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Materials with extreme mechanical anisotropy are designed to work near amaterial instability threshold where they display stress channelling and strainlocalization, effects that can be exploited in several technologies. Extremecouple stress solids are introduced and for the first time systematicallyanalyzed in terms of several material instability criteria:positive-definiteness of the strain energy (implying uniqueness of the mixedb.v.p.), strong ellipticity (implying uniqueness of the b.v.p. with prescribedkinematics on the whole boundary), plane wave propagation, ellipticity, and theemergence of discontinuity surfaces. Several new and unexpected features arehighlighted: (i.) Ellipticity is mainly dictated by the 'Cosserat part' of theelasticity and (ii.) its failure is shown to be related to the emergence ofdiscontinuity surfaces; (iii.) Ellipticity and wave propagation are notinterdependent conditions (so that it is possible for waves not to propagatewhen the material is still in the elliptic range and, in very special cases,for waves to propagate when ellipticity does not hold). The proof that loss ofellipticity induces stress channelling, folding and faulting of an elasticCosserat continuum (and the related derivation of the infinite-body Green'sfunction under antiplane strain conditions) is deferred to Part II of thisstudy.

Stress channelling in extreme couple-stress materials Part I: Strong ellipticity, wave propagation, ellipticity, and discontinuity relations