Locally resonant acoustic metamaterials (LRAMs) are engineered materials that exhibit interesting properties for wave control, e.g. frequency bandgaps, where, for certain frequencies, waves can not propagate due to the interaction with microscale dynamics. Bandgaps are typically characterised in infinite domains, which are rather far from realistic applications where devices have boundaries and complex loading conditions. However, finite-sized domains create an issue of computational affordability due to an excessively large number of degrees of freedom. Reduced-order computational homogenisation, e.g. Sridhar et al. [1], which is a linear multiscale technique, provides an equivalent macroscale continuum with non-standard constitutive relations retrieved from microstructural simulations. It allows not only to overcome computational cost issues but also to find an effective continuum for an LRAM by appropriately transferring the microscale inertia to the macroscale model. However, so far, this multiscale method has been limited to the linear elastic regime. On the other hand, dissipation is greatly enhanced by resonant phenomena, revealing a clear need to incorporate irreversible processes into multiscale frameworks, paving the way towards developing high-fidelity models for metamaterials. In this contribution, a reduced-order computational homogenisation framework is proposed capable of upscaling microscale inertia and dissipative phenomena of solid and/or fluid LRAMs, i.e., going beyond the elastic regime. Numerical examples demonstrate the efficiency and suitability of the proposed multiscale approach. [1] Sridhar, A., Kouznetsova, V.G. and Geers, M.G.D. Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. Comput. Mech. (2016) 57(3): 423--435.