Nonlinear vibration characteristics of refined higher-order multi-phase piezo-magnetic nanobeams Raad M. Fenjan, Ridha A. Ahmed and Nadhim M. Faleh Eur. Phys. J. Plus, 135 5 (2020) 439 Published online: 28 May 2020 DOI: 10.1140/epjp/s13360-020-00399-4