Three-Dimensional Free Vibration Analysis of Hyperelastic Structures under Bending Load Using the VDQ-Transformed Method

Using the ideas of variational differential quadrature (VDQ) technique and position transformation, an efficient numerical approach is developed herein in order to address the free vibration problem of compressible and nearly-incompressible solid bodies with arbitrary deformed shape within the framework of 3D hyperelasticity. The 3D hyperelasticity is first formulated by vector-matrix relations with the purpose of applying in coding process. An energy principle together with the Neo-Hookean strain energy function is also employed in the derivation of governing equations. The proposed numerical method is capable of addressing problems with irregular domains. Simple application, being free from the locking problem, and fast convergence rate are the key features of the approach. Hyperelastic rectangular/sector plates and cylindrical panel subjected to bending load are selected as test problems whose free vibrations are analyzed. The developed numerical method is found to be capable of yielding accurate solutions to the considered problems. Moreover, the effects of mode transition and geometrical properties are investigated in the numerical examples.