Bulky auxeticity, tensile buckling and deck-of-cards kinematics emerging from structured continua

Complex mechanical behaviours are generally met in macroscopically homogeneous media as effects of inelastic responses or as results of unconventional material properties, which are postulated or due to structural systems at the meso/micro-scale. Examples are strain localization due to plasticity or damage and metamaterials exhibiting negative Poisson’s ratios resulting from special porous, eventually buckling, sub-structures. In this work, through ad hoc conceived mechanical paradigms, we show that several non-standard behaviours can be obtained simultaneously by accounting for kinematical discontinuities, without invoking inelastic laws or initial voids. By allowing mutual sliding among rigid tesserae connected by pre-stressed hyperelastic links, we find several unusual kinematics such as localized shear modes and tensile buckling-induced instabilities, leading to deck-of-cards deformations—uncapturable with classical continuum models—and unprecedented ‘bulky’ auxeticity emerging from a densely packed, geometrically symmetrical ensemble of discrete units that deform in a chiral way. Finally, after providing some analytical solutions and inequalities of mechanical interest, we pass to the limit of an infinite number of tesserae of infinitesimal size, thus transiting from discrete to continuum, without the need to introduce characteristic lengths. In the light of the theory of structured deformations, this result demonstrates that the proposed architectured material is nothing else than the first biaxial paradigm of structured continuum —a body that projects, at the macroscopic scale, geometrical changes and disarrangements occurring at the level of its sub-macroscopic elements.

Equilibrium Thermodynamics

The ideas which we shall present in the remainder of this book are intimately connected with thermodynamics. In order to describe the various transport processes in structured continua, one must first build a solid foundation of equilibrium thermodynamics upon which to base further development. In approaching transport phenomena from an energetic viewpoint, one must first define what is meant by various thermodynamic variables as, for example, the temperature and pressure, in terms of the primitive variables used to characterize the system under investigation. In this chapter, we present a brief overview of equilibrium thermodynamics tailored to the needs of this book. Specifically, we want to define explicitly the thermodynamic quantities which are used in subsequent chapters. For clarity and completeness, we shall re-derive, rather than merely state, some of the standard thermodynamic relationships. Of course, the experienced reader may proceed directly to the next chapter and use this chapter as a reference to notation as the need arises. The starting point for our discussion of equilibrium thermodynamics is the axiomatic foundation of the description of macroscopic equilibria on certain fundamental principles. First, the macroscopic equilibrium of a closed system is completely described through the specification of a number of extensive (i.e., proportional to the total mass or volume of the system, and additive between systems) or intensive (i.e., independent of the total mass or volume of the system) parameters. This is a very important point which is usually overlooked in the traditional thermodynamic development. The extensive nature of the primary variables of the system introduces an additional relationship which acts on the allowed variations of the differentials, which, as we shall see, is tantamount to the Gibbs/Duhem relation. This implies, as we shall demonstrate in §4.3, that the density formalism, where every extensive quantity is reported on a unit volume basis, is a much more natural framework for describing the system in that it avoids a number of pitfalls of the traditional formalism.