Hardness and fracture toughness models by symbolic regression

Abstract

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 including more non-cubic systems. 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.

Appendix: Summary of experimental data on hardness and fracture toughness

Table 4 A summary of material name, shear modules (G, in GPa), bulk modules (B, in GPa), experimental hardness ($H^{\textrm{exp}}_V$, in GPa), predicted hardness using the Chen’s model ($H^{\textrm{Chen}}_V$, in GPa) and the model in Eq. (4) ($H^{\textrm{R2}}_V$, in GPa), and structure type for the dataset $T^A$

Table 5 Same as Table 4 but for the dataset $T^B$. This dataset contains additional cubic systems that were not included in the Chen’s work [32]. The values of shear and bulk moduli with asterisk are taken from the Materials Project [57]

Table 6 Same as Table 4 but for the dataset $T^C$. This dataset contains only non-cubic systems. The values of shear and bulk moduli with asterisk are taken from the Materials Project [57]

Table 7 The predicted fracture toughness (in MPa$\cdot$m$^{1/2}$) of covalent and ionic crystals using the Niu’s model Eq. (2) and the model in Eq. (5) as compared to the experimental data. The data of elastic moduli (G and B) and atomic volume ($V_0$) are taken from Ref. [52]