A new application of quadrilateral finite element model incorporating the discrete shear projection technique for free vibration response of CNT reinforced plates

Abstract

This study presents an advanced quadrilateral finite element model for analyzing the free vibration behavior of functionally graded carbon nanotube-reinforced (FG-CNTRC) nanocomposite plates. The proposed element, derived from the classical four-node Q4 element, incorporates a discrete shear projection method to evaluate transverse shear deformation using precise interpolation techniques. This approach effectively captures the complex mechanical behavior of nanocomposite plates while avoiding the computational complexity associated with higher-order shear deformation models. The developed Q4γ element, based on first-order shear deformation plate theory, features five degrees of freedom per node and maintains inter-element continuity through C⁰ continuity for kinematic variables. Isoparametric coordinates generate elementary stiffness and mass matrices, enhancing the formulation’s accuracy. Notably, the model mitigates shear locking without resorting to sophisticated numerical techniques. Governing equations are derived using the weak form of the variational principle. The mechanical properties of FG-CNTRC plates are modeled to vary gradually across the thickness, accounting for different distribution patterns and CNT volume fractions. The model’s performance is rigorously validated against analytical solutions and established finite element models, demonstrating excellent accuracy without requiring excessive mesh refinement. Comprehensive numerical investigations explore the influence of material and geometric configurations on the free vibration response of FG-CNTRC plates. The Q4γ element proves particularly effective in capturing both in-plane and out-of-plane responses in advanced composite structures. Results indicate that optimized reinforcement distribution patterns can significantly enhance computational efficiency. This research provides a valuable tool for designing and optimizing CNT-reinforced nanocomposite structures, with potential applications in aerospace and automotive industries where multiphysics environmental impacts are critical. The model’s ability to accurately predict vibrational behavior while maintaining computational efficiency represents a significant advancement in nanocomposite structural analysis.