up to 20 MPa, can be identified for the test set at  = 200 °C.Therefore, these results demonstrate the reliability of the model in representing the isothermal cyclic stress-strain response of LPBF HX for a wide range of temperatures and strain rates.The next section aims to evaluate the predictiveness of the model for non-isothermal loading conditions.

Model validation for non-isothermal loading
The calibrated constitutive model has been integrated into the UMAT subroutine and provided in Supplementary Materials to enable readers to deploy our model in the finite element solver of ABAQUS.As detailed in Appendix A, implementing the model into UMAT involved the analytical derivation of the consistent Jacobian matrix.It should be noted that the application of the constitutive model to uniaxial non-isothermal loading conditions in this section did not necessarily require the UMAT code and FE analysis, and could alternatively be programmed in, e.g., MATLAB.
The established UMAT subroutine has been employed to compute the non-isothermal model response for the measured temperature and  The increase in stress ceases around 700-790 °C, followed by a reduction in compressive stress, culminating in a stress level of 45-83 MPa in compression at 1000 °C, which serves as the starting condition for the next cycle.Upon alternating between high-temperature and lowtemperature phases during loading cycles, non-isothermal plasticitycreep interactions are expected to affect the observed stress-strain response.While, in principle, the unified approach of the model allows for capturing such effects, experimental investigation of this requires specially designed tests, which were not included in the scope of this work.There is generally a fair level of consistency between the experimental and model stress profiles, particularly for the test with a strain range of 0.75%.However, there are noticeable mismatches between the model and experimental data, which are worthy of further discussion.Particularly, the model fails to accurately describe the development of significantly smaller peak stress levels during nonisothermal loading compared with isothermal conditions at 200 °C.As discussed earlier, straining at elevated temperatures (up to 1000 °C) during non-isothermal testing does not harden the material as much as straining at a constant temperature of 200 °C during isothermal testing.Hence, lower stresses are expected from non-isothermal testing.This mechanism as well as plasticity-creep interactions are inherently included in the model formulation, which is why the model also predicts lower stress levels under non-isothermal conditions.However, the difference in stress at 200 °C between isothermal and non-isothermal testing is much higher in the experimental data than what the model predicts.Indeed, the model overestimates both the peak and valley stress during non-isothermal testing.This discrepancy could partly be due to uncertainties in the experimental setup.For instance, the use of thermocouples spot-welded to the sample's surface to control and monitor the temperature, coupled with forced air cooling, might lead to higher core temperatures than the surface temperature, resulting in a weaker mechanical response than anticipated.It is important to note that the calibration of the constitutive model was based on isothermal data using a resistive heating system without air cooling.However, it is believed that the inconsistency most likely stems from the fact that the employed model formulation, primarily Eqs. ( 4) and (5), may not fully capture the effects of temperature history on the alloy's hardening response.Previous discussions by Ahmed and Hassan (2017) indicate that exposure of the material to significantly high temperatures could influence its general and low-temperature responses more than what the Armstrong-Frederick rule accounts for, due to microstructural changes.In LPBF HX alloys, expected microstructural changes at 1000 °C include dislocation annihilation, cellular structure formation and growth, and possibly partial recrystallization.These mechanisms, which soften the material, are not considered in the current model formulation.While it is in principle possible to add additional terms to the model to address these phenomena (thus leading to a more complex model with a larger number of parameters), such an extension should be based on far more comprehensive non-isothermal testing data than is presented in this study to develop a robust understanding of the influential parameters affecting the alloy's response in non-isothermal tests.Therefore, such an extension has not been pursued in this study.
Despite the aforementioned discrepancies, the model provides a fairly consistent representation of the alloy behaviour during nonisothermal experiments.Notably, by explicitly considering the temperature dependence of the model parameters, we avoided abrupt jumps in the non-isothermal stress-strain curves reported by Morch et al. (2021) when model parameters were determined independently for each temperature.
The comparisons of model and experimental observations for isothermal and non-isothermal loading conditions show the reliability of the proposed temperature-dependent material model for representing the cyclic viscoplastic response of LPBF HX alloy over the temperature range of 22-1000 • C for various strain rates, and its for consideration in both mechanical assessment of components during service operation and thermomechanical simulation of the LPBF process.It should however be acknowledged that the proposed model formulation and the underlying data employed for its calibration International Journal of Solids and Structures 305 (2024) 113047 do not account for the moderate level of anisotropy in the mechanical response of LPBF HX, as previously mentioned in Section 2.2.

Concluding remarks
This study has investigated the cyclic mechanical behaviour of Hastelloy X, produced through LPBF, across a temperature range of 22-1000 °C and at various strain rates.Observations from 28 isothermal mechanical experiments were utilized to develop a robust viscoplastic constitutive model.The model is based on the widely accepted Chaboche model and incorporates a non-linear kinematic and isotropic hardening/softening formulation.The model is not entirely new but is rather based on a selective combination of existing formulations.A key extension is the explicit consideration of the temperature dependence of model parameters, which enables the use of a single set of parameters to describe the material's response across the examined temperature range, rather than determining individual parameters for each temperature.A dedicated calibration strategy has been developed for the systematic and reliable determination of the 39 model parameters, based on an inverse analysis approach that considers all 28 isothermal experiments concurrently.Comparing the model and experimental results under these conditions demonstrated the model's representativeness for isothermal loading conditions.Additionally, the model's effectiveness in non-isothermal scenarios was assessed by evaluating its performance in predicting the alloy's response during three independent non-isothermal cyclic benchmark tests.To facilitate practical applications in mechanical assessments of components during service operations or in thermomechanical simulations of the LPBF process, the model has been implemented as a UMAT subroutine for use within the finite element software ABAQUS.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
=   ′ (A.8) where  ′ is the deviatoric part of the relative trial stress tensor and  is a scalar value that can be interpreted as the multiplicative viscoplastic corrector: Ultimately, the solution to the problem at hand is to determine the value of    that serves as the root of the below residual function: The dependence of  and  on    are given by Eqs.(A.10) and (A.6), respectively, necessitating the need for an iterative approach.Once, the true    is obtained, the full viscoplastic strain tensor, and ultimately, the stress tensor can be calculated based on Eqs.(A.8), (A.5), (A.4) and (A.2), respectively.To this end, the Newton-Raphson (NR) method is employed which exhibits quadratic convergence for the situation where the initial guess is sufficiently close to the true solution, i.e. within the so-called radius of convergence.Benefiting from the smoothness of the problem and to assure robustness and quadratic convergence, the initial guess for    is based on considering the same equivalent viscoplastic strain rate as for the previous increment.For subsequent NR iterations, a refinement of    based on the iterative search for the true    is performed:

𝛥𝜀 𝑣𝑝
, =   ,(−1) − (  ,(−1) ) (A.12) where   , and   ,(−1) are the values for the equivalent viscoplastic strain increment from the current and the previous NR iteration, respectively, and     is the residual gradient at    ,(−1) .To increase computational efficiency, the analytical formulation of the residual gradient term of     has been derived and implemented in the provided UMAT code.
It should be noted that  ′ , and consequently,   , are not entirely independent from    .Therefore,  ′ and   need to be updated in each NR iteration.The iterations continue until the convergence criterion || < 1 −10 is met.
In summary, for each call of UMAT, at first, the elastic prediction with    = 0 is examined and its solution is accepted if  ≤ 0. For the viscoplastic regime, i.e.  > 0, the NR method looks for a positive    , starting from an estimate based on the equivalent viscoplastic strain rate from the previous increment.A maximum of 50 NR iterations is carried out and if the convergence is not achieved, the iteration scheme switches to the very robust but slow converging bisection method (Ehiwario and Aghamie, 2014).If the converged solution is not found within another 50 iterations, a cutback for the time increment is requested, i.e. 'PNEWDT < 1'.
When converged, the outcome is an accepted approximation for the true value of the equivalent viscoplastic strain increment    which is ultimately used to calculate the updated stress tensor.Finally, some of the calculated quantities such as   ,   , ,   ,   and ε  are stored as 'STATEV' in UMAT to be accessible in the subsequent increments or for post-processing.

Derivation of consistent material tangent
This describes the derivation of the consistent material tangent matrix ('DDSDDE') for the developed constitutive model.Inserting Eq. (A.1) into Hooke's law yields: =   −  ∶   (A.14) where  is the elastic stiffness tensor.The consistent material tangent  is a fourth order tensor and is defined as: The partial differentiation terms in Eqs.(A.16), (A.17) and (A.18) can all be derived straightforwardly, but are lengthy and hence are not presented here.For more details, we refer the reader to the provided UMAT code.Assurance of the reliability of the derived analytical consistent material tangent matrix and its implementation has been verified by comparing it with the numerical tangent obtained through the perturbation method.

Fig. 2 .
Fig. 2. A schematic illustrating the fabrication of dog-bone testpiece geometry for mechanical experiments (dimensions are in millimeters).

Fig. 3 .
Fig. 3. (a) Testing setup for isothermal mechanical experiments, (b) schematics of applied loading cycle for the performed isothermal mechanical experiments.

Fig. 4 .
Fig. 4. (a) Testing setup for non-isothermal mechanical experiments, (b) schematics of applied mechanical strain and temperature cycle for the performed non-isothermal mechanical experiments.

Fig. 8 .
Fig. 8. Comparing the monotonic isothermal deformation responses of samples tested at different temperatures, under the strain rate of 0.1% s −1 , indicating slight sample-to-sample variability in the mechanical response of HX.

Fig. 10 .
Fig. 10.Temperature dependence of the Chaboche model parameters for representing the mechanical response of LPBF HX.

Fig. 11 .
Fig. 11.Comparison of experimentally observed (coloured) and model represented (black) isothermal hysteresis response of LPBF HX at different temperatures and strain rates.

Fig. 12 .
Fig. 12.Comparison of experimentally observed (circles) and model represented (solid line) stress amplitudes of LPBF HX under isothermal loading at different temperatures and strain rates.

Table 2
Details for the conducted isothermal experiments.

Table 4
Calibrated values of  1−39 for representing the temperature dependence of constitutive model parameters.
15)Partial differentiation of Eq. (A.14) with respect to   gives: =  −  were  can be interpreted as viscoplastic corrector to the elastic stiffness matrix  and hence needs to be calculated only for elasticviscoplastic conditions: =  ∶