Strong-form PDDO based solution of high-order beam and plate theories

Carrera unified formulation (CUF) solves the elasticity problem by generating hierarchical approximated solution based on the 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 then 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 peridynamic solution for the refined beam and plate theories using peridynamic differential operator (PDDO). The refined 1D and 2D structural theories are established by means of CUF in strong form, where the cross-sectional (for 1D refined beam theories) and through-thickness (for 2D refined plate theories) characteristics are approximated by Taylor expansion in conjunction with Lagrange expansion. The displacement field is obtained through a formal expression regardless of the order of the theory, which is considered as the only input of the CUF analysis. The local differential operators are replaced by PDDO. PDDO is based on the concept of peridynamic interactions and constructed through Taylor series expansion with orthogonality property. It lessens the requirements over the symmetry of horizon and the smoothness of displacement field. The domain can therefore be divided into two regions concerning the inner domain and boundary layer. Displacement and traction boundary conditions can be applied directly on the boundary layer without any supplementary assumptions or additional means. The present method is verified through static analyses of beams with different cross-sectional shapes and plates with different thicknesses under various boundary conditions including displacement, stress and concentrated force boundaries. The 3D deformation details are unveiled in good accordance with results computed by Abaqus-FEM and CUF-FEM.