  We analytically derive the elastic, dielectric, piezoelectric, and the
flexoelectric phenomenological coefficients as functions of microscopic model
parameters such as ionic positions and spring constants in the two-dimensional
square-lattice model with rock-salt-type ionic arrangement. Monte-Carlo
simulation reveals that a difference in the given elastic constants of the
diagonal springs, each of which connects the same cations or anions, is
responsible for the linear flexoelectric effect in the model. We show the
quadratic flexoelectric effect is present only in non-centrosymmetric systems
and it can overwhelm the linear effect in feasibly large strain gradients.
