  Mechanical metamaterials feature engineered microstructures designed to
exhibit exotic, and often counter-intuitive, effective behaviour. Such a
behaviour is often achieved through instability-induced transformations of the
underlying periodic microstructure into one or multiple patterning modes. Due
to a strong kinematic coupling of individual repeating microstructural cells,
non-local behaviour and size effects emerge, which cannot easily be captured by
classical homogenization schemes. In addition, the individual patterning modes
can mutually interact in space as well as in time, while at the engineering
scale the entire structure can buckle globally. For efficient numerical
macroscale predictions, a micromorphic computational homogenization scheme has
recently been developed. Although this framework is in principle capable of
accounting for spatial and temporal interactions between individual patterning
modes, its implementation relied on a gradient-based quasi-Newton solution
technique. This solver is suboptimal because (i) it has sub-quadratic
convergence, and (ii) the absence of Hessians does not allow for proper
bifurcation analyses. Given that mechanical metamaterials often rely on
controlled instabilities, these limitations are serious. To address them, a
full Newton method is provided in detail in this paper. The construction of the
macroscopic tangent operator is not straightforward due to specific model
assumptions on the decomposition of the underlying displacement field pertinent
to the micromorphic framework, involving orthogonality constraints. Analytical
expressions for the first and second variation of the total potential energy
are given, and the complete algorithm is listed. The developed methodology is
demonstrated with two examples in which a competition between local and global
buckling exists and where multiple patterning modes emerge.
