Imagine a material in which shear bands and other instabilities may occur well inside the elastic range and far from failure. A material that can be designed to produce shear bands with a desired inclination, or in which shear bands are the first instability occurring at increasing stress, or in which the anisotropy (not imperfections) allows the formation of only one shear band. Imagine that this material would be characterized by rigorously determined elastic constitutive laws (thus avoiding complications such as the double branch of the incremental constitutive laws of plasticity) and would be, at least in principle, a material realizable (for instance via 3D printing technology) and testable in laboratory conditions. This material would be ideal not only to theoretically analyze instabilities, but also to practically realize the porous architected materials which are preconized to yield extreme mechanical properties such as foldability, channelled response, and surface effects [1–2].
Works of Ponte Castañeda and Triantafyllidis [1-7] addressing homogenization of composites are generalized to rigorously show that prestressed elastic lattices can be made equivalent to elastic materials and metamaterials capable of extreme mechanical performances, to be used for advanced applications.
References
[1] K. Bertoldi Harnessing Instabilities to Design Tunable Architected Cellular Materials. Annu. Rev. Mater. Res. 47.1 (2017) 51–61.
[2] D.M. Kochmann and K. Bertoldi Exploiting Microstructural Instabilities in Solids and Structures: From Metamaterials to Structural Transitions. Appl. Mech. Rev 69.5 (2017).
[3] N. Triantafyllidis and B. N. Maker On the Comparison Between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites. J. Appl. Mech 52.4 (1985) 794–800.
[4] G. Geymonat, S. Müller, and N. Triantafyllidis Homogenization of Nonlinearly Elastic Materials, Microscopic Bifurcation and Macroscopic Loss of Rank-One Convexity. Arch. Rational Mech. Anal. 122.3 (1993) 231–290.
[5] M. P. Santisi d’Avila, N. Triantafyllidis, and G. Wen Localization of Deformation and Loss of Macroscopic Ellipticity in Microstructured Solids. J. Mech. Phys. Solids 97 (2016) 275–298.
[6] P. Ponte Castañeda Second-Order Homogenization Estimates for Nonlinear Composites Incorporating Field Fluctuations: I-Theory. In: J. Mech. Phys. Solids 50 (2002) 737–757.
[7] P. Ponte Castañeda Second-Order Homogenization Estimates for Nonlinear Composites Incorporating Field Fluctuations: II-Applications. In: J. Mech. Phys. Solids 50 (2002) 759–782.