On the theories of plates and shells at the nanoscale

Abstract

During the last 50 years the nanotechnology is established as one of the advanced technologies manipulating matter on an atomic and molecular scale. New materials, devices or other structures possessing at least one dimension sized from 1-100nm are developed. The question arises how structures composed of nanomaterials should be modeled. In addition, if it is necessary to perform a structural analysis what kind of theory should be used. Two approaches are suggested—theories which take into account quantum mechanical effects since they are important at the quantum-realm scale and theories which are based on the classical continuum mechanics adapted to nanoscale problems. Here the second approach will be discussed in detail. It will be shown that the classical continuum mechanics is enough for a sufficient description of the mechanical behavior of nanomaterials and-structures if surface stresses are taken into account. There are also other approaches in the literature, but they are note discussed in detail in this paper. The basic equations for nanosized plates and shells will be discussed. It is shown that for this class of objects with the help of suggested equations such effects like the size-dependent changes of the stiffness parameters can be described in a proper manner. In contrast to the results for the size-dependence of the Young’s modulus (the Young’s modulus increases when the specimen diameter becomes very thin) the plate stiffness parameters can increase or decrease when the plate thickness is in the range of several nm. Finally, the theory of plates with surface stresses will be compared with the theory of three-layered plates.