Strain gradient theory in orthogonal curvilinear coordinates

In this short note, general formulations of the Toupin–Mindlin strain gradient theory in orthogonal curvilinear coordinate systems are derived, and are then specified for the cases of cylindrical coordinates and spherical coordinates. Expressions convenient for practical use are presented for the corresponding equilibrium equations, boundary conditions, and the physical components for strains and strain gradients in the two coordinate systems. The results obtained in this paper are general and complete, and can be useful for a wide range of applications, such as asymptotic crack tip field analysis, cylindrical and spherical cavity expansion problems, and the interpretation of micro/nano indentation tests and bending/twisting tests on small scales.