The role of material and geometric nonlinearities and damping effects in designing mechanically tunable acoustic metamaterials

•The roles of material and geometric nonlinearities and damping effects in the process of elastic wave manipulation are revealed.•The geometrical nonlinearity of the AM can manipulate the locally resonant and Bragg scattering band gaps.•The nonlinearity of coating material can manipulate the locally resonant and Bragg scattering band gaps, but the nonlinearity of the matrix material mainly affects the Bragg scattering band gaps.•The damping effect (Rayleigh damping and linear viscoelastic damping) in the elastomeric matrix and coating materials can lead to the appearance of a new band region and the changes in the position and width of the band region.•The material damping can suppress the elastic wave propagation in the AM structures, and excessive damping makes it hard to identify the band gaps from the transmittance curves.•The AM with holes of large size, thin coating and hard core of large radius benefits enhance its tunability of band gap.