This paper presents free vibration analyses and nonlinear dynamic responses of new complex edge contour non-rectangular plate structures with a part-through surface crack which the authors name as temporarily double-variable-edge (DVE) plates. The analytical results and numerical simulation are investigated in the Part I and it is expected that the experimental approach will be shown in Part II. The cracked plate made of a functionally graded porous (FGP) matrix structure and reinforced by graphene nano-platelets (GNPs) is covered by two piezoelectric layers and rested on a Winker-Pasternak elastic foundation. The material properties are determined by using the Halpin-Tsai micro-mechanical model, Gaussian Random Field (GRF) proposal, and the rules of mixtures. The governing equations are established based on the classical plate theory (CPT) and Galerkin's method which are convenient in analyzing the behaviors of complex edge contour plates. The obtained results are compared with those obtained from the existing Finite Element Method (FEM) to ensure the reliability and accuracy. In addition, the effects of geometrical shapes, crack parameters, applied voltages of piezoelectric layers, elastic foundations as well as thermal impacts are scrutinized. The obtained outcomes in the paper are promising for applications in the fields of aerospace, civil, and electronic engineering.