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Resonances are fundamentally important in the field of nano-photonics and optics. Thus, it is of great interest to know what are the limits to which they can be tuned. The bandwidth of the resonances in materials is an important feature, which is commonly characterized by using the Q-factor. We present tight bounds correlating the peak absorption with the Q-factor of two-phase quasi-static metamaterials and plasmonic resonators evaluated at a given peak frequency by introducing an alternative definition for the Q-factor in terms of the complex effective permittivity of the composite material. This composite may consist of well-separated clusters of plasmonic particles, and, thus, we obtain bounds on the response of a single cluster as governed by the polarizability. Optimal metamaterial microstructure designs achieving points on the bounds are presented. The most interesting optimal microstructure is a limiting case of doubly coated ellipsoids that attains points on the lower bound. We also obtain bounds on Q for three dimensional, isotropic, and fixed volume fraction two-phase quasi-static metamaterials and particle clusters with an isotropic polarizability. Some almost optimal isotropic microstructure geometries are identified. 

We consider the problem of finding a net that supports prescribed point forces, yet avoids certain obstacles, with all the elements of the net being under compression (or all being under tension), and being confined within a suitable bounding box. In the case of masonry structures, when described through the simple, no-tension constitutive model, this consists, for instance, in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region Γ available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region Γ ′ available to the obstacle with Γ ⊂ Γ ′ . The case where some of the forces are reactive is also treated. 

Truss structures composed of members that work exclusively in tension or in compression appear in several problems of science and engineering, e.g., in the study of the resisting mechanisms of masonry structures, as well as in the design of spider web-inspired web structures. This work generalizes previous results on the existence of cable webs that are able to support assigned sets of nodal forces under tension. We extend such a problem to the limit analysis of compression-only “strut nets” subjected to fixed and variable nodal loads. These systems provide discrete element models of masonry bodies, which lie inside the polygon/polyhedron with vertices at the points of application of the given forces (“underlying masonry structures”). It is assumed that fixed nodal forces are combined with variable forces growing proportionally to a scalar multiplier (load multiplier), and that the supporting strut net is subjected to kinematic constraints at given nodal positions. 

The analysis of wave patterns in a structure which possesses periodicity in the spatial and temporal dimensions is presented. The topic of imperfect chiral interfaces is also considered. Although causality is fundamental for physical processes, natural wave phenomena can be observed when a wave is split at a temporal interface. A wave split at a spatial interface is a more common occurrence; however, when the coefficients of the governing equations are time-dependent, the temporal interface becomes important. Here, the associated frontal waves are studied, and regimes are analysed where the growth of the solution in time is found. Imperfect interfaces, across which the displacements are discontinuous, are also considered in the vector case of chiral elastic systems. Analytical study and asymptotic approximations are supplied with illustrative numerical examples. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’. 

Changing the microstructure properties of a space–time metamaterial while a wave is propagating through it, in general, requires addition or removal of energy, which can be of an exponential form depending on the type of modulation. This limits the realization and application of space–time metamaterials. We resolve this issue by introducing a mechanism of conserving energy at temporal metasurfaces in a non-linear setting. The idea is first demonstrated by considering a wave-packet propagating in a discrete medium of a one-dimensional (1D) chain of springs and masses, where using our energy conserving mechanism, we show that the spring stiffness can be incremented at several time interfaces and the energy will still be conserved. We then consider an interesting application of time-reversed imaging in 1D and two-dimensional (2D) spring–mass systems with a wave packet traveling in the homogenized regime. Our numerical simulations show that, in 1D, when the wave packet hits the time-interface, two sets of waves are generated, one traveling forward in time and the other traveling backward. The time-reversed waves re-converge at the location of the source, and we observe its regeneration. In 2D, we use more complicated initial shapes and, even then, we observe regeneration of the original image or source. Thus, we achieve time-reversed imaging with conservation of energy in a non-linear system. The energy conserving mechanism can be easily extended to continuum media. Some possible ideas and concerns in experimental realization of space–time media are highlighted in conclusion and in the supplementary material. 

Abstract There is growing demand in industrialized and developing countries to provide people and structures with effective earthquake protection. Here, we employ architectured material concepts and a bio-inspired approach to trail-blaze a new path to seismic isolation. We develop a novel seismic isolator whose unit cell is formed by linkages that replicate the bones of human limbs. Deformable tendons connect the limb members to a central post carrying the vertical load, which can slide against the bottom plate of the system. While the displacement capacity of the device depends only on the geometry of the limbs, its vibration period is tuned by dynamically stretching the tendons in the nonlinear stress–strain regime, so as to avoid resonance with seismic excitations. This biomimetic, sliding–stretching isolator can be scaled to seismically protect infrastructure, buildings, artworks and equipment with customized properties and sustainable materials. It does not require heavy industry or expensive materials and is easily assembled from metallic parts and 3D-printed components.