Formulation of Toupin–Mindlin strain gradient theory in prolate and oblate spheroidal coordinates

•We formulate the Toupin–Mindlin theory in orthogonal curvilinear coordinates.•Formulae of the theory in prolate and oblate spheroidal coordinates are given.•Basic equations are derived in prolate and oblate spheroidal coordinates.•The previous results in cylindrical and spherical coordinates are amended.

The Toupin–Mindlin strain gradient theory is reformulated in orthogonal curvilinear coordinates, and is then applied to prolate and oblate spheroidal coordinates for the first time. The basic equations, boundary conditions, the gradient of the displacement, strain and strain gradient tensors of this theory are derived in terms of physical components in these two coordinate systems, which have a potential significance for the investigation of micro-inclusion and micro-void problems. As an example, using these formulae, we formulate and discuss the boundary-value problem of a spheroidal cavity embedded in a strain gradient elastic medium subjected to uniaxial tension. In addition, the previous results given by Zhao and Pedroso (Int. J. Solids. Struct. (2008) 45, 3507–3520) in cylindrical and spherical coordinates are amended.