A FEM-based iterative method for Mindlin integral nonlocal plates

Shear-deformable nanoplates are investigated within a constitutive framework combining local and stress-driven nonlocal elastic phases. Size effects are incorporated through integral convolutions of the stress resultants. The resulting governing equations are solved by a FEM-based iterative procedure, consisting of a sequence of linear boundary-value problems terminated upon satisfaction of a convergence criterion. By setting the local phase to zero, the proposed formulation reduces to the purely nonlocal model, which has been widely and successfully employed in nonlocal analysis of one-dimensional nanostructures, such as nanobeams. Although iterative solution schemes are known to be effective for purely nonlocal one-dimensional problems, the present study shows that convergence issues arise when such approaches are applied to purely nonlocal two-dimensional structures. By contrast, the proposed mixed local–nonlocal model exhibits robust numerical behavior, ensuring rapid, stable convergence of the iterative scheme. Using the mixed model, convergence is achieved to arbitrary accuracy, up to machine precision, for both moderately thick and thin nanostructured elements (plates and beams) with arbitrary geometry, boundary conditions, loading, and nonlocal convolution kernels. This robustness across structurally and physically diverse nanostructures highlights the generality of the proposed approach and supports its applicability to a broad class of nanoengineered systems. Representative numerical examples are provided to illustrate and support these findings.