  We derive the atomistic representations of the elastic tensors appearing in
the linearized theory of first strain-gradient elasticity for an arbitrary
multi-lattice. In addition to the classical (2nd-Piola) stress and elastic
moduli tensors, these include the rank-three double-stress tensor, the
rank-five tensor of mixed elastic moduli, and the rank-six tensor of
strain-gradient elastic moduli. The atomistic representations are closed-form
analytical expressions in terms of the first and second derivatives of the
interatomic potential with respect to interatomic distances, and dyadic
products of relative atomic positions. Moreover, all expressions are local, in
the sense that they depend only on the atomic neighborhood of a lattice site.
Our results emanate from the condition of energetic equivalence between
continuum and atomistic representations of a crystal, when the kinematics of
the latter is governed by the Cauchy-Born rule. Using the derived expressions,
we prove that the odd-order tensors vanish if the lattice basis admits
central-symmetry. The analytical expressions are implemented as a KIM compliant
algorithm to compute the strain gradient elastic tensors for various materials.
Numerical results are presented to compare representative interatomic
potentials used in the literature for cubic crystals, including simple lattices
and multi-lattices. We observe that central potentials exhibit generalized
Cauchy relations for the rank-six tensor of strain-gradient elastic moduli. In
addition, this tensor is found to be indefinite for many potentials. We discuss
the relationship between indefiniteness and material stability. Finally, the
atomistic representations are specialized to central potentials in simple
lattices. These expressions are used with analytical potentials to study the
sensitivity of the elastic tensors to the choice of the cutoff radius.
