Nonlinear Free Vibration Analysis of Sandwich Double-Curved Shallow Shells with Auxetic Core and Functionally Graded Face Sheets Resting on Linear and Nonlinear Elastic Substrates

Sandwich structures with auxetic cores possess significant properties due to a negative Poisson ratio that can be utilized in high-tech industries. This paper investigates the nonlinear free vibrations analysis of sandwich double-curved shallow (SDCS) shells with auxetic core and functionally graded (FG) face sheets. The SDCS shell rests on two linear foundations and one nonlinear fifth-degree substrate. First, the strain and kinetic energies of the shell are obtained considering a displacement field based on first-order shear deformation theory (FSDT) and von Karman nonlinear strain–displacement relations. The nonlinear dynamic equations are derived from the Hamilton principle. The partial differential equations are transformed into ordinary differential equations employing the Galerkin method. Then, the nonlinear equation is analytically solved using the harmonic balance method. Eventually, the effect of critical parameters of the SDCS shell, such as auxetic material, geometric ratios, FG gradient index and elastic foundations stiffness, on the structure’s linear and nonlinear natural frequency and hardening behavior is assessed.