This paper presents an investigation into the free vibration and nonlinear dynamic analyses of new complex edge contour plate structures, known as single-variable-edge (SVE) plates. Cracks emerge at the center of the plate's surface, and the effects of edge crack orientation and depth on the dynamical characteristics of the laminated nanocomposite SVE plates have been considered in this study. The functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) material was utilized for the edge-cracked plate that rests on a Winker-Pasternak elastic foundation. The Halpin-Tsai micromechanical model is used to predict the material characteristics, which are presumptively temperature-dependent. The rule of mixtures is employed to determine Poisson's ratio, density mass, and thermal expansion. The governing equations are established based on the theory of elasticity, and Von Karman-Donnell geometric nonlinearity assumption. To confirm the reliability and accuracy of this approach, the acquired findings are validated with those produced using Finite Element Analysis (FEA). Furthermore, the geometrical and material parameters, fracture properties, and environmental (temperature and elastic foundation) factors are also investigated in the study. The research findings hold significant implications for applications in aerospace and aeronautical, civil, and mechanical engineering.