A unified reproducing kernel gradient smoothing Galerkin meshfree approach to strain gradient elasticity

A unified reproducing kernel gradient smoothing formulation is presented for efficient Galerkin meshfree analysis of strain gradient elasticity problems with particular reference to high order basis functions, such as cubic and quartic basis functions. The integration constraint is elaborated for the Galerkin meshfree formulation of strain gradient elasticity, where the possible inconsistency issue associated with the separate first and second order gradient smoothing formalism is completely resolved. In this approach, the first order smoothed gradients of meshfree shape functions are expressed as a reproduced kernel form and the second order smoothed gradients are deduced from their first order counterparts through direct differentiation. Thereafter, the unknown coefficients in the first and second order smoothed gradients are simultaneously solved from the integration constraint. The resulting smoothed gradients automatically meet the integration consistency by construction. Subsequently, it is proven that the reproducing kernel gradient smoothing quadrature rules for the conventional elasticity problems are well suitable for the strain gradient elasticity problems. Numerical results clearly demonstrate the superior performance of the proposed approach regarding convergence, accuracy, as well as efficiency, in comparison with the Gauss integration-based meshfree formulation for strain gradient elasticity problems.