Broadband dynamic elastic moduli of honeycomb lattice materials: A generalized analytical approach

Abstract

A generic analytical framework is proposed to obtain the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The dynamic deformation behaviour of the individual beam elements of a lattice is distinct from the behaviour under a static condition. This leads to a completely different global deformation pattern of the lattice material and subsequently opens up a tremendous opportunity to modulate amplitude and phase of the elastic properties of lattices as a function of the ambient vibration. The dynamic stiffness approach proposed in this article precisely captures the sub-wavelength scale dynamics of the periodic network of beams in a lattice material using a single beam-like member. Here the dynamic stiffness matrix of a damped beam element based on the Timoshenko beam theory along with axial stretching is coupled with the unit cell-based approach to derive the most general closed-form analytical formulae for the elastic moduli of lattice materials across the whole frequency range. It is systematically shown how the general expressions of dynamic elastic moduli can be reduced to different special cases by neglecting axial and shear deformations under dynamic as well as classical static conditions. The significance of developing the dynamic stiffness approach compared to conventional dynamic finite element approach is highlighted by presenting detailed analytical derivations and representative numerical results. Further, it is shown how the analytical framework can be readily extended to lattices with non-prismatic beam elements with any spatial variation in geometry and intrinsic material properties. In general, research activities in the field of lattice metamaterials dealing with elastic properties revolve around intuitively designing the microstructural geometry of the lattice structure. Here we propose to couple the physics of deformation as a function of vibrating frequency along with the conventional approach of designing microstructural geometry to expand the effective design space significantly. The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex-valued elastic moduli due to the presence of damping in the constituent material. The amplitude, as well as the phase of effective elastic properties of lattice materials, can be quantified using the proposed approach. The dependence of Poisson's ratio on the intrinsic material physics in case of a geometrically regular lattice is found to be in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only the function of microstructural geometry. The generic analytical approach for analysing the elastic moduli is applicable to any form of two- or three-dimensional lattices, and any profile of the constituent beam-like elements (different cross-sections as well as spatially varying geometry and intrinsic material properties) through a wide range of frequency band. The closed-form expressions of elastic moduli derived in this article can be viewed as the broadband dynamic generalisation of the well-established classical expressions of elastic moduli under static loading, essentially adding a new exploitable dimension in the metamaterials research in terms of dynamics of the intrinsic material.