Analytical methods for the pyramidal truss metamaterial

Abstract

Mechanical metamaterials are a subclass of metamaterials which exhibit exotic effective mechanical proper- ties. These may include a negative bulk modulus, a negative Poisson's ratio, or a negative mass density. Subse- quently, the kinematics of these materials often defy intuition. Auxetics are a class of mechanical metamaterials that demonstrate this by widening in the direction perpendicular to the loading, as opposed to thinning. Addi- tionally, mechanical metamaterials can be characterized by their local, highly geometric structure which takes the form of a unit cell. Thus, the metamaterial can be seen as an amalgam of unit cells, with the exotic me- chanical behavior a result of the aggregate effects of unit cells and their interactions with each other. Precisely formulating how this occurs can prove challenging, due to nonlinearities in unit cells kinematics, along with higher order effects due to the interaction of unit cells. The work presented carries out this exact task, where we examine a specific mechanical metamaterial whose unit cell geometry is that of a square pyramid. To be precise, we explore surfaces which are composed of square pyramidal unit cells. We detail the kinematics of the unit cell and use these results to obtain a set of vector partial differential equations which dictate the behavior of our surface. Out of the kinematicaly admissible surfaces formed by our unit cell, we spend the bulk of the paper detailing the case of surfaces of revolution. We notice through simulations of the metamaterial that surfaces of this type possess oscillations along their axial direction, whose amplitude and wavelength are scale dependent. We use perturbation theory to show analytically that the profiles of these oscillations along the axial direction satisfy the nonlinear pendulum equation. Approximate results for the oscillation amplitude and wave length are given. Weakly curved surfaces that only deviate slightly from the planar state are also explored, with asymptotic methods used to obtain solutions for the displacements.