Abstract

Emerging generations of nanomaterials and biomaterials use ultrathin nanoplatelets or membrane sheets as reinforcements. In the talk, we discuss the use of the theories of material surfaces for modeling such materials. The theories describe an infinite isotropic elastic medium that contains a material surface treated as either a membrane or a shell of vanishing thickness. The material surface is characterized by its own elastic stiffness as well as residual surface tension The governing equations of the theories are reviewed and challenges associated with the theoretical analysis and numerical solutions of resulting non-classical boundary value problems are discussed. For solving the problems in the plane strain setting, the displacements in the matrix are sought in the complex variables form of a single layer elastic potential. The density of the potential represents the jump in complex tractions across the surface. Exact expressions for the elastic fields in the matrix are provided in terms of complex integral representations. We present solutions for the two-dimensional case of a surface along a straight segment. The dimensionless parameters that govern the problems are identified and the numerical examples, that illustrate their influence on the local elastic  fields in the material, are discussed.