Three-dimensional nonlinear primary resonance of functionally graded rectangular small-scale plates based on strain gradeint elasticity theory

Abstract

Abstract In this paper, a size-dependent three-dimensional (3D) nonlinear weak formulation is provided to examine the nonlinear primary resonance problem for functionally graded rectangular small-scale plates. The small-scale factors are taken into formulation by choosing the Mindlin's strain gradeint theory (SGT). According to the variational differential quadrature (VDQ) method, first, the displacement field, nonlinear strain-displacement and constitutive relations as well as the potential and kinetic energies are expressed as the vector and matrix forms. Then, by applying the discretized form of differential operators obtained via the generalized differential quadrature (GDQ) method, the discretized form of aforementioned relations is achieved. Finally, Hamilton's principle is employed to access the weak form of 3D nonlinear governing equations of thick rectangular small-scale plates. The achieved formulation is solved via a multi-step numerical technique to address the size-dependent nonlinear primary resonance of considered system under the harmonic lateral force. In addition to reducing the run time, computational effort and CPU usage, the feature of proposed weak form formulation is that one can employ it in other solution approaches such as finite element method. Also, the use of this formulation provides the possibility of recovering models on the basis of other types of size-dependent theories such as modified strain gradient and modified couple stress theories (MSGT and MCST). In the numerical results, the effects of boundary conditions, small-scale parameter, material index and geometry are examined.