On the equations of the surface elasticity model based on the theory of polymeric brushes

Motivating by theory of polymers, in particular, by the models of polymeric brushes we present here the homogenized (continual) two-dimensional (2D) model of surface elasticity. A polymeric brush consists of an system of almost aligned rigid polymeric chains. The interaction between chain links are described through Stockmayer potential, which take into account also dipole-dipole interactions. The presented 2D model can be treated as an highly anisotropic 2D strain gradient elasticity. The surface strain energy contains both first and second derivatives of the surface field of displacements. So it represents an intermediate class of 2D models of the surface elasticity such as Gurtin-Murdoch and Steigmann-Ogden ones.