In this paper, we give a review of the state of the art in the study of mechanical metamaterials. The very attractive property of having a microstructure capable of determining exotic and specifically tailored macroscopic behaviour makes the study of metamaterials a field that is actually in expansion, from both a theoretical and a technological point of view. This work is divided into two sections, describing the phenomenological and theoretical aspects of metamaterials. We first give an overview of some existing metamaterials, such as pentamode materials, auxetic materials, materials with negative mechanical constitutive coefficients and materials with enhanced mechanical properties. We also focus on some emerging areas, such as origami. Then, we present some theoretical studies in the field of mechanical metamaterials, such as those related to first- and second-gradient theories.

In this paper, we exploit some results in the theory of irreversible phenomena to address the study of quasi-static brittle fracture propagation in a two-dimensional isotropic continuum. The elastic strain energy density of the body has been assumed to be geometrically nonlinear and to depend on the strain gradient. Such generalized continua often arise in the description of microstructured media. These materials possess an intrinsic length scale, which determines the size of internal boundary layers. In particular, the non-locality conferred by this internal length scale avoids the concentration of deformations, which is usually observed when dealing with local models and which leads to mesh dependency. A scalar Lagrangian damage field, ranging from zero to one, is introduced to describe the internal state of structural degradation of the material. Standard Lamé and second-gradient elastic coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. Numerical solutions of the model are provided in the case of an obliquely notched rectangular specimen subjected to monotonous tensile and shear loading tests, and brittle fracture propagation is discussed.