Mechanics of elastic networks

Abstract

We consider a periodic lattice structure in d = 2 or 3 dimensions with unit cell comprising Z thin elastic members emanating from a similarly situated central node. A general theoretical approach provides an algebraic formula for the effective elasticity of such frameworks. The method yields the effective cubic elastic constants for three-dimensional spacefilling lattices with Z = 4, 6, 8, 12 and 14, the last being the 'stiffest' lattice proposed by Gurtner & Durand (Gurtner & Durand 2014 Proc. R. Soc. A 470, 20130611. (doi:10.1098/rspa.2013.0611)). The analytical expressions provide explicit formulae for the effective properties of pentamode materials, both isotropic and anisotropic, obtained from the general formulation in the stretch-dominated limit for Z = d + 1.

On the Use of Mechanical Metamaterials for Innovative Seismic Isolations Systems

“…A recent study by A. N. Norris [11] presents a discrete-to-continuum approach to the elastic moduli of different periodic structures, including pentamode, simple cubic, BCC, FCC octet truss and tetrakaidecahedral lattices. We hereafter summarize the main results of such a study, with reference to the unit cell of a d-dimensional lattice that is composed of a number Z of elastic members (rods) and occupies a volume V.…”

“…With reference to the unit cells examined in [11], it is easily shown that the elastic stiffness tensor that characterize the lattice response at the mesoscopic scale shows cubic symmetry with independent stiffness coefficients 11 12 , C C and 44 C (Voigt notation). Let us introduce the following notations…”

“…This means that such a lattice material can be regarded as a "solid water" [3,8,9], which is well suited for the manufacturing of next generation seismic isolation devices. Table 1: Unit cells and elastic properties of three-dimensional lattice materials [11].…”

“…These may include: exceptional strength-and stiffness-to-weight ratios; excellent strain recoverability; very soft and/or very stiff deformation modes; auxetic behavior; phononic bandgaps; sound control ability; negative effective mass density; negative effective stiffness; negative effective refraction index; superlens behavior; and/or localized confined waves, to name some examples (cf. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein). The category of "extremal materials" has been introduced in [3] to define materials that simultaneously show very soft and very stiff deformation modes (unimode, bimode, trimode, quadramode and pentamode materials, depending on the number of soft modes).…”

“…The present study with the use of pentamode lattices as tunable seismic isolation devices, making profit from the control of the soft modes of such materials through the tuning of the bending moduli of members and junctions [8][9][10][11]. An illustrative numerical example shows a practical implementation of a pentamode lattice as a base isolation device, establishing a comparison of the elastic moduli and the force-displacement response of such a system with those exhibited by a conventional rubber bearing.…”