Bordiga Giovanni, Bigoni Davide, Piccolroaz Andrea
Department of Civil, Environmental, and Mechanical Engineering, University of Trento, Trento, Italy.
Philos Trans A Math Phys Eng Sci. 2022 Sep 5;380(2231):20210388. doi: 10.1098/rsta.2021.0388. Epub 2022 Jul 18.
Homogenization of the incremental response of grids made up of elastic rods leads to homogeneous effective continua which may suffer macroscopic instability, occurring at the same time in both the grid and the effective continuum. This instability corresponds to the loss of ellipticity in the effective material and the formation of localized responses as, for instance, shear bands. Using lattice models of elastic rods, loss of ellipticity has always been found to occur for stress states involving compression of the rods, as these structural elements buckle only under compression. In this way, the locus of material stability for the effective solid is unbounded in tension, i.e. the material is always stable for a tensile prestress. A rigorous application of homogenization theory is proposed to show that the inclusion of sliders (constraints imposing axial and rotational continuity, but allowing shear jumps) in the grid of rods leads to loss of ellipticity in tension so that the locus for material instability becomes . This result explains (i) how to design elastic materials subject to localization of deformation and shear banding for all radial stress paths; and (ii) how for all these paths a material may fail by developing strain localization and without involving cracking. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.
由弹性杆组成的网格的增量响应均匀化会导致均匀的有效连续体，该连续体可能会遭受宏观不稳定性，这种不稳定性会同时在网格和有效连续体中出现。这种不稳定性对应于有效材料中椭圆性的丧失以及局部响应的形成，例如剪切带。使用弹性杆的晶格模型，人们总是发现，对于涉及杆压缩的应力状态，椭圆性会丧失，因为这些结构元件仅在压缩下会发生屈曲。这样，有效固体的材料稳定性轨迹在拉伸时是无界的，即材料在拉伸预应力下总是稳定的。本文提出了对均匀化理论的严格应用，以表明在杆的网格中包含滑块（施加轴向和旋转连续性但允许剪切跳跃的约束）会导致拉伸时椭圆性的丧失，从而使材料不稳定性轨迹变为…… 这一结果解释了：（i）如何针对所有径向应力路径设计易发生变形局部化和剪切带化的弹性材料；以及（ii）对于所有这些路径，材料如何通过发展应变局部化而不涉及开裂的情况下失效。本文是主题为“多尺度复杂介质和结构化超材料中的波产生与传播（第1部分）”的一部分。
