Effect of non-dimensional length scale in element free Galerkin method for classical and strain driven nonlocal elasto-static problems

Abstract

Meshfree methods, such as the Element-Free Galerkin (EFG) Method, offer flexibility in approximating field variables and their derivatives through various tunable parameters. Non-dimensional length scale ($d_{max}$) is one such parameter which govern the size of local support domain. The extend of local support domain is a measure of nonlocality in the meshfree approximations. This inherent flexibility to adjust nonlocality by changing the value of $d_{max}$ can be used for simulation of nonlocal elasticity problems using EFG method. This study formulates EFG method for one-dimensional nonlocal Euler–Bernoulli beams based on Eringen’s strain-driven nonlocal model. Numerical analyses are conducted on elasto-static problems in classical and strain-driven nonlocal Euler–Bernoulli beams with standard boundary conditions to evaluate the influence of the $d_{max}$ in EFG method. The results provides a clear guidance for choosing value of $d_{max}$ for accurate approximation of field variable and derivatives in classical and nonlocal elastic beam problems. For classical beams, the commonly accepted range of $d_{max}$ between two and four fails to adequately ensure force and moment equilibrium at supports. Similar trends are observed in nonlocal analysis, where even larger local supports are necessary due to the complexity introduced by the nonlocal constitutive modeling. The challenge in enforcing boundary conditions is handled by a modified approach, combining scaled transformations and Lagrange multipliers. Constitutive boundary conditions of nonlocal beams required for the satisfaction of equilibrium requirements is effectively incorporated using this method.