  This paper presents a homogenization framework for elastomeric metamaterials
exhibiting long-range correlated fluctuation fields. Based on full-scale
numerical simulations on a class of such materials, an ansatz is proposed that
allows to decompose the kinematics into three parts, i.e. a smooth mean
displacement field, a long-range correlated fluctuating field, and a local
microfluctuation part. With this decomposition, a homogenized solution is
defined by ensemble averaging the solutions obtained from a family of
translated microstructural realizations. Minimizing the resulting homogenized
energy, a micromorphic continuum emerges in terms of the average displacement
and the amplitude of the patterning long-range microstructural fluctuation
fields. Since full integration of the ensemble averaged global energy (and
hence also the corresponding Euler--Lagrange equations) is computationally
prohibitive, a more efficient approximative computational framework is
developed. The framework relies on local energy density approximations in the
neighbourhood of the considered Gauss integration points, while taking into
account the smoothness properties of the effective fields and periodicity of
the microfluctuation pattern. Finally, the implementation of the proposed
methodology is briefly outlined and its performance is demonstrated by
comparing its predictions against full scale simulations of a representative
example.
