Size-dependent nonlinear free vibration of bi-directionally functionally graded graphene platelet-reinforced composite nanoplates via isogeometric analysis: Quantitative insight into the softening-hardening interplay

This study establishes a novel isogeometric analysis (IGA) framework, integrating the nonlocal strain gradient theory (NSGT) and sinusoidal shear deformation theory (SSDT), to investigate the geometrically nonlinear free vibration of bi-directionally functionally graded graphene platelet-reinforced composite (BFG-GPLRC) nanoplates. The spatial distribution of graphene platelets (GPLs) is tailored along the length and thickness directions, with effective material properties characterized by the modified Halpin-Tsai model. The governing equations incorporate both nonlocal softening and strain-gradient hardening mechanisms, discretized using Non-Uniform Rational B-Splines (NURBS) basis functions that satisfy the rigorous higher-order continuity requirements of NSGT. Following validation against benchmarks, parametric studies on square and annular nanoplates examine how geometric and material parameters, size effects, and vibration amplitude influence nonlinear response. A critical finding reveals amplitude-dependent evolution of the softening-hardening interplay in which nonlocal softening exhibits weak amplitude dependence, whereas strain-gradient hardening is significantly amplified by geometric nonlinearity. Quantitative assessment establishes that for square nanoplates, the length-to-thickness ratio emerges as the primary parameter governing the amplitude-dependent softening-hardening balance, with thick plates exhibiting a transition from softening-dominated to hardening-dominated behavior. For annular nanoplates, the bidirectional material gradient indexes play the dominant role, with the radial and thickness gradient indexes exerting opposing effects of nearly equal magnitude.