We perform a parametric study of the analytic model of Liu et al. [ Z. Liu et al. Phys. Rev. B 71 014103 (2005)] describing the mechanical response of a core-shell particle to an acoustic excitation in order to help in selecting the constitutive materials and in designing innovative processes of fabrication of downsized core-shell resonators, which are key constituents of locally resonant acoustic metamaterials. We show that the value of the first Lamé coefficient of the material constituting the shell has no marked influence on the value of the resonance frequency of the core-shell resonator, that is, it does not necessarily need to be small for satisfying the condition of subwavelength resonator dimension at resonance. Moreover, we show that the larger the density contrast between the core and the shell and the thinner the shell, the broader is the frequency band over which the effective density of the resonator suspension is negative, but that it is practically useless to decrease the dimensionless shell thickness below 0.6. Finally, we show that the dissipation is also less perceptible the thinner is the shell and the larger is the density contrast. The effect of the density contrast between the core and the shell and of the dissipation on the resonance width are explained by comparing with the harmonic oscillator and the mass-in-mass 1D lattice.Read less <