Mixed meshless local Petrov-Galerkin (MLPG) collocation methods for gradient elasticity theories of Helmholtz type

Two mixed meshless collocation methods for solving problems by considering a linear gradient elasticity theory of the Helmholtz type are proposed. The solution process is facilitated by employing operator-split procedures, splitting the original 4th-order problem into two uncoupled second-order sub-problems, which are then solved in a staggered manner by applying the mixed meshless local Petrov-Galerkin collocation strategy. Thereby, identical nodal pattern is used for the discretization of both lower-order problems, while the approximation of all unknown field variables is performed by applying the Moving Least Squares functions with interpolatory conditions. The performance of the derived methods is tested by some suitable numerical examples, dealing with elasticity problems that are often treated by gradient theories. Therein, it is demonstrated that both proposed methods are able to capture the size effect and that the strain-based method is able to produce the non-singular strain field around the crack tip, similar to the nonlocal elasticity approach of Eringen. It has been found out that the obtained results agree well with available analytical and numerical solutions.