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

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: positive-definiteness 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.


Introduction
A so-called 'extreme material' possesses mechanical properties (typically an extreme degree of anisotropy) designed in such a way as to keep the material in a state close to an instability threshold (for instance failure of ellipticity), so that ultimate mechanical effects are nstrained rance. In principle with a non-vanishing couple traction, and a strain-energy density that depends upon both the strain and the gradient of rotation. Such assumptions are appropriate for materials with granular and layered structure, where the interaction between adjacent elements may introduce internal moments. In this way, characteristic material lengths appear that can be related with the material microstructure (Yang and Lakes, 1982;Zhang and Sharma, 2005;Maranganti and Sharma, 2007). The presence of these characteristic lengths implies that the couple-stress theory encompasses the analytical possibility of size effects which are absent in the classical theory. Therefore, the couple-stress theory and, more in general gradient and polar type theories, have been often advocated (Triantafyllidis and Aifantis, 1986;Mühlhaus and Vardoulakis, 1987;Gao et al. 1999;Huang et al. 2000;Radi, 2003;Radi and Gei, 2004;Dal Corso and Willis, 2011;Fleck et al., 2014) as possible remedies to constitutive formulations where ellipticity is lost in the classical sense and so strain localization into vanishing narrow bands is predicted to occur (with the related well-known mesh sensitivity in finite element formulations, Needleman and Tvergaard, 1984;Petryk, 1997) Despite the large number of articles devoted to the analysis of gradient effects in localization problems, the condition of failure of ellipticity has neither been obtained for constrained Cosserat elasticity, nor has it been linked to the possible emergence of localized solutions of boundary value problems or to the condition of planar wave propagation. More in general, with the exception of the 'Kirchhoff-like' uniqueness statement (which was related to loss of positive definiteness of the strain energy by Mindlin and Tiersten, 1962; see also Grentzelou and Georgiadis, 2005), material instability thresholds -for instance, strong ellipticity -have never been analyzed for constrained Cosserat elasticity. However, mention should be made to related works in the context of micropolar (unconstrained Cosserat) theory, where the ellipticity and strong ellipticity conditions have been derived, the latter providing a criterion for the existence of acceleration waves (Eremeyev, 2005;Altenbach et al., 2010;Eremeyev et al., 2013).
With reference to an anisotropic theory of couple-stress elasticity, the purpose of the Part I of the present article is: (i.) to introduce the notion of strong ellipticity and (ii.) to motivate this through an extension of the van Hove theorem (van Hove, 1947;Hayes, 1966;Bigoni and Zaccaria, 1992) as the condition of uniqueness for the homogeneous boundary value problem with prescribed kinematical conditions; (iii.) to obtain the condition for propagation of planar waves; (iv.) to derive the condition of ellipticity related to the structure of the differential operator of the governing field equations; (v.) to establish a hierarchy between the mentioned material stability criteria; (vi.) to obtain the condition holding at a discontinuity plane and to show how these conditions are related to the failure of ellipticity; (vii.) to analyze the particular case of antiplane elasticity where a classification into Elliptic-Imaginary, Elliptic-Complex, Hyperbolic, and Parabolic regimes is introduced; viii.) to provide examples of extreme materials where waves can propagate but ellipticity is lost.
In order to complete the picture of material instabilities in Cosserat anisotropic elasticity and to relate this to the design of extreme materials, Part II of this article is devoted to the derivation of infinite-body Green's functions that will be shown to be related to the condition of wave propagation. The pertinent Green's functions will then be used to show, in proximity of the border of ellipticity loss, the emergence of folding and faulting of a continuum in simple and cross geometries.

Fundamentals of constrained couple-stress anisotropic elasticity
In this Section, the equations governing the linearized elastic mechanical response are introduced for anisotropic couple-stress solids. Detailed presentations of the couple-stress theory have been given by Toupin (1962), and Mindlin and Tiersten (1962) (see also Muki and Sternberg, 1965;Gourgiotis and Piccolroaz, 2014). An extension of the couple-stress theory to finite deformations has been recently given by Srinivasa and Reddy (2013).
Couple-stress elasticity assumes that: (i.) each material particle has three degrees of freedom, (ii.) an augmented form of the Euler-Cauchy principle holds in which a nonvanishing couple traction prevails, and (iii.) the strain-energy density depends upon both strain and the rotation gradient.
In the absence of inertia and rotary inertia effects, the balance laws for the linear and angular momentum read where pq  is the strain tensor, q  the rotation vector, and pq  is the curvature tensor (i.e. the transpose of the gradient of rotation    κ ω -note that the notation of Mindlin and Tiersten (1962) is adopted in the present work) which by definition is traceless, 0 pp   . Accordingly, the compatibility equations for the above kinematical fields are (Naghdi, 1965) , 0 , has to be prescribed independently on the surface when defining kinematical boundary conditions. Therefore, the number of the kinematical boundary conditions that can be specified on the boundary are five: the three components of the displacement q u and the two tangential components of the rotation q  . Accordingly, the traction boundary conditions at any point on a smooth boundary, consist of the following three reduced force-tractions and two tangential couple-tractions (Mindlin and Tiersten, 1962;Koiter, 1964)  in the field equations (9) or in the boundary conditions (12). It follows that this quantity remains indeterminate in the constrained Cosserat theory. Following Muki and Sternberg (1965), and without loss in generality, we shall henceforth adopt the normalization: For linear constitutive behavior, the strain-energy density assumes the following general quadratic form for centrosymmetric materials where pqmn C and pqmn B are the elasticity tensors with the following symmetries (Mindlin and Tiersten, 1962) pqmn mnpq qpnm The last equality follows directly from the fact that the curvature tensor is deviatoric ( 0 pp   ). Accordingly, tensor C is defined through 21 independent components, whereas B has 36 independent components. The corresponding constitutive equations are It is remarked that tensor C defines a Cauchy (or 'classical') elastic behavior, which is recovered when tensor B , defining a 'purely Cosserat behavior', vanishes.
The necessary and sufficient conditions for the strain energy density W in Eq. (13) to be positive definite (PD) are 0 pqmn pq mn where Sym denotes the set of all symmetric tensors, Dev is the set of all deviatoric tensors, and 0 denotes the null element, which is excluded from the definition of positive definiteness. The condition of (PD) is sufficient for unconditional stability and uniqueness of the solution of the mixed boundary value problem (Mindlin and Tiersten, 1962;Grentzelou and Georgiadis, 2005). However, in classical elasticity lack of positive definiteness of the strain energy density is a subject of increasing attention (Lakes and Drugan, 2002;Kochmann and Drugan, 2009) as it can model situations in which prestressed solids release energy (and thus apparently violate thermodynamic requirements).
Finally, incorporating the constitutive equations (15) into Eqs. (9) and using the geometric relations in (10) yields the equations of equilibrium in terms of the displacement components (the counterpart of the Navier-Cauchy equations of the classical theory)

Strong ellipticity and van Hove uniqueness theorem
The following definitions of strong ellipticity (SE) for the elasticity tensors C and B are introduced, respectively to be satisfied for every unit vector n and non-null vector q . In view of the properties of the elasticity tensors defined by Eqs. (14), the conditions of (SE) for the classical and couplestress elasticity tensors can be equivalently written as respectively, which, according to Eqs. (16), allow for the conclusion that if the elasticity tensors are positive definite (PD) they are also strongly elliptic A constrained Cosserat material is defined to be strongly elliptic if both inequalities in Eqs.
(19) are simultaneously satisfied. Note that by replacing '  ' with '  ' , the conditions (19) define the 'semi-strong ellipticity' (SSE) conditions for C and B .
It will now be proven that: the (SE) conditions (19) van Hove conditions'). This statement represents the extension of the van Hove's theorem ( van Hove, 1947) to the context of the constrained Cosserat theory.
Following Hayes (1966) and Gurtin, (1972), and invoking superposition, if two solutions for displacements and rotations are possible in the van Hove conditions, the solution produced by the difference fields would correspond to homogeneous conditions at the boundary (for both displacement and rotations) and would make the strain energy   Therefore, to prove uniqueness it suffices to show that imply that  u 0 (and consequently  ω 0) on B . Note that since the surface displacements are zero, the normal component of the surface rotation vector is also zero and, thus, the condition      I n n ω 0 is equivalent to  ω 0 on B  .
The definitions of u and ω can be extended from the closed set B to the whole Euclidean space  simply by defining:  u 0 and  ω 0 on \ B  . The resulting fields are piecewise continuous on  , whereas the gradient of the displacement u and the gradient of the rotation ω may be discontinuous on B  . In particular, in view of Eq. (10) 2 and bearing in mind that the surface rotation is zero, it can be readily shown that only the part Moreover, since the body is homogeneous, the generalized Parseval theorem yields (Davies, 2002)   3 with U and Ω being the complex conjugates of U and Ω , respectively. Note that Eq. (25) where the indices R and I stand for the real and imaginary parts, respectively. A substitution of Eqs. (28) into Eqs. (26) and (27) yields Assume now that the fields u and ω do not vanish identically in B , then by the uniqueness of the Fourier transforms, u  and ω  cannot vanish identically on  . Hence, in view of the Eqs. (29) and (30), if the elasticity tensors C and B are strongly elliptic then , which contradicts the original assumption in Eq. (21). Therefore, when (SE) holds for a homogeneous Cosserat material, uniqueness of the kinematical b.v.p. follows. The importance of the van Hove theorem lies in the fact that it provides conditions to reach the limit of stress channelling (or shear banding, in the context of elastoplasticity), without encountering prior bifurcations (see Ryzhak, 1987;Bigoni and Zaccaria, 1992).
In the next Section, the (SE) conditions (19) will be proven to be sufficient but not necessary conditions to ensure propagation of plane waves in a couple-stress medium. This is in marked contrast with the classical elasticity case where the (SE) of the Cauchy elasticity tensor C is a necessary and sufficient condition for the positive definiteness of the acoustic tensor which, in turn, implies that all waves can travel with real and positive propagation speed (Gurtin, 1972).

Wave propagation and acoustic tensor
The propagation of plane waves in couple-stress elasticity is now considered. In particular, a plane harmonic wave is defined through a vector field of the form where t denotes time, d the wave amplitude vector, n the unit propagation vector, and k the wavenumber (in general complex). Moreover, the vector x denotes the position vector, kV   is the angular frequency assumed to be always real, and V is the phase velocity. The equations of motion in couple-stress elasticity follow from Eqs. (17), augmenting the latter with the standard inertia term , , 1 4 pqmn n mp pqk smn rkts n mtrp q u e e u u where 0   is the constant mass density, a superposed dot denotes time differentiation, and null body forces and moments are assumed. Note that for the purposes of the present analysis, the micro-rotational inertia effects are neglected.
A substitution of Eq. (31) into the equations of motion leads to the propagation where the Cosserat acoustic tensor A can be decomposed into a classical part ( ) A C and an additional couple-stress part ( ) A B in the following way (Toupin, 1962) Note that the symmetries of the elasticity tensors C and B imply also that ( ) A C and ( ) A B are symmetric second-order tensors. The symmetry of ( ) . Therefore, the acoustic tensor in couple-stress theory is symmetric, Moreover, it is remarked that the components of the acoustic tensor are non-homogeneous polynomials of fourth-degree with respect to the wave number k . Hence, contrary to the classical elasticity case, the frequency and the phase velocity depend on the wave number, which implies, in general, that waves are dispersive in the context of the couple-stress theory.
The condition (33) dictates that the amplitude d must be an eigenvector of the acoustic tensor A while its eigenvalues 2  (to within a multiplicative constant  ) are always real and the respective eigenspaces orthogonal. A non-trivial solution exists for the eigenvalue problem (33) when the determinant of the coefficients q d vanishes, An important property of the acoustic tensor is that its couple-stress part ( ) A B is singular, so that the propagation vector n is always an eigenvector of ( ) and consequently, which implies that for every couple-stress anisotropy rkts B at least one direction of propagation n exists such that the wave characteristics are governed only by the Cauchyelastic part of the constitutive equations. This direction coincides with the direction of propagation of purely longitudinal P -waves in a classical anisotropic medium. In fact, as it was shown by Truesdell (1966) (see also Kolodner, 1966) in the context of the classical theory, there always exist at least three distinct directions along which longitudinal waves can propagate. Therefore: regardless to the degree of anisotropy, a longitudinal P -wave always exists propagating in a Cosserat elastic material without dispersion and without displaying

Cosserat effects (provided this wave can propagate in the underlying Cauchy material).
A necessary and sufficient condition for plane waves to propagate with positive speed and for all real wavenumbers k is that the acoustic tensor is positive definite, which is now defined as the (WP) condition for every unit vector n and non-null vector p . Thus, if the wave propagation (WP) condition holds, the squared speeds (i.e. the eigenvalues of the acoustic tensor) corresponding to each real acoustical axis are positive. Accordingly, three linearly independent plane waves always exist for a given direction of propagation n and wavenumber k . It is apparent from Eq. (39), that for small wavenumbers 0 k  (low frequencies) the classical part ( ) A C dominates the behavior of the acoustic tensor, whereas for large wavenumbers k   (high frequencies) the behavior of the acoustic tensor is determined by its couple-stress part ( ) A B . Therefore, by taking into account that Eq. (39) must hold for all real non-zero wavenumbers, the (WP) condition in couple-stress elasticity is equivalent to the following pair of inequalities so that both ' 0  ' cannot simultaneously apply in Eqs. (40) 1 and (40) 2 . In fact, the above conditions imply that p cannot be an eigenvector corresponding to a null eigenvalue of both the classical part ( ) A C and the couple-stress part ( ) A B of the acoustic tensor. An immediate consequence of this statement is that when p is parallel to the propagation vector n , Eqs.
(40) reduce to which excludes the possibility that n be an eigenvector of ( ) A C corresponding to a null eigenvalue.
On the other hand, in view of the inequality (18) 1 , it can be readily shown that the (SE) of the classical Cauchy elasticity tensor is equivalent to the condition that ( ) Further, by setting   q n p and taking into account Eq. (35) 2 , the (SE) condition of the couple-stress elasticity tensor, Eq. (18) so that, since ( ) A B is symmetric and possesses always one null eigenvalue associated with the vector n (say, 1 0   ), the condition   SE B is equivalent to the fact that the remaining two eigenvalues must be strictly positive (i.e. 2 0   and 3 0   ). Replacing the strict inequalities in Eqs. (42) and (43) with "  " , the above conditions become the semi-strong ellipticity (SSE) conditions, holding for every p Note that the inequality in (44) 2 has been extended now to include the case where p is parallel to the propagation vector n .
The above results show that the (SE) of the elasticity tensors is a sufficient condition to guarantee the propagation of plane waves (WP). The necessary and sufficient conditions for (WP) are expressed through conditions (40), which are equivalent to the positive definiteness of the acoustic tensor A . These conditions are weaker than those implied by (SE). It will be shown in Part II that the (WP) plays a major role for the derivation of the infinite body Green's function.

Wave propagation in an isotropic Cosserat Material
To illustrate the constraints imposed on the elasticities by the (WP) condition, the simple case of an isotropic couple-stress material is analyzed. In this case, the elasticity tensors defined in Eqs. (14), assume the following form (Mindlin and Tiersten, 1962;Koiter, 1964)  where the moduli  and  have the same meaning as the Lamé constants of the classical elasticity theory, with the dimensions of 'stress', and the moduli  and   account for the couple-stress effects and are expressed in dimensions of 'force'. Upon substituting Eq. (45) into Eq. (35), the following expressions for the classical and couple-stress parts of the acoustic tensor are obtained, respectively so that the acoustic tensor can be written as Note that  and  are double eigenvalues for the tensors ( ) The first inequality is the same as in the classical theory and is an immediate consequence of Eq. (41). In this case, the propagation vector n is an eigenvector of the acoustic tensor corresponding to a non-dispersive longitudinal P-wave with phase velocity The second inequality involves also the wave number k , so that either Therefore, for an isotropic couple-stress material, the (WP) condition requires one of the following set of inequalities to hold On the other hand, it can be readily shown that the (SE) conditions (19) require that

Ellipticity
The definition of ellipticity (E) is now introduced in a way appropriate for the system of partial differential equations (17). Assuming zero body forces and moments, the governing equations of equilibrium can be written as where the fourth-order differential operator L has the following form   2 In the previous discussion regarding the propagation of plane waves and the associated (WP) condition, the wavenumber was taken to be a real quantity. However, as Mindlin and Tiersten (1962) where k  k n is an arbitrary real vector. The (total) symbol is thus identified with the , where its principal part P l is related to the highest (fourthorder) derivatives of the operator L , with P 4 ( ) . An immediate consequence of Eq.
(37) 1 is that showing that the principal part of the symbol is degenerate, so that the system of PDEs in couple-stress elasticity is not elliptic in the standard sense. To expose the degeneracy of the couple-stress operator, the divergence is applied to the governing system (52), yielding where the higher-order operator does not appear, as L u B is solenoidal. Equation (56) shows that the divergence of the fourth-order couple-stress operator degenerates into a scalar thirdorder PDE. Moreover, applying the gradient operator to Eq. (56), a fourth-order system of where a is a non-zero arbitrary constant. Note that, in view of Eq. (57), the fourth-order operators L and L  are equivalent. The modified principal symbol is defined as where 2  and 3  are the non-vanishing eigenvalues of ( ) A B) and   , , n t s are the respective eigenvectors forming an orthonormal basis. The classical part of the acoustic tensor is represented in the same orthonormal basis, so that 20 The condition of ellipticity for a constrained Cosserat material is that Note that two different, but equivalent ways to derive the above conditions of ellipticity are given in Appendix B.
The fact that only the conditions (63) 2 and (63) 3 refer to the Cosserat moduli, whereas condition (63) 1 involves the classical (Cauchy) moduli, is attributed to the degeneracy of the principal part of the symbol in couple-stress elasticity, Eq. (55). This degeneracy is directly related to the fact that in the constrained Cosserat theory the strain energy depends upon the gradient of the rotation (8 independent components) and not upon the complete gradient of strain (18 independent components). In strain gradient elasticity, where both rotation and stretch gradients are taken into account, the principal symbol is not degenerate, so that the ellipticity condition should involve only the strain gradient moduli, a problem that will be addressed elsewhere. Finally, in micropolar (unconstrained Cosserat) theory the equilibrium equations for the displacement and rotation components are of the second-order. The resulting principal symbol of the matrix operator is therefore not degenerate since the rotation vector is independent from the displacement field. The condition of ellipticity serves as a criterion for the existence of acceleration waves in a micropolar continuum (Altenbach et al., 2010;Eremeyev et al., 2013).
It is worth noting that for an isotropic couple-stress material, the (E) condition requires A comparison of the above relations with the respective (WP) conditions (50) shows that:

Maxwell's compatibility conditions and discontinuity surfaces
Through a generalization of Thomas (1957) and Hill (1961), the compatibility conditions on discontinuity surfaces are now introduced for couple-stress elasticity. Similarly to the 'classical' context of classical Cauchy elasticity, discontinuity surfaces will be shown to be possible at ellipticity loss.
A smooth surface  is considered with a unit normal n , defining the common boundary of two open regions D  and D  of a solid body. Moreover,   φ x denotes any field which is continuous in the interior of D  and D  , and approaches definite limit values  φ and  φ as It is assumed that φ and all of its derivatives exist in D D    and have finite limiting values on  . For simplicity, the surface of discontinuity  is assumed to be a planar surface.
From equilibrium considerations and imposing continuity of displacements across  the following conditions are obtained where the tractions ( ) n q P and ( ) n q R are given by Eqs. (12), respectively. Note that Eqs. (66) do not include the condition of continuity of rotations, which are left for generality unprescribed across the discontinuity surface.
Denoting the normal and the surface gradients respectively as and employing the Hadamard lemma (Thomas, 1957;Truesdell and Toupin, 1960;Hill, 1961), the Maxwell geometrical compatibility relations can be derived for the jumps of the first and the higher-order gradients of the displacement field in the form

 
(3) (2) , n mtr r t m n r t m r m t t m r n u nnn g n n D n n D n n D g

r m r m t m t r n n D D n D D n D D g
where ( )    is the discontinuity vector of the p -order normal derivative.
Taking now into account Eqs. (8), (12) and (15)  (with , 1,2,3 p j  ). It is worth noting that in the classical elasticity case, the differential system (73) degenerates into an algebraic system for the jump in the deformation gradient, providing the known condition of failure of ellipticity, ( ) (1) 0 qn n A g  C (Hill, 1961).
To further investigate the discontinuity relations, it is expedient to decompose the discontinuity vectors into their normal and tangential parts on the singular surface  as is a tangential vector on the surface  .

The case of loss of ellipticity
An interesting situation occurs at failure of (E) in the couple-stress medium. Consider for simplicity an intrinsic Cartesian coordinate system where the normal n at a given point on  coincides with the local 1 x -axis, so that corresponding to a null eigenvalue (say 2 0   and 3 0   ), so that according to Eq. (63), (E) is lost and In view of the above, the differential system (73) assumes the following explicit form with (3) (2) 3 3 3 1 t g g   .
The differential system comprises now of 5 unknowns, namely (1) Therefore: at loss of ellipticity the differential system becomes determinate. The fact that the system becomes determinate proves that discontinuous solutions can be possible at loss of ellipticity. This statement will be validated with precise examples in Part II of this study, where it will be shown that at ellipticity loss folding of a continuum (in which the displacement gradient becomes discontinuous but remains finite) occurs under a concentrated force, and faulting (in which the displacement exhibits a finite discontinuity) occurs under a concentrated moment.

Antiplane strain
The general equations for an anisotropic couple-stress material and the relevant notions of ellipticity (E) and strong ellipticity (SE) are particularized in the present Section to the case of an orthotropic material under antiplane strain conditions. The case of antiplane strain is chosen for its comparative simplicity and since localization phenomena can be captured such as stress channelling, but also folding, and faulting.

Basic equations for an orthotropic material
For a body occupying a region in the   , x y -plane, when antiplane strain conditions prevail, the displacement field reduces to Note that in Eq. (83) and henceforth, the coordinates   1 2 3 , , x x x are replaced by   , , x y z , respectively, so that z denotes the axis corresponding to the out-of-plane direction. By virtue of Eq. (10), the non-vanishing components of strain, rotation, and curvature are given in the Further, considering an orthotropic centrosymmetric material, the constitutive equations (15) reduce to (for details refer to Appendix A) represent the bending stiffnesses in the principal y -and x -directions, 1 2 b the principal twisting stiffness, and 3 2 b the stiffness associated with the effects of secondary bending (Lekhnitskii, 1963). Moreover, assuming the strain energy density W to be positive definite (PD), the material moduli must satisfy the following inequalities Enforcing equilibrium yields a single PDE of the fourth-order for the out-of-plane displacement component where the differential operator L is defined as   and 0 is a material parameter that accounts for both torsion and secondary bending effects. Note that neglecting the body forces and body moments, the equilibrium equation (91) is of the same form as the equation of bending of thin orthotropic plates with prestress (Lekhnitskii, 1963). In particular, the principal part cs L w is associated with the deflection of the plate, whereas the classical (lower) part cl L w plays the role of the prestress. , the above equations transform to those governing isotropic couple-stress elasticity for antiplane strain deformations (Lubarda, 2003). In that case, the scalar equation of equilibrium (91) reduces to 2 4

Ellipticity
It is well-known for classical Cauchy materials that loss of ellipticity may result in the emergence of various kinds of discontinuities. Therefore, since conditions of stress To investigate the restrictions that condition (94) imposes on the couple-stress moduli, the principal operator cs L of the governing PDE (91) is examined (Renardy and Rogers, 2004). In particular, a solution to the equation 0 cs L w  is assumed in the form (Bigoni, 2012) where F is an analytical function and  is a complex constant satisfying the following bi- which are real or complex depending on the values of 2 b , 4 b and 0 b , or equivalently the dimensionless parameters  and  . In what follows, unless otherwise stated, it is assumed The roots q  defined by Eq. (97) change their nature according to the values taken by parameters  and  , so that they can be classified as follows.

Elliptic complex regime (EC)
If 0 (96) admits four complex conjugate roots with If 0   , Eq. (96), admits two real and two imaginary roots, namely holding for 4 0 b  . Equivalently, the ellipticity conditions can be written as 0   and     (Fig. 3). Therefore, ellipticity (E) can be lost in two ways: either (i.) at the (EI/P) boundary (red line in Fig. 3 corresponding to the following components of the unit vector n normal to the discontinuity surfaces 4 2 4 The implications of both cases of ellipticity loss on the behavior of the solution and the emergence of the associated discontinuities in the components of the displacement and its gradient will be examined in Part II. As a conclusion from the above-discussion, note that loss of uniqueness, as related to failure of (PD), for a problem of antiplane deformation can be lost simultaneously with (E). Therefore, a material can be designed to work in antiplane conditions and display extreme behaviors (such as stress channelling and emergence of localized folding and faulting), but still preserving uniqueness of the solution and wave propagation.

SH waves and the (WP) condition
Antiplane shear (i.e. horizontally polarized or SH) motions are now examined in a homogeneous orthotropic constrained Cosserat medium. To this purpose, the governing equation (91) is augmented with the classical inertia term w   . Assuming zero body forces and moments, the equation of motion for the out-of-plane displacement becomes Substituting into the equation of motion a plane wave harmonic solution of the form For an orthotropic material under antiplane motions, the only non-vanishing out-of-plane component of the acoustic tensor is The (WP) condition requires that SH waves propagate with real non-zero velocities for all real wavenumbers k in every direction n , which, in turn, implies that    (Graff, 1975). Moreover, it is remarked that in the special case in which   4 2 2 4 3 2 0 4 2 0 ellipticity is lost, but waves may still propagate. For example, assuming that 2 0 b  , 0 0 b  , and 4 0 b  , the condition of (E) fails since in the direction   It follows, as expected, that (SE) implies both the (WP) and (E) conditions. In the classical elasticity case, the notions of (SE) and (WP) coalesce and the pertinent conditions become 55 0 c  and 44 0 c  .

Extreme Cosserat materials
An extreme Cosserat material is defined in a way that its mechanical properties are close to ellipticity loss. An interesting case occurs when both the response of the Cosserat material and of the underlying Cauchy solid (obtained setting the Cosserat stiffness to zero) are simultaneously close to a failure of ellipticity. In all these cases, the extreme material is on the verge of a material instability, but still within the elliptic range. Although these extreme materials will be chosen so that waves can still propagate -the (WP) condition, it will be shown through specific antiplane strain examples (deferred to Part II of this study) that stress channelling is related to the 'distance' to failure of ellipticity. In this context, localized folding and faulting will also emerge.
For this material (PD) is lost (as related to the loss of shear stiffness), but ellipticity is preserved and waves can propagate with positive speed. For instance, assuming that B is isotropic, the phase velocities of a longitudinal and shear waves become, respectively where S V coincides with the speed of flexural waves in a beam (Graff, 1975

Conclusions
The definition of anisotropic extreme Cosserat materials and their material stability analysis has led to several results defining the behavior of these materials, which can be listed as follows.
i.) Positive definiteness of the strain energy (PD) is related to both (PD) C of the Cauchy and (PD) B of the Cosserat parts of the elasticity, so that loss of (PD) for a Cauchy material cannot be restored by augmenting the constitutive equation with a Cosserat part.
ii.) A notion of strong ellipticity (SE) has been introduced (reducing to the 'usual' notion for classical Cauchy elasticity), which implies uniqueness for a problem with kinematics prescribed on the whole boundary of a homogeneous solid, that has been proven through an extension of the van Hove theorem.
iii.) Strong ellipticity (SE) implies that planar waves can propagate with real speed (WP) and that the partial differential operator governing equilibrium is elliptic (E). A necessary condition for (WP) is that the Cauchy part of elasticity allows propagation of pressure waves.
iv.) Ellipticity (E) and wave propagation (WP) condition are not interdependent conditions, so that it can happen that a wave cannot propagate when the material is still in the elliptic range, or vice versa that waves propagate when the material is at the boundary of ellipticity loss. where ip R are the orthogonal tensors that characterize the three mutually perpendicular planes of reflective symmetry (Cowin, 2013). In particular, it can be shown that the couplestress elasticity tensor B has 12 independent components in the orthotropic case and can be represented in a matrix form in the following way The constitutive equations follow from Eq. (15) 2 , where it can be easily verified that 0 pp m  .
Note that for micropolar elasticity (the so-called 'unconstrained Cosserat theory'), not considered here, an orthotropic material is characterized by 15 independent components since 0 ijkk  B (Ilcewicz et al., 1986).
In the special case of antiplane strain, all the in-plane degrees of freedom are zero, and thus: 0 so that the original governing system of equations (52) assumes the equivalent form The above system comprises of elements which have different orders, in such cases, ellipticity can be defined in the general Douglis-Nirenberg sense, where terms of different orders may be included in the principal part. Note that such an approach could not be employed in the original system (52), since all elements of the original operator qn L were of the fourth-order.
The condition of ellipticity (E) for a constrained Cosserat material is that the determinant of the principal symbol is different from zero, which reduces to the same conditions provided by Eqs. (63) and Eqs. (B1). It can be therefore stated that the couple-stress operator is elliptic in the more general Douglis-Nirenberg sense.