  The antiplane strain Green's functions for an applied concentrated force and
moment are obtained for Cosserat elastic solids with extreme anisotropy, which
can be tailored to bring the material in a state close to an instability
threshold such as failure of ellipticity. It is shown that the wave propagation
condition (and not ellipticity) governs the behavior of the antiplane strain
Green's functions. These Green's functions are used as perturbing agents to
demonstrate in an extreme material the emergence of localized (single and
cross) stress channelling and the emergence of antiplane localized folding (or
creasing, or weak elastostatic shock) and faulting (or elastostatic shock) of a
Cosserat continuum, phenomena which remain excluded for a Cauchy elastic
material. During folding some components of the displacement gradient suffer a
finite jump, whereas during faulting the displacement itself displays a finite
discontinuity.
