The Hardness Calculator is based on the article "Vickers hardness prediction from machine learning methods" by V. Dovale-Farelo, P. Tavadze, L. Lang, A. Bautista-Hernandez, and A. H. Romero, published in Scientific Reports (2022). The study investigates how to predict Vickers hardness from elastic properties using both classic empirical models and modern machine learning techniques.
The model was trained with an experimental Vickers hardness database of 143 materials, assuring various kinds of compounds. For each material, the bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν) were calculated from the theoretical elastic tensor. These four mechanical parameters serve as input features for all models.
Two main approaches are developed. In the Classic Calculator, several well-known empirical hardness formulas are combined with simple physical descriptors (crystal system, bandgap, and density) to select the most appropriate relation for a given material. In the Machine Learning Calculator, a Gradient Boosting Regressor (GBR) predicts Vickers hardness directly from (B, G, E, ν), while a Gradient Boosting Classifier (GBC) chooses the best empirical hardness formula for an indirect prediction. Among all tested models, GBR achieves the highest accuracy, with a mean absolute error of about 1.3 GPa and an R2 close to 0.99, followed by GBC and the best classic model.
The Hardness Calculator was applied to materials from the Materials Project with available elastic tensors, identifying candidates for hard and superhard behavior. This screening recovers compounds already known or previously proposed to be hard, and highlights sixteen additional materials as new hard or superhard candidates.
Note on correction: In the original paper, the fourth hardness model H4 was written using the bulk modulus due to an error that propagated from the reference [1] used for this relation. The correct expression uses Young’s modulus and should read: \(\mathbf{H_{4}=\frac{(1-2\nu)}{6(1+\nu)}E}\). The implementation in this website uses the corrected formula. The correct expression can be found in [2].