  In this paper we model the size-effects of metamaterial beams under bending
with the aid of the relaxed micromorphic continuum. We analyze first the
size-dependent bending stiffness of heterogeneous fully discretized
metamaterial beams subjected to pure bending loads. Two equivalent loading
schemes are introduced which lead to a constant moment along the beam length
with no shear force. The relaxed micromorphic model is employed then to
retrieve the size-effects. We present a procedure for the determination of the
material parameters of the relaxed micromorphic model based on the fact that
the model operates between two well-defined scales. These scales are given by
linear elasticity with micro and macro elasticity tensors which bound the
relaxed micromorphic continuum from above and below, respectively. The micro
elasticity tensor is specified as the maximum possible stiffness that is
exhibited by the assumed metamaterial while the macro elasticity tensor is
given by standard periodic first-order homogenization. For the identification
of the micro elasticity tensor, two different approaches are shown which rely
on affine and non-affine Dirichlet boundary conditions of candidate unit cell
variants with the possible stiffest response. The consistent coupling condition
is shown to allow the model to act on the whole intended range between macro
and micro elasticity tensors for both loading cases. We fit the relaxed
micromorphic model against the fully resolved metamaterial solution by
controlling the curvature magnitude after linking it with the specimen's size.
The obtained parameters of the relaxed micromorphic model are tested for two
additional loading scenarios.
