  We investigate the Casimir-Lifshitz force (CLF) between two identical
graphene strip gratings, laid on finite dielectric substrates, by using the
scattering matrix (S-matrix) approach derived from the Fourier Modal Method
with Local Basis Functions (FMM-LBF). We fully take into account the high-order
electromagnetic diffractions, the multiple scattering and the exact 2D feature
of the graphene strips. We show that the non-additivity, which is one of the
most interesting features of the CLF in general, is significantly high and can
be modulated in situ, without any change in the actual material geometry and
this by varying the graphene chemical potential. We discuss the nature of the
geometrical effects and show the relevance of the geometric parameter d/D (i.e.
the ratio between separation and grating period), which allows to explore the
regions of parameters where the additive result is fully acceptable or where
the full calculation is needed. This study can open to deeper experimental
exploration of the non-additive features of the CLF with micro- or
nano-electromechanical graphene-based systems.
