A Combined Microbeam and Phase-Field Approach to Identify the Toughness and Ultimate Strength of Amorphous Silica
This work presents a new approach to evaluating the toughness, described by the critical energy release rate ($G_c$), and ultimate tensile strength ($\sigma_c$) of amorphous silica (SiO$_2$ glass), combining microbeam tests and phase-field calculations. The latter provides a numerical route to brittle fracture without prescribing explicit fracture surfaces \textit{a priori}, enabling crack initiation and propagation to be tracked. Single-notched microbeams and newly designed bone-shaped microbeams with a notch-free gauge section were fabricated by Focused Ion Beam (FIB) milling nd tested under bending in air, probing the brittle-fracture and strength-controlled regimes, respectively. Both geometries were modeled by Finite Element Analysis (FEA) coupled with a phase-field formulation. We found $G_c = 5.1$~J/m$^2$ (critical stress intensity factor $K_{IC} = 0.61$~MPa$\cdot$m$^{1/2}$), an intrinsic material length scale $\ell_c = 9.1$~nm, and $\sigma_c = 6.8$~GPa, consistent with previously reported brittle properties of silica glass. Through a parametric study, we show the effect of notch geometry on the fracture response of the microbeams and the impact of dimensional measurement error on the determined toughness. Unlike conventional micromechanical methods that yield only $K_{IC}$, our combined microbeam geometries and phase-field approach simultaneously deliver $G_c$ and $\sigma_c$, bridging brittle-fracture characterization and the strength-controlled regime inaccessible to toughness-only techniques.