Recently, Shang et al. (Angew Chem Int Ed 57(3):774–778, 2018) have developed a method to synthesize ultrathin (around 1.9 nm) graphyne nanosheets. We reported here the mixed-mode I–II fracture toughnesses and crack growth angles of single-layer graphyne sheets using molecular dynamics (MD) simulations and the finite element (FE) method based on the boundary layer model, respectively. The various carbon–carbon bonds of graphyne sheets in the FE method are equated with the nonlinear Timoshenko beams based on the Tersoff–Brenner potential, where all the parameters of the nonlinear beams are completely determined based on the continuum modeling. All the results from the present FE method are reasonable in comparison with those from our MD simulations using the REBO potential. The present results show that both the critical stress intensity factors (SIFs) and the crack growth angle strongly depend on the chirality and loading angle 