  We present a new first-order approach to strain-engineering of graphene's
electronic structure where no continuous displacement field $\mathbf{u}(x,y)$
is required. The approach is valid for negligible curvature. The theory is
directly expressed in terms of atomic displacements under mechanical load, such
that one can determine if mechanical strain is varying smoothly at each unit
cell, and the extent to which sublattice symmetry holds. Since strain deforms
lattice vectors at each unit cell, orthogonality between lattice and reciprocal
lattice vectors leads to renormalization of the reciprocal lattice vectors as
well, making the $K$ and $K'$ points shift in opposite directions. From this
observation we conclude that no $K-$dependent gauges enter on a first-order
theory. In this formulation of the theory the deformation potential and
pseudo-magnetic field take discrete values at each graphene unit cell. We
illustrate the formalism by providing strain-generated fields and local density
of electronic states on graphene membranes with large numbers of atoms. The
present method complements and goes beyond the prevalent approach, where strain
engineering in graphene is based upon first-order continuum elasticity.
