  Flexoelectricity phenomenon is the response of electric polarization to an
applied strain gradient and is developed as a consequence of crystal symmetry
in all materials. In this study, we show that the presence of strain gradient
in non-piezoelectric graphene sheet does not only affect the ionic positions,
but also the asymmetric redistribution of the electron density, which induce
strong polarization in the graphene sheet. Using quantum mechanics
calculations, the resulting axial and normal piezoelectric coefficients of the
graphene sheet were determined using two loading conditions: (i) a graphene
sheet containing non-centrosymmetric pores subjected to an axial load, and (ii)
a pristine graphene sheet subjected to a bending moment. Particular emphases
were placed on the role of edge and corner states of pores arising due to the
functionalization. We also investigated the electronic structure of graphene
sheet under different in-plane strain distributions using quantum mechanics
calculations and tight-binding approach. The findings of our work reveal that
the respective axial and normal electromechanical couplings in graphene can be
engineered by changing the size of non-centrosymmetric pores and radii of
curvature. Our fundamental study highlights the possibility of using graphene
sheets in nanoelectromechanical systems as sensors or actuators
