Element-free Galerkin method for static analysis of nonlocal elastic solids using differential and integral straindriven models

The element-free Galerkin (EFG) method is formulated and applied to analyse the bending mechanics of Euler-Bernoulli beams, utilising Eringen’s strain-driven constitutive model. EFG, a meshless numerical method, is particularly well-suited for simulating non-local solids due to its inherent ability to handle non-local interactions. The formulation effectively incorporates long-range interactions within the entire domain through carefully constructed weighting functions, allowing for a more accurate representation of the physical behaviour of materials with non-local effects. This study examines the bending response of nano/micro beams subjected to standard boundary conditions and a set of non-local parameters. The analysis uses Eringen’s differential and original integral forms of the non-local constitutive models. A discrete set of governing equations for the integral model is obtained by minimising the potential energy of the system, which captures the non-local effects through integral operators that account for interactions over finite distances. The boundary conditions, crucial for accurately capturing the beam’s response, are implemented using a scaled transformation method and Lagrange multipliers. The study investigates key structural responses under various boundary conditions, including deflections, slopes, bending moments, and shear forces. These results are then compared against available analytical solutions and numerical results from other methods.