  Willis elasticity is an effective medium theory for linearly elastic
composites that incorporates an unusual coupling between stress and velocity,
as well as between momentum and strain. Interest in the theory peaked following
the discovery that its formulation is invariant under curvilinear changes of
coordinates and that, consequently, it can be used to inverse-design
``invisibility'' cloaks for elastodynamics. That said, the
microstructure-property relationships in Willis elasticity are poorly
understood and, in particular, the mechanics that underlie the coupling are
largely unknown. Thus, no such cloaks were constructed.
  Here, we put forward the idea that Willis elasticity is a particular
microcontinuum field theory where the (generalized) micro-displacements have
been eliminated in favor of the macroscopic displacement field as if by Schur
completion. The field theory is special in that it features an inertial
coupling between the micro- and macro-displacements that, upon completion,
re-emerges as the coupling term in Willis elasticity. Concretely, we analyze an
asymptotic regime where mechanical lattices exhibit a kinematic enrichment with
a strong (leading-order) inertial coupling. We provide, in closed-form, the
resulting microstructure-property relationships. As an application, and in
light of the gained insights, we design an ``invisibility'' cloak resolved into
Willis-elastic mechanical lattices.
