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

Abstract

Materials with extreme mechanical anisotropy are designed to work near a material instability threshold where they display stress channelling and strain localization, effects that can be exploited in several technologies. Extreme couple stress solids are introduced and for the first time systematically analyzed in terms of several material instability criteria: positivedefiniteness of the strain energy (implying uniqueness of the mixed b.v.p.), strong ellipticity (implying uniqueness of the b.v.p. with prescribed kinematics on the whole boundary), plane wave propagation, ellipticity, and the emergence of discontinuity surfaces. Several new and unexpected features are highlighted: (i.) Ellipticity is mainly dictated by the 'Cosserat part' of the elasticity and (ii.) its failure is shown to be related to the emergence of discontinuity surfaces; (iii.) Ellipticity and wave propagation are not interdependent conditions (so that it is possible for waves not to propagate when 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 of ellipticity induces stress channelling, folding and faulting of an elastic Cosserat continuum (and the related derivation of the infinite-body Green's function under antiplane strain conditions) is deferred to Part II of this study.

Real wave propagation in the isotropic-relaxed micromorphic model

“…This fact seems to have been appreciated also in the study of the Cosserat model [47][48][49][50][51].…”

The dynamics of folding instability in a constrained Cosserat medium

“…A detailed presentation of the couple-stress theory (called also ‘constrained Cosserat theory’) can be found in [7] (see also [4,8]).…”

“…The corresponding constitutive equations are It is remarked that the Cauchy elastic behaviour is recovered when tensor , defining a ‘purely Cosserat behaviour’, is set to zero. The conditions for positive definiteness of the strain energy density, strong ellipticity of the elasticity tensors and the related van Hove uniqueness theorem were given recently in [4]. Moreover, assuming that the kinetic energy density is positive definite implies, in turn, that is also positive definite.…”

“…As shown in Gourgiotis & Bigoni [4], is a singular tensor that always possesses one null eigenvalue corresponding to the eigenvector n , i.e. .…”

“…Note that the acoustic tensor is symmetric. Furthermore, for a continuum without microinertia (), the acoustic tensor reduces to , a case that has been thoroughly examined in [4]. …”