Effective mass and effective stiffness of finite and infinite metamaterial lattices

In this paper, we explore the dynamic behaviour of a metamaterial lattice which is composed of elastically connected concentrated masses coupled with local resonators (mass-in-mass lattice also referred to as the Kelvin lattice). This kind of metamaterial lattice is known to be associated with interesting phenomena such as a mass reduction effect, as compared to an equivalent Lagrange lattice composed of elastically connected effective masses. The effective mass of this metamaterial lattice is exactly calculated for finite and infinite metamaterial structures. For finite systems, the effective mass is identified from the eigenfrequencies spectrum of the metamaterial lattice. These eigenfrequencies are obtained for various boundary conditions of the metamaterial lattice, by solving a linear difference eigenvalue problem. For infinite systems, the wave dispersion of the metamaterial lattice is compared to the effective Born-Karman dispersion relation. It is shown for both the finite and the infinite systems, that the effective mass related to each branch (acoustical and optical branches) is proportional to the square of frequency associated with the complementary branch. An asymptotic expansion confirms that the effective mass tends towards the sum of each mass (the primary mass and the mass of the resonator) for the long-wave approximation of the acoustical branch. The effective mass is greater than the sum of each mass for the acoustical branch and is significantly reduced for the optical branch (with eventually a negligible reduced mass). An asymptotic analysis confirms the parametric study for each branch of the lattice spectrum. The effective stiffness is also calculated for both the finite and the infinite systems. A nonlocal continuous approach is developed to approximate the metamaterial lattice response in terms of frequency spectrum or effective mass variation. This study is concluded by a design-oriented methodology for the calibration of the effective mass from the metamaterial lattice properties.