<p>The main goal of this study is an application of the R-functions theory to investigate the free linear and geometrically nonlinear vibrations of the sandwich auxetic composite plates. The sandwich plate consists of three layers. The central layer is auxetic honeycomb structure with negative Poisson’s ratio. The face sheets are made of functionally graded materials (FGM). Mathematical statement of the problem is fulfilled by the first-order shear deformation theory (FSDT) with the geometrical nonlinear in von Karman. Nonlinear vibrations are studied using the Ritz method, Galerkin’s procedure and Runge–Kutta method based on the original approach proposed. Rectangular plates with special type of the boundary conditions (partially clamped and partially simply supported along each side) are studied. The corresponding system of the admissible functions is constructed by the R-functions theory. The influence of the geometrical parameters of the cell, types of FGM, thickness of the core on the natural frequencies and nonlinear behavior of sandwich auxetic honeycomb plates are analyzed.</p>