Abstract Nonlinear dynamic behavior of double curved shallow shells with negative Poisson’s ratios in auxetic honeycombs on elastic foundations subjected blast, mechanical and damping loads is investigated in the present article. This study considers double curved shallow shells with auxetic core which have three layers in which the top and the bottom outer skins are isotropic aluminum materials; the central layer has honeycomb structure using the same aluminum material. Based on the analytical solution, Reddy’s third order shear deformation theory (TSDT) with the geometrical nonlinear in von Karman and Airy stress functions method, Galerkin method and the fourth-order Runge-Kutta method, the resulting equations are solved to obtain expressions for nonlinear motion equations. The effects of geometrical parameters, material properties, elastic foundations, imperfections, blast loads, mechanical and damping loads on the nonlinear dynamic analysis of double curved shallow shells with negative Poisson’s ratios in auxetic honeycombs are studied.