Finite element modeling of free vibration of cracked nanoplates with flexoelectric effects Duc Hong Doan, Ashraf M. Zenkour and Do Van Thom Eur. Phys. J. Plus, 137 4 (2022) 447 Published online: 09 April 2022 DOI: 10.1140/epjp/s13360-022-02631-9