Second strain gradient theory in orthogonal curvilinear coordinates: Prediction of the relaxation of a solid nanosphere and embedded spherical nanocavity

In this paper, Mindlin’s second strain gradient theory is formulated and presented in an arbitrary orthogonal curvilinear coordinate system. Equilibrium equations, generalized stress-strain constitutive relations, components of the strain tensor and their first and second gradients, and the expressions for three different types of traction boundary conditions are derived in any orthogonal curvilinear coordinate system. Subsequently, for demonstration, Mindlin’s second strain gradient theory is represented in the spherical coordinate system as a highly-practical coordinate system in nanomechanics. Second strain gradient elasticity have been developed mainly for its ability to capture the surface effects in the presence of micro-/nano- structures. As a numeric illustration of the theory, the surface relaxation of spherical domains in Mindlin’s second strain gradient theory is considered and compared with that in the framework of Gurtin–Murdoch surface elasticity. It is observed that Mindlin’s second strain gradient theory predicts much larger value for the radial displacement just near the surface in comparison to Gurtin–Murdoch surface elasticity.