The development of metallic structural materials with superior mechanical properties is vital for various applications, particularly those involving constant or static loads. High-entropy alloys (HEAs), characterized by their excellent mechanical properties and extensive compositional space, have emerged as promising candidates. This study aims to establish a framework for the joint optimization of mechanical properties, focusing initially on maximizing ultimate tensile strength (US). The research acknowledges the inherent trade-offs among different properties (e.g., strength versus ductility) and considers various factors influencing US, including composition, microstructure, processing parameters, and defect levels. Unlike traditional alloys with limited elemental compositions, HEAs offer a significantly broader compositional space, making data analytics and machine learning (ML) tools potentially valuable for expediting the identification of alloys with specific properties of interest. However, the current scarcity of experimental data on HEAs with superior strength necessitates a strategy that effectively leverages existing physical understanding and data. The approach adopted in this study prioritizes incorporating physics-based dependencies into the prediction model to overcome limitations inherent in traditional ML methods when applied to datasets with a limited number of data points. This research builds upon previous findings, demonstrating that even simple linear regression can yield accurate predictions, comparable to those from complex ML algorithms, when suitable consideration is given to the underlying physical principles influencing the desired material characteristics. Ultimately, this study aims to introduce a prediction model that maintains consistency among predictions, empirical thermodynamic rules, and experimental results, despite limited data availability.

The literature review discusses the existing research on high-entropy alloys (HEAs) and their mechanical properties. It highlights the work of Agrawal et al. (2014) on predicting fatigue strength of stainless steel using various techniques, including linear regression and artificial neural networks, showing that simple models can yield high accuracy. The study also references works by Zhang et al. (2008) and Feng et al. (2016) on solid-solution phase formation rules and the design of lightweight HEAs. Furthermore, it cites Senkov et al. (2019), Wu et al. (2014), Maresca et al. (2020), and others' work on the temperature dependence of yield strength in various alloy systems and the underlying physical mechanisms. These references establish a baseline understanding of existing modeling techniques and their limitations, informing the methodology proposed in the current research. Existing models often utilize single exponential functions to model temperature dependence of yield strength, lacking the break temperature identified as critical in this study.

The study utilized a dataset comprising 24 HEA compositions exhibiting high ultimate tensile strength (US) at room temperature. Two feature sets, A and B, were derived from this dataset, incorporating different elemental compositions. Data curation was performed to ensure data quality and consistency across the dataset. The study initially employed multivariate linear regression to predict US at room temperature using the two feature vectors, solving an unconstrained optimization problem. This model addressed situations with limited data sets, suggesting the use of pseudo-inverse if matrix inversion was problematic. For high-temperature applications, a bilinear log model was proposed to account for the non-linear temperature dependence of US. This model incorporates two separate exponential functions, one for low-temperature and one for high-temperature regimes, separated by a break temperature (Tbreak). Global optimization was used to fit the model parameters. The model also incorporates a constraint based on diffusion processes and a continuity constraint across the break temperature. A comparison was made between the two-exponential bilinear log model and a single-exponential model, and error analysis was performed to compare modeling accuracy. Experimental validation involved the synthesis of the predicted compositions through arc-melting and subsequent compression testing. Microstructural characterization techniques including Energy Dispersive X-ray Spectroscopy (EDX) mapping and X-ray diffraction (XRD) were used to examine the microstructure and phases of the synthesized alloys. The methodology also includes sections detailing data curation, model selection, optimization techniques, and experimental validation. Specific optimization techniques such as the Matlab's function fminunc were employed for solving the unconstrained minimization problem of finding the best-fit model parameters.

The study successfully developed a bilinear log model to predict the temperature-dependent ultimate strength (US) of BCC high-entropy alloys. The model incorporates a break temperature (Tbreak) that distinguishes between low- and high-temperature regimes. Global optimization of the model parameters improved prediction accuracy, particularly concerning Tbreak. The model demonstrated significantly improved accuracy compared to existing single-exponential models. Experimental validation confirmed that the predicted compositions exhibited superior strength compared to the reference compositions. Specifically, the predicted compositions Al0.5Mo0.5Nb1.5Ta0.5Zr1.5 and Mo1.25Nb1.25Ti0.5V0.5Zr1.25 showed higher yield strength and maximum strength than their respective reference compositions. Microstructural analysis showed a dendrite-interdendrite structure in both predicted alloys, with evidence of elemental segregation and the presence of two BCC phases, consistent with the expected enhancement of strength through solid-solution and second-phase strengthening mechanisms. The analysis of 21 compositions revealed that the mean squared error (MSE) in the log domain was significantly lower for the bilinear model (0.003) than for a single-exponential model (0.195). This improvement translates to a substantial difference in the linear domain as well. The MoNbTaW composition, comprising refractory elements, showed the highest Tbreak of 1124 °C, while MoNbTaVW, including a weakly refractory element (V), showed the lowest slope in the high-temperature region, highlighting the impact of element selection on high-temperature strength. The study emphasizes that elements with higher melting points tend to yield higher break temperatures.

The findings demonstrate the effectiveness of the proposed bilinear log model in accurately predicting the temperature-dependent ultimate strength of BCC high-entropy alloys. The incorporation of Tbreak as a key parameter is a significant contribution to the field, providing a more nuanced understanding of high-temperature material behavior. The superior performance of the bilinear model compared to single-exponential models highlights the importance of accounting for the distinct low and high-temperature regimes in strength prediction. The experimental validation strongly supports the model's accuracy and predictive capability. This work contributes to improved alloy design strategies by enabling a more accurate and informed selection of compositions for high-temperature applications. The joint optimization framework provides a pathway to consider multiple desired properties simultaneously, moving beyond solely optimizing ultimate tensile strength. The results reinforce the importance of considering factors such as elemental segregation, phase formation, and diffusion mechanisms when designing HEAs for specific applications.

This study successfully developed and validated a bilinear log model for predicting temperature-dependent ultimate strength in BCC high-entropy alloys. The model's incorporation of a break temperature (Tbreak) and its improved accuracy over single-exponential models constitute significant advancements. Experimental validation confirmed the predictive power of the model. Future research could expand this framework to include other mechanical properties and explore the application of more advanced machine learning techniques with larger and more diverse datasets. Investigating the influence of additional factors such as grain size and defect densities could further refine the model's accuracy and predictive power. The joint optimization framework provides a foundation for future studies aimed at optimizing multiple material properties simultaneously.

The study's reliance on a relatively limited dataset may affect the generalizability of the model to a broader range of HEA compositions. The experimental validation focused on compression testing; future work should include tensile testing to confirm the model's applicability to different loading conditions. While the model incorporates physical dependencies into the prediction, some complexities of microstructural evolution and phase transformations may not be fully captured. Further investigation into specific microstructural features and their contribution to material strength is needed. Additionally, the data curation process was vital for achieving reliable results. The selection of the feature vectors is somewhat subjective and might affect the accuracy of the model.