Iterative nonlocal elasticity for Kirchhoff plates

• An iterative nonlocal elasticity model is proposed to study the bending behavior of nano Kirchhoff plates.
• The proposed nonlocal model is based up on an iterative procedure of the type local prediction/nonlocal correction.
• In each iterative step the local field problem is solved and corrected for the nonlocal field.
• The nonlocal field is diffused to the local field by imposed-like correction strain energy.
• In the iterative-based nonlocal model, the local boundary conditions are applied instead of the nonlocal ones.

Recently, the nonlocal elasticity theories have been used in studying the different behaviors of micro/nanostructures. However, there is a complicity in applying the natural boundary conditions in the context of the nonlocal differential elasticity models. Also, the nonlocal integral elasticity could provide a suitable remedy for this type of problems but with paying highly computational efforts.In the present work, an iterative nonlocal elasticity model is proposed to study the bending behavior of nano-sized Kirchhoff plates, having both essential and natural boundary conditions with less computational efforts compared with conventional nonlocal elasticity methods. The proposed iterative nonlocal model is based up on an iterative procedure of the type local prediction/nonlocal correction, in which the nonlocal field is diffused to the local field by imposed-like correction strain energy. The most distinguished feature of this iterative procedure-based model is that in each iterative step the local field problem is solved and corrected for the nonlocal field. Therefore, the local boundary conditions are applied, in each iterative step, instead of the nonlocal ones resolving the troubles existing in most of other nonlocal solutions.Some numerical examples are solved for nonlocal Bernoulli–Euler beams and Kirchhoff plates with simple supported, clamped and clamped-free boundary conditions. The obtained results reflected the high efficiency of the proposed iterative procedure thus it converged in few iterations.