A free vibration analysis of carbon nanotube reinforced magneto-electro-elastic nanoplates using nonlocal strain gradient theory

This study presents a combination approach of the higher-order shear deformation theory, nonlocal strain gradient theory (NSGT) and isogeometric analysis (IGA) for the free vibration of carbon nanotube-reinforced (CNT) magneto-electro-elastic (MEE) nanoplates. To account size-dependent effects at the nanoscale, the classical theory model is extended with two additional scale parameters. However, this extended model necessitates at least the third derivative of the approximation function, which is incompatible with the standard finite element method. So, IGA with NURBS offers higher-order continuity through its basis functions, making it well-suited for this size-dependent model. To simplify computations, a power-law scheme is employed to represent the material properties. Various distribution types of carbon nanotubes (CNTs) including UD, FG-X, FG-O and FG-V are incorporated to investigate their effects on mechanical behaviors of CNT-MEE nanoplates. The governing equations of motion are derived in their weak form using the principle of extended virtual displacement and then solved by isogeometric analysis (IGA). The impact of the magnetic, electric and elastic fields on the coupling behaviors of CNT-MEE nanoplates are studied. Specially, parametric studies are conducted to analyze the influence of geometrical parameters, CNT distributions, CNT volume fraction, matrix volume fraction, electric voltage, magnetic potential, nonlocal and strain gradient parameters on the natural frequencies of the CNT-MEE nanoplates. Comparisons between the results obtained using NSGT and the classical theory reveal significant findings. The natural frequencies calculated by NSGT exhibit dependence on the relative values of the nonlocal and strain gradient parameters.