3D dynamic analysis of elastically restrained multi-directional FGMs rectangular parallelepiped

In this research, the dynamic features of three-directional functionally graded materials (3DFGMs) rectangular parallelepiped with classic/elastic restraints are investigated based on 3D elastic theory. The general boundary conditions are implemented by introducing artificial displacement springs on the chosen surfaces of rectangular solid. The gradient materials are distributed along the two in-plane and thickness directions of parallelepiped. By setting boundary constraints and geometric parameters, the 3DFGMs rectangular parallelepiped can be evolved into slender beam, thick or thin plate, or even a cuboidal solid. Lagrangian energy functions are formulated for parallelepiped-spring system. The free vibration characters of 3DFGMs rectangular parallelepiped are solved employing the Ritz method in conjunction with the Jacobi polynomials. For transient analysis, the analytical expressions of impulse responses are derived for different types of pulsed excitation. The presented modeling and solution methods are validated by comparing with the results from open literature, finite element analysis and experimental results. Numerical simulations are performed to reveal the effect mechanisms of material gradients, geometrical configuration and boundary restraints on the vibration characters of 3DFGMs parallelepiped. The results demonstrate that dynamic performance of rectangular parallelepiped depends critically on material gradient which may be regarded as regulatory factor to regulate the modal displacement distribution or modal sequence.