An asymptotic expansion method for multi-cell homogenization leading to strain gradient elasticity

Strain gradient elasticity theory, which incorporates intrinsic material length parameters into its constitutive relationships, is extensively utilized in modeling size-dependent mechanical behaviors of solids and structures in the engineering science. However, numerical challenges such as overflow phenomena are frequently encountered during solution procedures, whether through analytical solutions or numerical methods like the finite element method, boundary element method, and finite difference method. In this paper, we propose an asymptotic expansion method for addressing boundary value problems associated with strain gradient elasticity theory produced by a multi-cell homogenization. This method allows for the expansion of the displacement vector and frequency parameter in series form, thereby generating a sequence of new boundary value problems to be solved at varying powers of the intrinsic material length parameters. Analytical solutions for ordered frequency parameters are derived using newly proposed boundary conditions, integration by parts, and a unified identity introduced by the present authors. Two examples illustrating free vibration of strain gradient rods and beams are provided to demonstrate the efficiency of the asymptotic expressions developed within the theoretical framework of the asymptotic expansion method for strain gradient elasticity. It is anticipated that the asymptotic expansion method established in this work can serve as an effective computational tool within strain gradient theory, particularly in scenarios where conventional numerical methods, such as the finite element method, fail due to overflow issues.