Free vibration response of bidirectional functionally graded rotating micro-disk under mechanical and thermal loading

The present work studies the free vibration response of functionally graded rotating micro-disks subjected to transverse pressure and thermal loading based on the modified couple stress theory. The disk material is considered to be functionally graded along the radial and thickness directions, and its properties are assumed to be temperature-dependent following the Touloukian model. The mathematical formulation is based on an energy functional involving the von Kármán type non-linearity, in which appropriate displacement derivatives and its conjugate stress measures are used to define the strain energy of the micro-disk. The minimum potential energy principle is employed to develop the governing equations for determining the deformed configuration of the micro-disk under combined centrifugal, pressure and thermal loading. Further, the governing equations for free vibratory motion of the micro-disk are derived following Hamilton’s principle and incorporating the tangent stiffness of the deformed micro-disk. The governing equations are discretized and solved employing the Ritz method. The mathematical model is successfully validated with different reduced problems available in the literature. The influence of rotational speed, transverse pressure, thermal loading, size-dependent thickness and volume fraction indices are investigated for a wide range of parametric values. Some illustrative mode shapes along with the contour have also been presented. The present study is first of its kind and the presented results would definitely serve as benchmarks for any further study in this field.