Uniqueness of solution for plane deformations of a micropolar elastic solid with surface effects

We present a linear theory describing plane-strain deformations of a micropolar elastic solid which incorporates the additional contribution of surface micropolar elasticity. This theory provides a more comprehensive account of size effects in solids allowing for a more accurate representation of, for example, the mechanical properties of micro-/nanostructures. The surface model is represented by a thin micropolar reinforcing elastic shell of separate micropolar elasticity, perfectly bonded to the micropolar bulk and capable of both extension and bending. We present two different versions of the surface shell model: one which gives rise to a fourth-order surface theory and another which, via a particular form of the Kirchhoff–Love kinematic assumption, results in a second-order surface model. In each case, we formulate interior and exterior mixed boundary value problems and show that they can have at most one smooth solution despite the presence of boundary conditions of order equal to or higher than that of the governing equations. This result is essential in establishing that the corresponding mixed boundary value problems are well-posed in certain classes of smooth matrix functions.