Gap opening in graphene by simple periodic inhomogeneous strain

Using ab initio methods we show first that the uniform deformation either leaves graphene (semi)metallic or opens up a small gap (yet only beyond its mechanical breaking point), contrary to claims in the literature based on tight-binding (TB) calculations. It is possible however to open up a global gap by a sine-like inhomogeneous deformation applied along any direction but the armchair one, with the largest gap along the zigzag direction (~1.0$ eV) and without any electrostatic gating. The gap opening has a threshold character with very sharp rise when the ratio of the amplitude $A$ and the period of the sine wave deformation ${λ}$ exceeds (${A/{λ})}c{~}0.1$ and the inversion symmetry is preserved, while it is thresholdless when the symmetry is broken. The gap opening occurs in graphene mesh on boron-nitride substrate.