A mathematical framework of high-order surface stresses in three-dimensional configurations

The mathematical behavior of a curved interface between two different solid phases with surface or interface stress effects is often described by the generalized Young–Laplace (YL) equations. The generalized YL equations can be derived by considering force equilibrium of a thin interphase with membrane stresses along the interface. In this work, we present a refined mathematical framework by incorporating high-order surface or interface stresses between two neighboring media in three dimensions. The high-order interface stresses are resulting from the nonuniform surface stress across the layer thickness, and thereby effectively inducing a bending effect. In the formulation, the deformation of the thin interphase is approximated by the Kirchhoff–Love assumption of thin shell. The stress equilibrium conditions are fulfilled by consideration of balance for forces as well as stress couples. By simple geometric expositions, we derive in explicit form the stress jump conditions for high-order surface stresses. In illustrations, the bending deformation of nanoplates with high-order stresses is investigated and is compared with the results by the conventional YL equation.