Hardness and fracture toughness models by symbolic regression

Superhard materials with good fracture toughness have found wide industrial applications, which necessitates the development of accurate hardness and fracture toughness models for efficient materials design. Although several macroscopic models have been proposed, they are mostly semiempirical based on prior knowledge or assumptions, and obtained by fitting limited experimental data. Here, through an unbiased and explanatory symbolic regression technique, we built a macroscopic hardness model and fracture toughness model, which only require shear and bulk moduli as inputs. The developed hardness model was trained on an extended dataset, which not only includes cubic systems, but also contains non-cubic systems with anisotropic elastic properties. The obtained models turned out to be simple, accurate, and transferable. Moreover, we assessed the performance of three popular deep learning models for predicting bulk and shear moduli, and found that the crystal graph convolutional neural network and crystal explainable property predictor perform almost equally well, both better than the atomistic line graph neural network. By combining the machine-learned bulk and shear moduli with the hardness and fracture toughness prediction models, potential superhard materials with good fracture toughness can be efficiently screened out through high-throughput calculations.