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.

中文翻译：

---

非线性缩放结构中的刚度优化

机械超材料是微观结构的机械系统，显示出主要取决于其微观几何形状和微观本构特性的整体宏观行为。此外，它们的奇异特性通常对其微观结构中机械和几何特性的微小变化极为敏感。显然，结构优化的方法一旦与用于描述多尺度系统的技术相结合，有望显着提高新设计的超材料的质量。在本文中，我们仅考虑作为示范性示例，已证明在扩展方面非常坚固的平面受电弓结构，为了以有效的方式描述受电弓结构行为，已提出使用 Piola-Hencky 型拉格朗日模型，其中对所涉及的微变形过程的力学的理解允许制定有效的数字代码。在本文中，我们证明可以通过合适的宏观剪切刚度选择，以增加破裂现象的标准偏置试验中的容击膜结构的最大伸长率。用于此目的的基本工具是约束优化算法，该算法使用先前为确定平衡形状而开发的数值工具作为子程序。实际上，人们寻找剪切刚度分布，考虑到缩放结构的强加伸长率和使用的硬装置施加到它的力，该分布使总伸长能最小化。