Folding and faulting of an elastic continuum

Stress channeling and localization of strain are effects displayed by material with extreme anisootropy, that can be exploited in different technologies, for instance, in mechanical wave guiding, stress wave shielding, and invisibility cloaking. Theoretically, stress chan-nelling was pioneered by Pipkin (1984) and Spencer (1984), who demonstrated that stress does not diffuse in materials with extreme orthotropic stiffness ratio, and was found to occur in masonry models by Bigoni and Noselli (2010a; b). In the limit when the stiff-ness ratio between different material directions tends to zero, the equations governing equilibrium reach the elliptic boundary and the stress percolates through null-thickness deformation bands. In this situation, the material microstructure sets the percolation thickness and becomes a dominant factor. Our aim is to analyze stress channelling, strain localization, and the emergence of discontinuities in extremely anisotropic elastic materials governed by couple-stress elasticity. The theory of couple-stress elasticity, also known as Cosserat theory with constrained rotations, is the simplest gradient theory in which couple-stresses make their appearance. In particular, the couple-stress theory assumes an augmented form of the Euler-Cauchy 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 structure, where the interaction between adjacent elements may introduce internal moments. In this way,characteristic material lengths may appear representing the material microstructure. An investigation of loss of ellipticity, emergence of discontinuities (as pioneered by Hill, 1961) and use of the perturbative approach (in the spirit of Bigoni and Capuani,2002; 2005) shows that Cosserat effects can explain unexpected phenomena, such as for instance the single and cross folding of an elastic continuum, but also the formation of faults in single and cross geometries.