Stiffness optimization in nonlinear pantographic structures

Mechanical metamaterials are microstructured mechanical systems showing an overall macroscopic behaviour that depends mainly on their microgeometry and microconstitutive properties. Moreover, their exotic properties are very often extremely sensitive to small variations of mechanical and geometrical properties in their microstructure. Clearly, the methods of structural optimization, once combined with the techniques used to describe multiscale systems, are expected to determine a dramatic improvement in the quality of newly designed metamaterials. In this paper, we consider, only as a demonstrative example, planar pantographic structures which have proved to be extremely tough in extension, To describe pantographic structure behaviour in an efficient way, it has been proposed to use Piola–Hencky-type Lagrangian models, in which the understanding of the mechanics of involved microdeformation processes allows for the formulation of efficient numerical codes. In this paper, we prove that it is possible, via a suitable choice of the macroscopic shear stiffness, to increase the maximal elongation of pantographic structures, in the standard bias test, before the occurrence of rupture phenomena. The basic tool employed to this aim is a constrained optimization algorithm, which uses the numerical tool, previously developed for determining equilibrium shapes, as a subroutine. Actually, one looks for the shear stiffness distribution, which, given the imposed elongation of the pantographic structure and the force applied to it by the used hard device, minimizes the total elongation energy. The so-optimized shear stiffness distribution does prove able to extend the range of imposed elongations that the specimen can experience while remaining undamaged.