2D nonlinear dynamic analysis of hyperelastic structures resting on nonlinear elastic foundation using the VDQ-transform method

Abstract

This paper presents a new variational differential quadrature (VDQ)-transform method for the two-dimensional nonlinear dynamic analysis of hyperelastic structures resting on nonlinear elastic foundations. The developed formulation is established within Hamilton's variational framework using the compressible Neo-Hookean constitutive model, thereby accounting for both material and geometrical nonlinearities under finite deformation. The nonlinear foundation is modeled using an extended three-parameter Winkler-Pasternak formulation, incorporating shear interaction and nonlinear stiffness coupling. A major novelty of the study lies in the tensor-to-matrix transformation, which systematically converts higher-order tensor operations into compact matrix-vector relations for stresses and tangent moduli, ensuring computational efficiency and easy implementability across arbitrarily shaped geometries. Benchmark problems—including a curved beam and Cook's membrane—are provided to validate the proposed approach and to demonstrate its performance. The results highlight that nonlinear foundation parameters substantially modify the transient response and stability limits of hyperelastic structures, emphasizing the significance of combined geometrical and material nonlinearities in advanced flexible systems. © 2025 Elsevier Inc.