Peridynamic differential operator for solution of high-order beam and plate equations

Carrera unified formulation (CUF) solves the elasticity problem by generating hierarchical approximated solution based on fundamental nuclei, leading to the automatic implementation of any classical and refined structural theories. Refined beam and plate theories constructed on the basis of CUF provide dimensionally reduced models for the 3D analysis of structures with distinctive 1D or 2D characteristics, which is especially attractive to nonlocal meshless methods. This study presents a strong-form solution for the refined beam and plate theories using peridynamic differential operator (PDDO) and generates a unified 3D formulation of both 1D and 2D PD models. The CUF-PDDO governing equations are derived by combining the principle of virtual work and CUF, where the cross-sectional (for 1D refined beam theories) and through-thickness (for 2D refined plate theories) characteristics are approximated by Maclaurin expansion of the unknown variables. The requirements of symmetric horizon and smooth displacement field are removed by PDDO, which allows direct application of boundary conditions without any supplementary assumptions or additional constraints. The present approach is verified through static analysis of beams with different cross-sections and plates with different thicknesses under various boundary conditions including uniform displacement, stress and concentrated force. The 3D deformation details are unveiled in good agreement with the results computed by ABAQUS and CUF-FEM.