Folding and faulting of an elastic continuum

Abstract

Folding is a process in which bending is localized at sharp edges separated by almost undeformed elements. This process is rarely encountered in Nature, although some exceptions can be found in unusual layered rock formations (called ‘chevrons’) and seashell patterns (for instance Lopha cristagalli). In mechanics, the bending of a three-dimensional elastic solid is common (for example, in bulk wave propagation), but folding is usually not achieved. In this article, the route leading to folding is shown for an elastic solid obeying the couple-stress theory with an extreme anisotropy. This result is obtained with a perturbation technique, which involves the derivation of new two-dimensional Green's functions for applied concentrated force and moment. While the former perturbation reveals folding, the latter shows that a material in an extreme anisotropic state is also prone to a faulting instability, in which a displacement step of finite size emerges. Another failure mechanism, namely the formation of dilation/compaction bands, is also highlighted. Finally, a geophysical application to the mechanics of chevron formation shows how the proposed approach may explain the formation of natural structures.

Mechanical Behavior of Anisotropic Composite Materials as Micropolar Continua

“…This study highlights aspects that, in the Authors' knowledge, have not been previously investigated in materials with anisotropies. These aspects can be also interesting in the perspective of dealing with significant problems of loss of ellipticity followed by strain localization phenomena, with folding and/or fracture for instance, that affect both classical and constrained micropolar materials (Sluys et al, 1993;Nguyen et al, 2012;Bigoni and Gourgiotis, 2016;Gourgiotis and Bigoni, 2016).…”

“…To this regard, it is worth noting that the micropolar continuum, differently from the couple-stress continuum, that is a micropolar continuum with constrained rotations (Sokolowski, 1972;Masiani and Trovalusci, 1996 Appendix), and also from second gradient continuum (Bacigalupo and Gambarotta, 2011;Trovalusci and Pau, 2014), presents the peculiar strain measure of the relative rotation between the local rigid rotation (macrorotation) and the microrotation that is related to the skew symmetric part of strain and then, in terms of work expended, to the skew symmetric part of the stress, whose contribution has significant role in anisotropic media (Pau and Trovalusci, 2012;Trovalusci and Pau, 2014). It is worth noting that, the different behavior between micropolar without and with constrained rotations media, for which the relative rotation is null, is also of interest for investigating the loss of ellipticity of problems leading to strain localization phenomena (Bigoni and Gourgiotis, 2016;Gourgiotis and Bigoni, 2016). Such effects are expected to be governed by both material size (de Borst, 1991;Sluys et al, 1993) and also the degree of anisotropy.…”

The Hertz contact problem in couple-stress elasticity

“…We begin by giving a brief account of the theory of couple-stress elasticity. A thorough account of the anisotropic couple-stress theory is given in Bigoni and Gourgiotis, 2016) In the absence of inertia effects M A N U S C R I P T and body loads, the balance laws for the linear and angular momentum lead to the following force and moment equations of equilibrium (Mindlin and Tiersten, 1962) σ ji,j = 0, e ijk σ jk + m ji,j = 0,…”

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].…”